CFRG A. Faz-Hernandez Internet-Draft Cloudflare, Inc. Intended status: Informational S. Scott Expires: 14 May 2022 Cornell Tech N. Sullivan Cloudflare, Inc. R.S. Wahby Stanford University C.A. Wood Cloudflare, Inc. 10 November 2021 Hashing to Elliptic Curves draft-irtf-cfrg-hash-to-curve-13 Abstract This document specifies a number of algorithms for encoding or hashing an arbitrary string to a point on an elliptic curve. This document is a product of the Crypto Forum Research Group (CFRG) in the IRTF. Status of This Memo This Internet-Draft is submitted in full conformance with the provisions of BCP 78 and BCP 79. Internet-Drafts are working documents of the Internet Engineering Task Force (IETF). Note that other groups may also distribute working documents as Internet-Drafts. The list of current Internet- Drafts is at https://datatracker.ietf.org/drafts/current/. Internet-Drafts are draft documents valid for a maximum of six months and may be updated, replaced, or obsoleted by other documents at any time. It is inappropriate to use Internet-Drafts as reference material or to cite them other than as "work in progress." This Internet-Draft will expire on 14 May 2022. Copyright Notice Copyright (c) 2021 IETF Trust and the persons identified as the document authors. All rights reserved. This document is subject to BCP 78 and the IETF Trust's Legal Provisions Relating to IETF Documents (https://trustee.ietf.org/ license-info) in effect on the date of publication of this document. Faz-Hernandez, et al. Expires 14 May 2022 [Page 1] Internet-Draft hash-to-curve November 2021 Please review these documents carefully, as they describe your rights and restrictions with respect to this document. Code Components extracted from this document must include Simplified BSD License text as described in Section 4.e of the Trust Legal Provisions and are provided without warranty as described in the Simplified BSD License. Table of Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . 5 1.1. Requirements Notation . . . . . . . . . . . . . . . . . . 6 2. Background . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.1. Elliptic curves . . . . . . . . . . . . . . . . . . . . . 6 2.2. Terminology . . . . . . . . . . . . . . . . . . . . . . . 8 2.2.1. Mappings . . . . . . . . . . . . . . . . . . . . . . 8 2.2.2. Encodings . . . . . . . . . . . . . . . . . . . . . . 9 2.2.3. Random oracle encodings . . . . . . . . . . . . . . . 9 2.2.4. Serialization . . . . . . . . . . . . . . . . . . . . 10 2.2.5. Domain separation . . . . . . . . . . . . . . . . . . 10 3. Encoding byte strings to elliptic curves . . . . . . . . . . 11 3.1. Domain separation requirements . . . . . . . . . . . . . 12 4. Utility functions . . . . . . . . . . . . . . . . . . . . . . 13 4.1. The sgn0 function . . . . . . . . . . . . . . . . . . . . 15 5. Hashing to a finite field . . . . . . . . . . . . . . . . . . 17 5.1. Security considerations . . . . . . . . . . . . . . . . . 17 5.2. Efficiency considerations in extension fields . . . . . . 18 5.3. hash_to_field implementation . . . . . . . . . . . . . . 18 5.4. expand_message . . . . . . . . . . . . . . . . . . . . . 19 5.4.1. expand_message_xmd . . . . . . . . . . . . . . . . . 20 5.4.2. expand_message_xof . . . . . . . . . . . . . . . . . 22 5.4.3. Using DSTs longer than 255 bytes . . . . . . . . . . 22 5.4.4. Defining other expand_message variants . . . . . . . 23 6. Deterministic mappings . . . . . . . . . . . . . . . . . . . 24 6.1. Choosing a mapping function . . . . . . . . . . . . . . . 24 6.2. Interface . . . . . . . . . . . . . . . . . . . . . . . . 24 6.3. Notation . . . . . . . . . . . . . . . . . . . . . . . . 25 6.4. Sign of the resulting point . . . . . . . . . . . . . . . 25 6.5. Exceptional cases . . . . . . . . . . . . . . . . . . . . 25 6.6. Mappings for Weierstrass curves . . . . . . . . . . . . . 26 6.6.1. Shallue-van de Woestijne method . . . . . . . . . . . 26 6.6.2. Simplified Shallue-van de Woestijne-Ulas method . . . 27 6.6.3. Simplified SWU for AB == 0 . . . . . . . . . . . . . 28 6.7. Mappings for Montgomery curves . . . . . . . . . . . . . 30 6.7.1. Elligator 2 method . . . . . . . . . . . . . . . . . 30 6.8. Mappings for twisted Edwards curves . . . . . . . . . . . 31 6.8.1. Rational maps from Montgomery to twisted Edwards curves . . . . . . . . . . . . . . . . . . . . . . . 31 6.8.2. Elligator 2 method . . . . . . . . . . . . . . . . . 32 7. Clearing the cofactor . . . . . . . . . . . . . . . . . . . . 32 Faz-Hernandez, et al. Expires 14 May 2022 [Page 2] Internet-Draft hash-to-curve November 2021 8. Suites for hashing . . . . . . . . . . . . . . . . . . . . . 33 8.1. Implementing a hash-to-curve suite . . . . . . . . . . . 36 8.2. Suites for NIST P-256 . . . . . . . . . . . . . . . . . . 36 8.3. Suites for NIST P-384 . . . . . . . . . . . . . . . . . . 37 8.4. Suites for NIST P-521 . . . . . . . . . . . . . . . . . . 38 8.5. Suites for curve25519 and edwards25519 . . . . . . . . . 39 8.6. Suites for curve448 and edwards448 . . . . . . . . . . . 40 8.7. Suites for secp256k1 . . . . . . . . . . . . . . . . . . 41 8.8. Suites for BLS12-381 . . . . . . . . . . . . . . . . . . 42 8.8.1. BLS12-381 G1 . . . . . . . . . . . . . . . . . . . . 42 8.8.2. BLS12-381 G2 . . . . . . . . . . . . . . . . . . . . 43 8.9. Defining a new hash-to-curve suite . . . . . . . . . . . 44 8.10. Suite ID naming conventions . . . . . . . . . . . . . . . 45 9. IANA considerations . . . . . . . . . . . . . . . . . . . . . 46 10. Security considerations . . . . . . . . . . . . . . . . . . . 47 10.1. encode_to_curve: output distribution and indifferentiability . . . . . . . . . . . . . . . . . . 48 10.2. hash_to_field security . . . . . . . . . . . . . . . . . 49 10.3. expand_message_xmd security . . . . . . . . . . . . . . 49 10.4. Domain separation recommendations . . . . . . . . . . . 50 10.5. Target security levels . . . . . . . . . . . . . . . . . 53 11. Acknowledgements . . . . . . . . . . . . . . . . . . . . . . 54 12. Contributors . . . . . . . . . . . . . . . . . . . . . . . . 54 13. References . . . . . . . . . . . . . . . . . . . . . . . . . 54 13.1. Normative References . . . . . . . . . . . . . . . . . . 54 13.2. Informative References . . . . . . . . . . . . . . . . . 55 Appendix A. Related work . . . . . . . . . . . . . . . . . . . . 64 Appendix B. Hashing to ristretto255 . . . . . . . . . . . . . . 66 Appendix C. Hashing to decaf448 . . . . . . . . . . . . . . . . 67 Appendix D. Rational maps . . . . . . . . . . . . . . . . . . . 68 D.1. Generic Montgomery to twisted Edwards map . . . . . . . . 69 D.2. Weierstrass to Montgomery map . . . . . . . . . . . . . . 71 Appendix E. Isogeny maps for suites . . . . . . . . . . . . . . 71 E.1. 3-isogeny map for secp256k1 . . . . . . . . . . . . . . . 72 E.2. 11-isogeny map for BLS12-381 G1 . . . . . . . . . . . . . 73 E.3. 3-isogeny map for BLS12-381 G2 . . . . . . . . . . . . . 77 Appendix F. Straight-line implementations of deterministic mappings . . . . . . . . . . . . . . . . . . . . . . . . 79 F.1. Shallue-van de Woestijne method . . . . . . . . . . . . . 79 F.2. Simplified SWU method . . . . . . . . . . . . . . . . . . 80 F.2.1. sqrt_ratio subroutines . . . . . . . . . . . . . . . 81 F.3. Elligator 2 method . . . . . . . . . . . . . . . . . . . 85 Appendix G. Curve-specific optimized sample code . . . . . . . . 86 G.1. Interface and projective coordinate systems . . . . . . . 86 G.2. Elligator 2 . . . . . . . . . . . . . . . . . . . . . . . 87 G.2.1. curve25519 (q = 5 (mod 8), K = 1) . . . . . . . . . . 87 G.2.2. edwards25519 . . . . . . . . . . . . . . . . . . . . 88 G.2.3. curve448 (q = 3 (mod 4), K = 1) . . . . . . . . . . . 89 Faz-Hernandez, et al. Expires 14 May 2022 [Page 3] Internet-Draft hash-to-curve November 2021 G.2.4. edwards448 . . . . . . . . . . . . . . . . . . . . . 90 G.2.5. q = 3 (mod 4) . . . . . . . . . . . . . . . . . . . . 92 G.2.6. q = 5 (mod 8) . . . . . . . . . . . . . . . . . . . . 94 G.3. Cofactor clearing for BLS12-381 G2 . . . . . . . . . . . 95 Appendix H. Scripts for parameter generation . . . . . . . . . . 97 H.1. Finding Z for the Shallue-van de Woestijne map . . . . . 97 H.2. Finding Z for Simplified SWU . . . . . . . . . . . . . . 98 H.3. Finding Z for Elligator 2 . . . . . . . . . . . . . . . . 99 Appendix I. sqrt and is_square functions . . . . . . . . . . . . 99 I.1. q = 3 (mod 4) . . . . . . . . . . . . . . . . . . . . . . 100 I.2. q = 5 (mod 8) . . . . . . . . . . . . . . . . . . . . . . 100 I.3. q = 9 (mod 16) . . . . . . . . . . . . . . . . . . . . . 100 I.4. Constant-time Tonelli-Shanks algorithm . . . . . . . . . 101 I.5. is_square for F = GF(p^2) . . . . . . . . . . . . . . . . 102 Appendix J. Suite test vectors . . . . . . . . . . . . . . . . . 103 J.1. NIST P-256 . . . . . . . . . . . . . . . . . . . . . . . 103 J.1.1. P256_XMD:SHA-256_SSWU_RO_ . . . . . . . . . . . . . . 103 J.1.2. P256_XMD:SHA-256_SSWU_NU_ . . . . . . . . . . . . . . 105 J.2. NIST P-384 . . . . . . . . . . . . . . . . . . . . . . . 107 J.2.1. P384_XMD:SHA-384_SSWU_RO_ . . . . . . . . . . . . . . 107 J.2.2. P384_XMD:SHA-384_SSWU_NU_ . . . . . . . . . . . . . . 109 J.3. NIST P-521 . . . . . . . . . . . . . . . . . . . . . . . 111 J.3.1. P521_XMD:SHA-512_SSWU_RO_ . . . . . . . . . . . . . . 111 J.3.2. P521_XMD:SHA-512_SSWU_NU_ . . . . . . . . . . . . . . 114 J.4. curve25519 . . . . . . . . . . . . . . . . . . . . . . . 116 J.4.1. curve25519_XMD:SHA-512_ELL2_RO_ . . . . . . . . . . . 116 J.4.2. curve25519_XMD:SHA-512_ELL2_NU_ . . . . . . . . . . . 118 J.5. edwards25519 . . . . . . . . . . . . . . . . . . . . . . 120 J.5.1. edwards25519_XMD:SHA-512_ELL2_RO_ . . . . . . . . . . 120 J.5.2. edwards25519_XMD:SHA-512_ELL2_NU_ . . . . . . . . . . 122 J.6. curve448 . . . . . . . . . . . . . . . . . . . . . . . . 124 J.6.1. curve448_XMD:SHA-512_ELL2_RO_ . . . . . . . . . . . . 124 J.6.2. curve448_XMD:SHA-512_ELL2_NU_ . . . . . . . . . . . . 127 J.7. edwards448 . . . . . . . . . . . . . . . . . . . . . . . 129 J.7.1. edwards448_XMD:SHA-512_ELL2_RO_ . . . . . . . . . . . 129 J.7.2. edwards448_XMD:SHA-512_ELL2_NU_ . . . . . . . . . . . 132 J.8. secp256k1 . . . . . . . . . . . . . . . . . . . . . . . . 134 J.8.1. secp256k1_XMD:SHA-256_SSWU_RO_ . . . . . . . . . . . 134 J.8.2. secp256k1_XMD:SHA-256_SSWU_NU_ . . . . . . . . . . . 136 J.9. BLS12-381 G1 . . . . . . . . . . . . . . . . . . . . . . 138 J.9.1. BLS12381G1_XMD:SHA-256_SSWU_RO_ . . . . . . . . . . . 138 J.9.2. BLS12381G1_XMD:SHA-256_SSWU_NU_ . . . . . . . . . . . 140 J.10. BLS12-381 G2 . . . . . . . . . . . . . . . . . . . . . . 142 J.10.1. BLS12381G2_XMD:SHA-256_SSWU_RO_ . . . . . . . . . . 142 J.10.2. BLS12381G2_XMD:SHA-256_SSWU_NU_ . . . . . . . . . . 146 Appendix K. Expand test vectors . . . . . . . . . . . . . . . . 148 K.1. expand_message_xmd(SHA-256) . . . . . . . . . . . . . . . 149 K.2. expand_message_xmd(SHA-256) (long DST) . . . . . . . . . 153 Faz-Hernandez, et al. Expires 14 May 2022 [Page 4] Internet-Draft hash-to-curve November 2021 K.3. expand_message_xmd(SHA-512) . . . . . . . . . . . . . . . 157 K.4. expand_message_xof(SHAKE128) . . . . . . . . . . . . . . 162 K.5. expand_message_xof(SHAKE128) (long DST) . . . . . . . . . 166 K.6. expand_message_xof(SHAKE256) . . . . . . . . . . . . . . 170 Authors' Addresses . . . . . . . . . . . . . . . . . . . . . . . 174 1. Introduction Many cryptographic protocols require a procedure that encodes an arbitrary input, e.g., a password, to a point on an elliptic curve. This procedure is known as hashing to an elliptic curve, where the hashing procedure provides collision resistance and does not reveal the discrete logarithm of the output point. Prominent examples of cryptosystems that hash to elliptic curves include password- authenticated key exchanges [BM92] [J96] [BMP00] [p1363.2], Identity- Based Encryption [BF01], Boneh-Lynn-Shacham signatures [BLS01] [I-D.irtf-cfrg-bls-signature], Verifiable Random Functions [MRV99] [I-D.irtf-cfrg-vrf], and Oblivious Pseudorandom Functions [NR97] [I-D.irtf-cfrg-voprf]. Unfortunately for implementors, the precise hash function that is suitable for a given protocol implemented using a given elliptic curve is often unclear from the protocol's description. Meanwhile, an incorrect choice of hash function can have disastrous consequences for security. This document aims to bridge this gap by providing a comprehensive set of recommended algorithms for a range of curve types. Each algorithm conforms to a common interface: it takes as input an arbitrary-length byte string and produces as output a point on an elliptic curve. We provide implementation details for each algorithm, describe the security rationale behind each recommendation, and give guidance for elliptic curves that are not explicitly covered. We also present optimized implementations for internal functions used by these algorithms. Readers wishing to quickly specify or implement a conforming hash function should consult Section 8, which lists recommended hash-to- curve suites and describes both how to implement an existing suite and how to specify a new one. This document does not cover rejection sampling methods, sometimes referred to as "try-and-increment" or "hunt-and-peck," because the goal is to describe algorithms that can plausibly be computed in constant time. Use of these rejection methods is NOT RECOMMENDED, because they have been a perennial cause of side-channel vulnerabilities. See Dragonblood [VR20] as one example of this problem in practice. Faz-Hernandez, et al. Expires 14 May 2022 [Page 5] Internet-Draft hash-to-curve November 2021 This document represents the consensus of the Crypto Forum Research Group (CFRG). 1.1. Requirements Notation The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT", "SHOULD", "SHOULD NOT", "RECOMMENDED", "NOT RECOMMENDED", "MAY", and "OPTIONAL" in this document are to be interpreted as described in BCP 14 [RFC2119] [RFC8174] when, and only when, they appear in all capitals, as shown here. 2. Background 2.1. Elliptic curves The following is a brief definition of elliptic curves, with an emphasis on important parameters and their relation to hashing to curves. For further reference on elliptic curves, consult [CFADLNV05] or [W08]. Let F be the finite field GF(q) of prime characteristic p > 3. (This document does not consider elliptic curves over fields of characteristic 2 or 3.) In most cases F is a prime field, so q = p. Otherwise, F is an extension field, so q = p^m for an integer m > 1. This document writes elements of extension fields in a primitive element or polynomial basis, i.e., as a vector of m elements of GF(p) written in ascending order by degree. The entries of this vector are indexed in ascending order starting from 1, i.e., x = (x_1, x_2, ..., x_m). For example, if q = p^2 and the primitive element basis is (1, I), then x = (a, b) corresponds to the element a + b * I, where x_1 = a and x_2 = b. (Note that all choices of basis are isomorphic, but certain choices may result in a more efficient implementation; this document does not make any particular assumptions about choice of basis.) An elliptic curve E is specified by an equation in two variables and a finite field F. An elliptic curve equation takes one of several standard forms, including (but not limited to) Weierstrass, Montgomery, and Edwards. Faz-Hernandez, et al. Expires 14 May 2022 [Page 6] Internet-Draft hash-to-curve November 2021 The curve E induces an algebraic group of order n, meaning that the group has n distinct elements. (This document uses additive notation for the elliptic curve group operation.) Elements of an elliptic curve group are points with affine coordinates (x, y) satisfying the curve equation, where x and y are elements of F. In addition, all elliptic curve groups have a distinguished element, the identity point, which acts as the identity element for the group operation. On certain curves (including Weierstrass and Montgomery curves), the identity point cannot be represented as an (x, y) coordinate pair. For security reasons, cryptographic uses of elliptic curves generally require using a (sub)group of prime order. Let G be such a subgroup of the curve of prime order r, where n = h * r. In this equation, h is an integer called the cofactor. An algorithm that takes as input an arbitrary point on the curve E and produces as output a point in the subgroup G of E is said to "clear the cofactor." Such algorithms are discussed in Section 7. Certain hash-to-curve algorithms restrict the form of the curve equation, the characteristic of the field, or the parameters of the curve. For each algorithm presented, this document lists the relevant restrictions. The table below summarizes quantities relevant to hashing to curves: Faz-Hernandez, et al. Expires 14 May 2022 [Page 7] Internet-Draft hash-to-curve November 2021 +========+=====================+=======================+ | Symbol | Meaning | Relevance | +========+=====================+=======================+ | F,q,p | A finite field F of | For prime fields, q = | | | characteristic p | p; otherwise, q = p^m | | | and #F = q = p^m. | and m>1. | +--------+---------------------+-----------------------+ | E | Elliptic curve. | E is specified by an | | | | equation and a field | | | | F. | +--------+---------------------+-----------------------+ | n | Number of points on | n = h * r, for h and | | | the elliptic curve | r defined below. | | | E. | | +--------+---------------------+-----------------------+ | G | A prime-order | Destination group to | | | subgroup of the | which byte strings | | | points on E. | are encoded. | +--------+---------------------+-----------------------+ | r | Order of G. | r is a prime factor | | | | of n (usually, the | | | | largest such factor). | +--------+---------------------+-----------------------+ | h | Cofactor, h >= 1. | An integer satisfying | | | | n = h * r. | +--------+---------------------+-----------------------+ Table 1 2.2. Terminology In this section, we define important terms used throughout the document. 2.2.1. Mappings A mapping is a deterministic function from an element of the field F to a point on an elliptic curve E defined over F. In general, the set of all points that a mapping can produce over all possible inputs may be only a subset of the points on an elliptic curve (i.e., the mapping may not be surjective). In addition, a mapping may output the same point for two or more distinct inputs (i.e., the mapping may not be injective). For example, consider a mapping from F to an elliptic curve having n points: if the number of elements of F is not equal to n, then this mapping cannot be bijective (i.e., both injective and surjective) since the mapping is defined to be deterministic. Faz-Hernandez, et al. Expires 14 May 2022 [Page 8] Internet-Draft hash-to-curve November 2021 Mappings may also be invertible, meaning that there is an efficient algorithm that, for any point P output by the mapping, outputs an x in F such that applying the mapping to x outputs P. Some of the mappings given in Section 6 are invertible, but this document does not discuss inversion algorithms. 2.2.2. Encodings Encodings are closely related to mappings. Like a mapping, an encoding is a function that outputs a point on an elliptic curve. In contrast to a mapping, however, the input to an encoding is an arbitrary-length byte string. This document constructs deterministic encodings by composing a hash function Hf with a deterministic mapping. In particular, Hf takes as input an arbitrary string and outputs an element of F. The deterministic mapping takes that element as input and outputs a point on an elliptic curve E defined over F. Since Hf takes arbitrary- length byte strings as inputs, it cannot be injective: the set of inputs is larger than the set of outputs, so there must be distinct inputs that give the same output (i.e., there must be collisions). Thus, any encoding built from Hf is also not injective. Like mappings, encodings may be invertible, meaning that there is an efficient algorithm that, for any point P output by the encoding, outputs a string s such that applying the encoding to s outputs P. The instantiation of Hf used by all encodings specified in this document (Section 5) is not invertible. Thus, the encodings are also not invertible. In some applications of hashing to elliptic curves, it is important that encodings do not leak information through side channels. [VR20] is one example of this type of leakage leading to a security vulnerability. See Section 10 for further discussion. 2.2.3. Random oracle encodings A random-oracle encoding satisfies a strong property: it can be proved indifferentiable from a random oracle [MRH04] under a suitable assumption. Both constructions described in Section 3 are indifferentiable from random oracles [MRH04] when instantiated following the guidelines in this document. The constructions differ in their output distributions: one gives a uniformly random point on the curve, the other gives a point sampled from a nonuniform distribution. Faz-Hernandez, et al. Expires 14 May 2022 [Page 9] Internet-Draft hash-to-curve November 2021 A random-oracle encoding with a uniform output distribution is suitable for use in many cryptographic protocols proven secure in the random oracle model. See Section 10 for further discussion. 2.2.4. Serialization A procedure related to encoding is the conversion of an elliptic curve point to a bit string. This is called serialization, and is typically used for compactly storing or transmitting points. The inverse operation, deserialization, converts a bit string to an elliptic curve point. For example, [SEC1] and [p1363a] give standard methods for serialization and deserialization. Deserialization is different from encoding in that only certain strings (namely, those output by the serialization procedure) can be deserialized. In contrast, this document is concerned with encodings from arbitrary strings to elliptic curve points. This document does not cover serialization or deserialization. 2.2.5. Domain separation Cryptographic protocols proven secure in the random oracle model are often analyzed under the assumption that the random oracle only answers queries associated with that protocol (including queries made by adversaries) [BR93]. In practice, this assumption does not hold if two protocols use the same function to instantiate the random oracle. Concretely, consider protocols P1 and P2 that query a random oracle RO: if P1 and P2 both query RO on the same value x, the security analysis of one or both protocols may be invalidated. A common way of addressing this issue is called domain separation, which allows a single random oracle to simulate multiple, independent oracles. This is effected by ensuring that each simulated oracle sees queries that are distinct from those seen by all other simulated oracles. For example, to simulate two oracles RO1 and RO2 given a single oracle RO, one might define RO1(x) := RO("RO1" || x) RO2(x) := RO("RO2" || x) where || is the concatenation operator. In this example, "RO1" and "RO2" are called domain separation tags; they ensure that queries to RO1 and RO2 cannot result in identical queries to RO, meaning that it is safe to treat RO1 and RO2 as independent oracles. In general, domain separation requires defining a distinct injective encoding for each oracle being simulated. In the above example, "RO1" and "RO2" have the same length and thus satisfy this Faz-Hernandez, et al. Expires 14 May 2022 [Page 10] Internet-Draft hash-to-curve November 2021 requirement when used as prefixes. The algorithms specified in this document take a different approach to ensuring injectivity; see Section 5.4 and Section 10.4 for more details. 3. Encoding byte strings to elliptic curves This section presents a general framework and interface for encoding byte strings to points on an elliptic curve. The constructions in this section rely on three basic functions: * The function hash_to_field, {0, 1}^* x {1, 2, ...} -> (F, F, ...), hashes arbitrary-length byte strings to a list of one or more elements of a finite field F; its implementation is defined in Section 5. * The function map_to_curve, F -> E, calculates a point on the elliptic curve E from an element of the finite field F over which E is defined. Section 6 describes mappings for a range of curve families. * The function clear_cofactor, E -> G, sends any point on the curve E to the subgroup G of E. Section 7 describes methods to perform this operation. The two encodings (Section 2.2.2) defined in this section have the same interface and are both random-oracle encodings (Section 2.2.3). The difference between the two is that their outputs are sampled from different distributions: * encode_to_curve is a nonuniform encoding from byte strings to points in G. That is, the distribution of its output is not uniformly random in G: the set of possible outputs of encode_to_curve is only a fraction of the points in G, and some points in this set are more likely to be output than others. Section 10.1 gives a more precise definition of encode_to_curve's output distribution. encode_to_curve(msg) Input: msg, an arbitrary-length byte string. Output: P, a point in G. Steps: 1. u = hash_to_field(msg, 1) 2. Q = map_to_curve(u[0]) 3. P = clear_cofactor(Q) 4. return P Faz-Hernandez, et al. Expires 14 May 2022 [Page 11] Internet-Draft hash-to-curve November 2021 * hash_to_curve is a uniform encoding from byte strings to points in G. That is, the distribution of its output is statistically close to uniform in G. This function is suitable for most applications requiring a random oracle returning points in G, when instantiated with any of the map_to_curve functions described in Section 6. See Section 10 for further discussion. hash_to_curve(msg) Input: msg, an arbitrary-length byte string. Output: P, a point in G. Steps: 1. u = hash_to_field(msg, 2) 2. Q0 = map_to_curve(u[0]) 3. Q1 = map_to_curve(u[1]) 4. R = Q0 + Q1 # Point addition 5. P = clear_cofactor(R) 6. return P Each hash-to-curve suite in Section 8 instantiates one of these encoding functions for a specifc elliptic curve. 3.1. Domain separation requirements All uses of the encoding functions defined in this document MUST include domain separation (Section 2.2.5) to avoid interfering with other uses of similar functionality. Applications that instantiate multiple, independent instances of either hash_to_curve or encode_to_curve MUST enforce domain separation between those instances. This requirement applies both in the case of multiple instances targeting the same curve and in the case of multiple instances targeting different curves. (This is because the internal hash_to_field primitive (Section 5) requires domain separation to guarantee independent outputs.) Domain separation is enforced with a domain separation tag (DST), which is a byte string constructed according to the following requirements: 1. Tags MUST be supplied as the DST parameter to hash_to_field, as described in Section 5. Faz-Hernandez, et al. Expires 14 May 2022 [Page 12] Internet-Draft hash-to-curve November 2021 2. Tags MUST have nonzero length. A minimum length of 16 bytes is RECOMMENDED to reduce the chance of collisions with other applications. 3. Tags SHOULD begin with a fixed identification string that is unique to the application. 4. Tags SHOULD include a version number. 5. For applications that define multiple ciphersuites, each ciphersuite's tag MUST be different. For this purpose, it is RECOMMENDED to include a ciphersuite identifier in each tag. 6. For applications that use multiple encodings, either to the same curve or to different curves, each encoding MUST use a different tag. For this purpose, it is RECOMMENDED to include the encoding's Suite ID (Section 8) in the domain separation tag. For independent encodings based on the same suite, each tag SHOULD also include a distinct identifier, e.g., "ENC1" and "ENC2". As an example, consider a fictional application named Quux that defines several different ciphersuites. A reasonable choice of tag is "QUUX-V-CS-", where and are two-digit numbers indicating the version and ciphersuite, respectively, and is the Suite ID of the encoding used in ciphersuite . As another example, consider a fictional application named Baz that requires two independent random oracles to the same curve. Reasonable choices of tags for these oracles are "BAZ-V-CS- -ENC1" and "BAZ-V-CS--ENC2", respectively, where , , and are as described above. The example tags given above are assumed to be ASCII-encoded byte strings without null termination, which is the RECOMMENDED format. Other encodings CAN be used, but in all cases the encoding as a sequence of bytes MUST be specified unambiguously. 4. Utility functions Algorithms in this document use the utility functions described below, plus standard arithmetic operations (addition, multiplication, modular reduction, etc.) and elliptic curve point operations (point addition and scalar multiplication). For security, implementations of these functions SHOULD be constant time: in brief, this means that execution time and memory access patterns SHOULD NOT depend on the values of secret inputs, Faz-Hernandez, et al. Expires 14 May 2022 [Page 13] Internet-Draft hash-to-curve November 2021 intermediate values, or outputs. For such constant-time implementations, all arithmetic, comparisons, and assignments MUST also be implemented in constant time. Section 10 briefly discusses constant-time security issues. Guidance on implementing low-level operations (in constant time or otherwise) is beyond the scope of this document; readers should consult standard reference material [MOV96] [CFADLNV05]. * CMOV(a, b, c): If c is False, CMOV returns a, otherwise it returns b. For constant-time implementations, this operation must run in time independent of the value of c. * AND, OR, NOT, and XOR are standard bitwise logical operators. For constant-time implementations, short-circuit operators MUST be avoided. * is_square(x): This function returns True whenever the value x is a square in the field F. By Euler's criterion, this function can be calculated in constant time as is_square(x) := { True, if x^((q - 1) / 2) is 0 or 1 in F; { False, otherwise. In certain extension fields, is_square can be computed in constant time more quickly than by the above exponentiation. [AR13] and [S85] describe optimized methods for extension fields. Appendix I.5 gives an optimized straight-line method for GF(p^2). * sqrt(x): The sqrt operation is a multi-valued function, i.e., there exist two roots of x in the field F whenever x is square (except when x = 0). To maintain compatibility across implementations while allowing implementors leeway for optimizations, this document does not require sqrt() to return a particular value. Instead, as explained in Section 6.4, any function that calls sqrt also specifies how to determine the correct root. The preferred way of computing square roots is to fix a deterministic algorithm particular to F. We give several algorithms in Appendix I. * sgn0(x): This function returns either 0 or 1 indicating the "sign" of x, where sgn0(x) == 1 just when x is "negative". (In other words, this function always considers 0 to be positive.) Section 4.1 defines this function and discusses its implementation. Faz-Hernandez, et al. Expires 14 May 2022 [Page 14] Internet-Draft hash-to-curve November 2021 * inv0(x): This function returns the multiplicative inverse of x in F, extended to all of F by fixing inv0(0) == 0. A straightforward way to implement inv0 in constant time is to compute inv0(x) := x^(q - 2). Notice that on input 0, the output is 0 as required. Certain fields may allow faster inversion methods; detailed discussion of such methods is beyond the scope of this document. * I2OSP and OS2IP: These functions are used to convert a byte string to and from a non-negative integer as described in [RFC8017]. (Note that these functions operate on byte strings in big-endian byte order.) * a || b: denotes the concatenation of byte strings a and b. For example, "ABC" || "DEF" == "ABCDEF". * substr(str, sbegin, slen): for a byte string str, this function returns the slen-byte substring starting at position sbegin; positions are zero indexed. For example, substr("ABCDEFG", 2, 3) == "CDE". * len(str): for a byte string str, this function returns the length of str in bytes. For example, len("ABC") == 3. * strxor(str1, str2): for byte strings str1 and str2, strxor(str1, str2) returns the bitwise XOR of the two strings. For example, strxor("abc", "XYZ") == "9;9" (the strings in this example are ASCII literals, but strxor is defined for arbitrary byte strings). In this document, strxor is only applied to inputs of equal length. 4.1. The sgn0 function This section defines a generic sgn0 implementation that applies to any field F = GF(p^m). It also gives simplified implementations for the cases F = GF(p) and F = GF(p^2). The definition of the sgn0 function for extension fields relies on the polynomial basis or vector representation of field elements, and iterates over the entire vector representation of the input element. As a result, sgn0 depends on the primitive polynomial used to define the polynomial basis; see Section 8 for more information about this basis, and see Section 2.1 for a discussion of representing elements of extension fields as vectors. Faz-Hernandez, et al. Expires 14 May 2022 [Page 15] Internet-Draft hash-to-curve November 2021 sgn0(x) Parameters: - F, a finite field of characteristic p and order q = p^m. - p, the characteristic of F (see immediately above). - m, the extension degree of F, m >= 1 (see immediately above). Input: x, an element of F. Output: 0 or 1. Steps: 1. sign = 0 2. zero = 1 3. for i in (1, 2, ..., m): 4. sign_i = x_i mod 2 5. zero_i = x_i == 0 6. sign = sign OR (zero AND sign_i) # Avoid short-circuit logic ops 7. zero = zero AND zero_i 8. return sign When m == 1, sgn0 can be significantly simplified: sgn0_m_eq_1(x) Input: x, an element of GF(p). Output: 0 or 1. Steps: 1. return x mod 2 The case m == 2 is only slightly more complicated: sgn0_m_eq_2(x) Input: x, an element of GF(p^2). Output: 0 or 1. Steps: 1. sign_0 = x_0 mod 2 2. zero_0 = x_0 == 0 3. sign_1 = x_1 mod 2 4. s = sign_0 OR (zero_0 AND sign_1) # Avoid short-circuit logic ops 5. return s Faz-Hernandez, et al. Expires 14 May 2022 [Page 16] Internet-Draft hash-to-curve November 2021 5. Hashing to a finite field The hash_to_field function hashes a byte string msg of arbitrary length into one or more elements of a field F. This function works in two steps: it first hashes the input byte string to produce a uniformly random byte string, and then interprets this byte string as one or more elements of F. For the first step, hash_to_field calls an auxiliary function expand_message. This document defines two variants of expand_message: one appropriate for hash functions like SHA-2 [FIPS180-4] or SHA-3 [FIPS202], and another appropriate for extensible-output functions such as SHAKE128 [FIPS202]. Security considerations for each expand_message variant are discussed below (Section 5.4.1, Section 5.4.2). Implementors MUST NOT use rejection sampling to generate a uniformly random element of F, to ensure that the hash_to_field function is amenable to constant-time implementation. The reason is that rejection sampling procedures are difficult to implement in constant time, and later well-meaning "optimizations" may silently render an implementation non-constant-time. This means that any hash_to_field function based on rejection sampling would be incompatible with constant-time implementation. The hash_to_field function is also suitable for securely hashing to scalars. For example, when hashing to the scalar field for an elliptic curve (sub)group with prime order r, it suffices to instantiate hash_to_field with target field GF(r). 5.1. Security considerations The hash_to_field function is designed to be indifferentiable from a random oracle [MRH04] when expand_message (Section 5.4) is modeled as a random oracle (see Section 10.2). Ensuring indifferentiability requires care; to see why, consider a prime p that is close to 3/4 * 2^256. Reducing a random 256-bit integer modulo this p yields a value that is in the range [0, p / 3] with probability roughly 1/2, meaning that this value is statistically far from uniform in [0, p - 1]. To control bias, hash_to_field instead uses random integers whose length is at least ceil(log2(p)) + k bits, where k is the target security level for the suite in bits. Reducing such integers mod p gives bias at most 2^-k for any p; this bias is appropriate when targeting k-bit security. For each such integer, hash_to_field uses expand_message to obtain L uniform bytes, where Faz-Hernandez, et al. Expires 14 May 2022 [Page 17] Internet-Draft hash-to-curve November 2021 L = ceil((ceil(log2(p)) + k) / 8) These uniform bytes are then interpreted as an integer via OS2IP [RFC8017]. For example, for a 255-bit prime p, and k = 128-bit security, L = ceil((255 + 128) / 8) = 48 bytes. Note that k is an upper bound on the security level for the corresponding curve. See Section 10.5 for more details, and Section 8.9 for guidelines on choosing k for a given curve. 5.2. Efficiency considerations in extension fields The hash_to_field function described in this section is inefficient for certain extension fields. Specifically, when hashing to an element of the extension field GF(p^m), hash_to_field requires expanding msg into m * L bytes (for L as defined above). For extension fields where log2(p) is significantly smaller than the security level k, this approach is inefficient: it requires expand_message to output roughly m * log2(p) + m * k bits, whereas m * log2(p) + k bytes suffices to generate an element of GF(p^m) with bias at most 2^-k. In such cases, an applications MAY use an alternative hash_to_field function, provided it meets the following security requirements: * The function MUST output field element(s) that are uniformly random except with bias at most 2^-k. * The function MUST NOT use rejection sampling. * The function SHOULD be amenable to straight line implementations. For example, Pornin [P20] describes a method for hashing to GF(9767^19) that meets these requirements while using fewer output bits from expand_message than hash_to_field would for that field. 5.3. hash_to_field implementation The following procedure implements hash_to_field. The expand_message parameter to this function MUST conform to the requirements given in Section 5.4. Section 3.1 discusses the REQUIRED method for constructing DST, the domain separation tag. Faz-Hernandez, et al. Expires 14 May 2022 [Page 18] Internet-Draft hash-to-curve November 2021 hash_to_field(msg, count) Parameters: - DST, a domain separation tag (see discussion above). - F, a finite field of characteristic p and order q = p^m. - p, the characteristic of F (see immediately above). - m, the extension degree of F, m >= 1 (see immediately above). - L = ceil((ceil(log2(p)) + k) / 8), where k is the security parameter of the suite (e.g., k = 128). - expand_message, a function that expands a byte string and domain separation tag into a uniformly random byte string (see discussion above). Inputs: - msg, a byte string containing the message to hash. - count, the number of elements of F to output. Outputs: - (u_0, ..., u_(count - 1)), a list of field elements. Steps: 1. len_in_bytes = count * m * L 2. uniform_bytes = expand_message(msg, DST, len_in_bytes) 3. for i in (0, ..., count - 1): 4. for j in (0, ..., m - 1): 5. elm_offset = L * (j + i * m) 6. tv = substr(uniform_bytes, elm_offset, L) 7. e_j = OS2IP(tv) mod p 8. u_i = (e_0, ..., e_(m - 1)) 9. return (u_0, ..., u_(count - 1)) 5.4. expand_message expand_message is a function that generates a uniformly random byte string. It takes three arguments: 1. msg, a byte string containing the message to hash, 2. DST, a byte string that acts as a domain separation tag, and 3. len_in_bytes, the number of bytes to be generated. This document defines the following two variants of expand_message: * expand_message_xmd (Section 5.4.1) is appropriate for use with a wide range of hash functions, including SHA-2 [FIPS180-4], SHA-3 [FIPS202], BLAKE2 [RFC7693], and others. Faz-Hernandez, et al. Expires 14 May 2022 [Page 19] Internet-Draft hash-to-curve November 2021 * expand_message_xof (Section 5.4.2) is appropriate for use with extensible-output functions (XOFs) including functions in the SHAKE [FIPS202] or BLAKE2X [BLAKE2X] families. These variants should suffice for the vast majority of use cases, but other variants are possible; Section 5.4.4 discusses requirements. 5.4.1. expand_message_xmd The expand_message_xmd function produces a uniformly random byte string using a cryptographic hash function H that outputs b bits. For security, H must meet the following requirements: * The number of bits output by H MUST be b >= 2 * k, for k the target security level in bits, and b MUST be divisible by 8. The first requirement ensures k-bit collision resistance; the second ensures uniformity of expand_message_xmd's output. * H MAY be a Merkle-Damgaard hash function like SHA-2. In this case, security holds when the underlying compression function is modeled as a random oracle [CDMP05]. (See Section 10.3 for discussion.) * H MAY be a sponge-based hash function like SHA-3 or BLAKE2. In this case, security holds when the inner function is modeled as a random transformation or as a random permutation [BDPV08]. * Otherwise, H MUST be a hash function that has been proved indifferentiable from a random oracle [MRH04] under a reasonable cryptographic assumption. SHA-2 [FIPS180-4] and SHA-3 [FIPS202] are typical and RECOMMENDED choices. As an example, for the 128-bit security level, b >= 256 bits and either SHA-256 or SHA3-256 would be an appropriate choice. The hash function H is assumed to work by repeatedly ingesting fixed- length blocks of data. The length in bits of these blocks is called the input block size (s). As examples, s = 1024 for SHA-512 [FIPS180-4] and s = 576 for SHA3-512 [FIPS202]. For correctness, H requires b <= s. The following procedure implements expand_message_xmd. Faz-Hernandez, et al. Expires 14 May 2022 [Page 20] Internet-Draft hash-to-curve November 2021 expand_message_xmd(msg, DST, len_in_bytes) Parameters: - H, a hash function (see requirements above). - b_in_bytes, b / 8 for b the output size of H in bits. For example, for b = 256, b_in_bytes = 32. - s_in_bytes, the input block size of H, measured in bytes (see discussion above). For example, for SHA-256, s_in_bytes = 64. Input: - msg, a byte string. - DST, a byte string of at most 255 bytes. See below for information on using longer DSTs. - len_in_bytes, the length of the requested output in bytes, not greater than the lesser of (255 * b_in_bytes) or 2^16-1. Output: - uniform_bytes, a byte string. Steps: 1. ell = ceil(len_in_bytes / b_in_bytes) 2. ABORT if ell > 255 3. DST_prime = DST || I2OSP(len(DST), 1) 4. Z_pad = I2OSP(0, s_in_bytes) 5. l_i_b_str = I2OSP(len_in_bytes, 2) 6. msg_prime = Z_pad || msg || l_i_b_str || I2OSP(0, 1) || DST_prime 7. b_0 = H(msg_prime) 8. b_1 = H(b_0 || I2OSP(1, 1) || DST_prime) 9. for i in (2, ..., ell): 10. b_i = H(strxor(b_0, b_(i - 1)) || I2OSP(i, 1) || DST_prime) 11. uniform_bytes = b_1 || ... || b_ell 12. return substr(uniform_bytes, 0, len_in_bytes) Note that the string Z_pad is prefixed to msg when computing b_0 (step 7). This is necessary for security when H is a Merkle-Damgaard hash, e.g., SHA-2 (see Section 10.3). Hashing this additional data means that the cost of computing b_0 is higher than the cost of simply computing H(msg). In most settings this overhead is negligible, because the cost of evaluating H is much less than the other costs involved in hashing to a curve. It is possible, however, to entirely avoid this overhead by taking advantage of the fact that Z_pad depends only on H, and not on the arguments to expand_message_xmd. To do so, first precompute and save the internal state of H after ingesting Z_pad. Then, when computing b_0, initialize H using the saved state. Further details are implementation dependent, and beyond the scope of this document. Faz-Hernandez, et al. Expires 14 May 2022 [Page 21] Internet-Draft hash-to-curve November 2021 5.4.2. expand_message_xof The expand_message_xof function produces a uniformly random byte string using an extensible-output function (XOF) H. For security, H must meet the following criteria: * The collision resistance of H MUST be at least k bits. * H MUST be an XOF that has been proved indifferentiable from a random oracle under a reasonable cryptographic assumption. The SHAKE [FIPS202] XOF family is a typical and RECOMMENDED choice. As an example, for 128-bit security, SHAKE128 would be an appropriate choice. The following procedure implements expand_message_xof. expand_message_xof(msg, DST, len_in_bytes) Parameters: - H, an extensible-output function. H(m, d) hashes message m and returns d bytes. Input: - msg, a byte string. - DST, a byte string of at most 255 bytes. See below for information on using longer DSTs. - len_in_bytes, the length of the requested output in bytes. Output: - uniform_bytes, a byte string. Steps: 1. DST_prime = DST || I2OSP(len(DST), 1) 2. msg_prime = msg || I2OSP(len_in_bytes, 2) || DST_prime 3. uniform_bytes = H(msg_prime, len_in_bytes) 4. return uniform_bytes 5.4.3. Using DSTs longer than 255 bytes The expand_message variants defined in this section accept domain separation tags of at most 255 bytes. If applications require a domain separation tag longer than 255 bytes, e.g., because of requirements imposed by an invoking protocol, implementors MUST compute a short domain separation tag by hashing, as follows: * For expand_message_xmd using hash function H, DST is computed as Faz-Hernandez, et al. Expires 14 May 2022 [Page 22] Internet-Draft hash-to-curve November 2021 DST = H("H2C-OVERSIZE-DST-" || a_very_long_DST) * For expand_message_xof using extensible-output function H, DST is computed as DST = H("H2C-OVERSIZE-DST-" || a_very_long_DST, ceil(2 * k / 8)) Here, a_very_long_DST is the DST whose length is greater than 255 bytes, "H2C-OVERSIZE-DST-" is a 17-byte ASCII string literal, and k is the target security level in bits. 5.4.4. Defining other expand_message variants When defining a new expand_message variant, the most important consideration is that hash_to_field models expand_message as a random oracle. Thus, implementors SHOULD prove indifferentiability from a random oracle under an appropriate assumption about the underlying cryptographic primitives; see Section 10.2 for more information. In addition, expand_message variants: * MUST give collision resistance commensurate with the security level of the target elliptic curve. * MUST be built on primitives designed for use in applications requiring cryptographic randomness. As examples, a secure stream cipher is an appropriate primitive, whereas a Mersenne twister pseudorandom number generator [MT98] is not. * MUST NOT use rejection sampling. * MUST give independent values for distinct (msg, DST, length) inputs. Meeting this requirement is subtle. As a simplified example, hashing msg || DST does not work, because in this case distinct (msg, DST) pairs whose concatenations are equal will return the same output (e.g., ("AB", "CDEF") and ("ABC", "DEF")). The variants defined in this document use a suffix-free encoding of DST to avoid this issue. * MUST use the domain separation tag DST to ensure that invocations of cryptographic primitives inside of expand_message are domain separated from invocations outside of expand_message. For example, if the expand_message variant uses a hash function H, an encoding of DST MUST be added either as a prefix or a suffix of the input to each invocation of H. Adding DST as a suffix is the RECOMMENDED approach. * SHOULD read msg exactly once, for efficiency when msg is long. Faz-Hernandez, et al. Expires 14 May 2022 [Page 23] Internet-Draft hash-to-curve November 2021 In addition, each expand_message variant MUST specify a unique EXP_TAG that identifies that variant in a Suite ID. See Section 8.10 for more information. 6. Deterministic mappings The mappings in this section are suitable for implementing either nonuniform or uniform encodings using the constructions in Section 3. Certain mappings restrict the form of the curve or its parameters. For each mapping presented, this document lists the relevant restrictions. Note that mappings in this section are not interchangeable: different mappings will almost certainly output different points when evaluated on the same input. 6.1. Choosing a mapping function This section gives brief guidelines on choosing a mapping function for a given elliptic curve. Note that the suites given in Section 8 are recommended mappings for the respective curves. If the target elliptic curve is a Montgomery curve (Section 6.7), the Elligator 2 method (Section 6.7.1) is recommended. Similarly, if the target elliptic curve is a twisted Edwards curve (Section 6.8), the twisted Edwards Elligator 2 method (Section 6.8.2) is recommended. The remaining cases are Weierstrass curves. For curves supported by the Simplified SWU method (Section 6.6.2), that mapping is the recommended one. Otherwise, the Simplified SWU method for AB == 0 (Section 6.6.3) is recommended if the goal is best performance, while the Shallue-van de Woestijne method (Section 6.6.1) is recommended if the goal is simplicity of implementation. (The reason for this distinction is that the Simplified SWU method for AB == 0 requires implementing an isogeny map in addition to the mapping function, while the Shallue-van de Woestijne method does not.) The Shallue-van de Woestijne method (Section 6.6.1) works with any curve, and may be used in cases where a generic mapping is required. Note, however, that this mapping is almost always more computationally expensive than the curve-specific recommendations above. 6.2. Interface The generic interface shared by all mappings in this section is as follows: Faz-Hernandez, et al. Expires 14 May 2022 [Page 24] Internet-Draft hash-to-curve November 2021 (x, y) = map_to_curve(u) The input u and outputs x and y are elements of the field F. The affine coordinates (x, y) specify a point on an elliptic curve defined over F. Note, however, that the point (x, y) is not a uniformly random point. 6.3. Notation As a rough guide, the following conventions are used in pseudocode: * All arithmetic operations are performed over a field F, unless explicitly stated otherwise. * u: the input to the mapping function. This is an element of F produced by the hash_to_field function. * (x, y), (s, t), (v, w): the affine coordinates of the point output by the mapping. Indexed variables (e.g., x1, y2, ...) are used for candidate values. * tv1, tv2, ...: reusable temporary variables. * c1, c2, ...: constant values, which can be computed in advance. 6.4. Sign of the resulting point In general, elliptic curves have equations of the form y^2 = g(x). The mappings in this section first identify an x such that g(x) is square, then take a square root to find y. Since there are two square roots when g(x) != 0, this may result in an ambiguity regarding the sign of y. When necessary, the mappings in this section resolve this ambiguity by specifying the sign of the y-coordinate in terms of the input to the mapping function. Two main reasons support this approach: first, this covers elliptic curves over any field in a uniform way, and second, it gives implementors leeway in optimizing square-root implementations. 6.5. Exceptional cases Mappings may have exceptional cases, i.e., inputs u on which the mapping is undefined. These cases must be handled carefully, especially for constant-time implementations. Faz-Hernandez, et al. Expires 14 May 2022 [Page 25] Internet-Draft hash-to-curve November 2021 For each mapping in this section, we discuss the exceptional cases and show how to handle them in constant time. Note that all implementations SHOULD use inv0 (Section 4) to compute multiplicative inverses, to avoid exceptional cases that result from attempting to compute the inverse of 0. 6.6. Mappings for Weierstrass curves The mappings in this section apply to a target curve E defined by the equation y^2 = g(x) = x^3 + A * x + B where 4 * A^3 + 27 * B^2 != 0. 6.6.1. Shallue-van de Woestijne method Shallue and van de Woestijne [SW06] describe a mapping that applies to essentially any elliptic curve. (Note, however, that this mapping is more expensive to evaluate than the other mappings in this document.) The parameterization given below is for Weierstrass curves; its derivation is detailed in [W19]. This parameterization also works for Montgomery (Section 6.7) and twisted Edwards (Section 6.8) curves via the rational maps given in Appendix D: first evaluate the Shallue-van de Woestijne mapping to an equivalent Weierstrass curve, then map that point to the target Montgomery or twisted Edwards curve using the corresponding rational map. Preconditions: A Weierstrass curve y^2 = x^3 + A * x + B. Constants: * A and B, the parameter of the Weierstrass curve. * Z, a non-zero element of F meeting the below criteria. Appendix H.1 gives a Sage [SAGE] script that outputs the RECOMMENDED Z. 1. g(Z) != 0 in F. 2. -(3 * Z^2 + 4 * A) / (4 * g(Z)) != 0 in F. 3. -(3 * Z^2 + 4 * A) / (4 * g(Z)) is square in F. 4. At least one of g(Z) and g(-Z / 2) is square in F. Faz-Hernandez, et al. Expires 14 May 2022 [Page 26] Internet-Draft hash-to-curve November 2021 Sign of y: Inputs u and -u give the same x-coordinate for many values of u. Thus, we set sgn0(y) == sgn0(u). Exceptions: The exceptional cases for u occur when (1 + u^2 * g(Z)) * (1 - u^2 * g(Z)) == 0. The restrictions on Z given above ensure that implementations that use inv0 to invert this product are exception free. Operations: 1. tv1 = u^2 * g(Z) 2. tv2 = 1 + tv1 3. tv1 = 1 - tv1 4. tv3 = inv0(tv1 * tv2) 5. tv4 = sqrt(-g(Z) * (3 * Z^2 + 4 * A)) # can be precomputed 6. If sgn0(tv4) == 1, set tv4 = -tv4 # sgn0(tv4) MUST equal 0 7. tv5 = u * tv1 * tv3 * tv4 8. tv6 = -4 * g(Z) / (3 * Z^2 + 4 * A) # can be precomputed 9. x1 = -Z / 2 - tv5 10. x2 = -Z / 2 + tv5 11. x3 = Z + tv6 * (tv2^2 * tv3)^2 12. If is_square(g(x1)), set x = x1 and y = sqrt(g(x1)) 13. Else If is_square(g(x2)), set x = x2 and y = sqrt(g(x2)) 14. Else set x = x3 and y = sqrt(g(x3)) 15. If sgn0(u) != sgn0(y), set y = -y 16. return (x, y) Appendix F.1 gives an example straight-line implementation of this mapping. 6.6.2. Simplified Shallue-van de Woestijne-Ulas method The function map_to_curve_simple_swu(u) implements a simplification of the Shallue-van de Woestijne-Ulas mapping [U07] described by Brier et al. [BCIMRT10], which they call the "simplified SWU" map. Wahby and Boneh [WB19] generalize and optimize this mapping. Preconditions: A Weierstrass curve y^2 = x^3 + A * x + B where A != 0 and B != 0. Constants: * A and B, the parameters of the Weierstrass curve. * Z, an element of F meeting the below criteria. Appendix H.2 gives a Sage [SAGE] script that outputs the RECOMMENDED Z. The criteria are: Faz-Hernandez, et al. Expires 14 May 2022 [Page 27] Internet-Draft hash-to-curve November 2021 1. Z is non-square in F, 2. Z != -1 in F, 3. the polynomial g(x) - Z is irreducible over F, and 4. g(B / (Z * A)) is square in F. Sign of y: Inputs u and -u give the same x-coordinate. Thus, we set sgn0(y) == sgn0(u). Exceptions: The exceptional cases are values of u such that Z^2 * u^4 + Z * u^2 == 0. This includes u == 0, and may include other values depending on Z. Implementations must detect this case and set x1 = B / (Z * A), which guarantees that g(x1) is square by the condition on Z given above. Operations: 1. tv1 = inv0(Z^2 * u^4 + Z * u^2) 2. x1 = (-B / A) * (1 + tv1) 3. If tv1 == 0, set x1 = B / (Z * A) 4. gx1 = x1^3 + A * x1 + B 5. x2 = Z * u^2 * x1 6. gx2 = x2^3 + A * x2 + B 7. If is_square(gx1), set x = x1 and y = sqrt(gx1) 8. Else set x = x2 and y = sqrt(gx2) 9. If sgn0(u) != sgn0(y), set y = -y 10. return (x, y) Appendix F.2 gives a general and optimized straight-line implementation of this mapping. For more information on optimizing this mapping, see [WB19] Section 4 or the example code found at [hash2curve-repo]. 6.6.3. Simplified SWU for AB == 0 Wahby and Boneh [WB19] show how to adapt the simplified SWU mapping to Weierstrass curves having A == 0 or B == 0, which the mapping of Section 6.6.2 does not support. (The case A == B == 0 is excluded because y^2 = x^3 is not an elliptic curve.) This method applies to curves like secp256k1 [SEC2] and to pairing- friendly curves in the Barreto-Lynn-Scott [BLS03], Barreto-Naehrig [BN05], and other families. This method requires finding another elliptic curve E' given by the equation Faz-Hernandez, et al. Expires 14 May 2022 [Page 28] Internet-Draft hash-to-curve November 2021 y'^2 = g'(x') = x'^3 + A' * x' + B' that is isogenous to E and has A' != 0 and B' != 0. (See [WB19], Appendix A, for one way of finding E' using [SAGE].) This isogeny defines a map iso_map(x', y') given by a pair of rational functions. iso_map takes as input a point on E' and produces as output a point on E. Once E' and iso_map are identified, this mapping works as follows: on input u, first apply the simplified SWU mapping to get a point on E', then apply the isogeny map to that point to get a point on E. Note that iso_map is a group homomorphism, meaning that point addition commutes with iso_map. Thus, when using this mapping in the hash_to_curve construction of Section 3, one can effect a small optimization by first mapping u0 and u1 to E', adding the resulting points on E', and then applying iso_map to the sum. This gives the same result while requiring only one evaluation of iso_map. Preconditions: An elliptic curve E' with A' != 0 and B' != 0 that is isogenous to the target curve E with isogeny map iso_map from E' to E. Helper functions: * map_to_curve_simple_swu is the mapping of Section 6.6.2 to E' * iso_map is the isogeny map from E' to E Sign of y: for this map, the sign is determined by map_to_curve_simple_swu. No further sign adjustments are necessary. Exceptions: map_to_curve_simple_swu handles its exceptional cases. Exceptional cases of iso_map are inputs that cause the denominator of either rational function to evaluate to zero; such cases MUST return the identity point on E. Operations: 1. (x', y') = map_to_curve_simple_swu(u) # (x', y') is on E' 2. (x, y) = iso_map(x', y') # (x, y) is on E 3. return (x, y) See [hash2curve-repo] or [WB19] Section 4.3 for details on implementing the isogeny map. Faz-Hernandez, et al. Expires 14 May 2022 [Page 29] Internet-Draft hash-to-curve November 2021 6.7. Mappings for Montgomery curves The mapping defined in this section applies to a target curve M defined by the equation K * t^2 = s^3 + J * s^2 + s 6.7.1. Elligator 2 method Bernstein, Hamburg, Krasnova, and Lange give a mapping that applies to any curve with a point of order 2 [BHKL13], which they call Elligator 2. Preconditions: A Montgomery curve K * t^2 = s^3 + J * s^2 + s where J != 0, K != 0, and (J^2 - 4) / K^2 is non-zero and non-square in F. Constants: * J and K, the parameters of the elliptic curve. * Z, a non-square element of F. Appendix H.3 gives a Sage [SAGE] script that outputs the RECOMMENDED Z. Sign of t: this mapping fixes the sign of t as specified in [BHKL13]. No additional adjustment is required. Exceptions: The exceptional case is Z * u^2 == -1, i.e., 1 + Z * u^2 == 0. Implementations must detect this case and set x1 = -(J / K). Note that this can only happen when q = 3 (mod 4). Operations: 1. x1 = -(J / K) * inv0(1 + Z * u^2) 2. If x1 == 0, set x1 = -(J / K) 3. gx1 = x1^3 + (J / K) * x1^2 + x1 / K^2 4. x2 = -x1 - (J / K) 5. gx2 = x2^3 + (J / K) * x2^2 + x2 / K^2 6. If is_square(gx1), set x = x1, y = sqrt(gx1) with sgn0(y) == 1. 7. Else set x = x2, y = sqrt(gx2) with sgn0(y) == 0. 8. s = x * K 9. t = y * K 10. return (s, t) Appendix F.3 gives an example straight-line implementation of this mapping. Appendix G.2 gives optimized straight-line procedures that apply to specific classes of curves and base fields. Faz-Hernandez, et al. Expires 14 May 2022 [Page 30] Internet-Draft hash-to-curve November 2021 6.8. Mappings for twisted Edwards curves Twisted Edwards curves (a class of curves that includes Edwards curves) are given by the equation a * v^2 + w^2 = 1 + d * v^2 * w^2 with a != 0, d != 0, and a != d [BBJLP08]. These curves are closely related to Montgomery curves (Section 6.7): every twisted Edwards curve is birationally equivalent to a Montgomery curve ([BBJLP08], Theorem 3.2). This equivalence yields an efficient way of hashing to a twisted Edwards curve: first, hash to an equivalent Montgomery curve, then transform the result into a point on the twisted Edwards curve via a rational map. This method of hashing to a twisted Edwards curve thus requires identifying a corresponding Montgomery curve and rational map. We describe how to identify such a curve and map immediately below. 6.8.1. Rational maps from Montgomery to twisted Edwards curves There are two ways to select a Montgomery curve and rational map for use when hashing to a given twisted Edwards curve. The selected Montgomery curve and rational map MUST be specified as part of the hash-to-curve suite for a given twisted Edwards curve; see Section 8. 1. When hashing to a standardized twisted Edwards curve for which a corresponding Montgomery form and rational map are also standardized, the standard Montgomery form and rational map SHOULD be used to ensure compatibility with existing software. In certain cases, e.g., edwards25519 [RFC7748], the sign of the rational map from the twisted Edwards curve to its corresponding Montgomery curve is not given explicitly. In this case, the sign MUST be fixed such that applying the rational map to the twisted Edwards curve's base point yields the Montgomery curve's base point with correct sign. (For edwards25519, see [RFC7748] and [EID4730].) When defining new twisted Edwards curves, a Montgomery equivalent and rational map SHOULD also be specified, and the sign of the rational map SHOULD be stated explicitly. 2. When hashing to a twisted Edwards curve that does not have a standardized Montgomery form or rational map, the map given in Appendix D SHOULD be used. Faz-Hernandez, et al. Expires 14 May 2022 [Page 31] Internet-Draft hash-to-curve November 2021 6.8.2. Elligator 2 method Preconditions: A twisted Edwards curve E and an equivalent Montgomery curve M meeting the requirements in Section 6.8.1. Helper functions: * map_to_curve_elligator2 is the mapping of Section 6.7.1 to the curve M. * rational_map is a function that takes a point (s, t) on M and returns a point (v, w) on E, as defined in Section 6.8.1. Sign of t (and v): for this map, the sign is determined by map_to_curve_elligator2. No further sign adjustments are required. Exceptions: The exceptions for the Elligator 2 mapping are as given in Section 6.7.1. The exceptions for the rational map are as given in Section 6.8.1. No other exceptions are possible. The following procedure implements the Elligator 2 mapping for a twisted Edwards curve. (Note that the output point is denoted (v, w) because it is a point on the target twisted Edwards curve.) map_to_curve_elligator2_edwards(u) Input: u, an element of F. Output: (v, w), a point on E. 1. (s, t) = map_to_curve_elligator2(u) # (s, t) is on M 2. (v, w) = rational_map(s, t) # (v, w) is on E 3. return (v, w) 7. Clearing the cofactor The mappings of Section 6 always output a point on the elliptic curve, i.e., a point in a group of order h * r (Section 2.1). Obtaining a point in G may require a final operation commonly called "clearing the cofactor," which takes as input any point on the curve and produces as output a point in the prime-order (sub)group G (Section 2.1). The cofactor can always be cleared via scalar multiplication by h. For elliptic curves where h = 1, i.e., the curves with a prime number of points, no operation is required. This applies, for example, to the NIST curves P-256, P-384, and P-521 [FIPS186-4]. Faz-Hernandez, et al. Expires 14 May 2022 [Page 32] Internet-Draft hash-to-curve November 2021 In some cases, it is possible to clear the cofactor via a faster method than scalar multiplication by h. These methods are equivalent to (but usually faster than) multiplication by some scalar h_eff whose value is determined by the method and the curve. Examples of fast cofactor clearing methods include the following: * For certain pairing-friendly curves having subgroup G2 over an extension field, Scott et al. [SBCDK09] describe a method for fast cofactor clearing that exploits an efficiently-computable endomorphism. Fuentes-Castaneda et al. [FKR11] propose an alternative method that is sometimes more efficient. Budroni and Pintore [BP17] give concrete instantiations of these methods for Barreto-Lynn-Scott pairing-friendly curves [BLS03]. This method is described for the specific case of BLS12-381 in Appendix G.3. * Wahby and Boneh ([WB19], Section 5) describe a trick due to Scott for fast cofactor clearing on any elliptic curve for which the prime factorization of h and the structure of the elliptic curve group meet certain conditions. The clear_cofactor function is parameterized by a scalar h_eff. Specifically, clear_cofactor(P) := h_eff * P where * represents scalar multiplication. When a curve does not support a fast cofactor clearing method, h_eff = h and the cofactor MUST be cleared via scalar multiplication. When a curve admits a fast cofactor clearing method, clear_cofactor MAY be evaluated either via that method or via scalar multiplication by the equivalent h_eff; these two methods give the same result. Note that in this case scalar multiplication by the cofactor h does not generally give the same result as the fast method, and SHOULD NOT be used. 8. Suites for hashing This section lists recommended suites for hashing to standard elliptic curves. A hash-to-curve suite fully specifies the procedure for hashing byte strings to points on a specific elliptic curve group. Section 8.1 describes how to implement a suite. Applications that require hashing to an elliptic curve should use either an existing suite or a new suite specified as described in Section 8.9. Faz-Hernandez, et al. Expires 14 May 2022 [Page 33] Internet-Draft hash-to-curve November 2021 All applications using a hash-to-curve suite MUST choose a domain separation tag (DST) in accordance with the guidelines in Section 3.1. In addition, applications whose security requires a random oracle that returns uniformly random points on the target curve MUST use a suite whose encoding type is hash_to_curve; see Section 3 and immediately below for more information. A hash-to-curve suite comprises the following parameters: * Suite ID, a short name used to refer to a given suite. Section 8.10 discusses the naming conventions for suite IDs. * encoding type, either uniform (hash_to_curve) or nonuniform (encode_to_curve). See Section 3 for definitions of these encoding types. * E, the target elliptic curve over a field F. * p, the characteristic of the field F. * m, the extension degree of the field F. If m > 1, the suite MUST also specify the polynomial basis used to represent extension field elements. * k, the target security level of the suite in bits. (See Section 10.5 for discussion.) * L, the length parameter for hash_to_field (Section 5.1). * expand_message, one of the variants specified in Section 5.4 plus any parameters required for the specified variant (for example, H, the underlying hash function). * f, a mapping function from Section 6. * h_eff, the scalar parameter for clear_cofactor (Section 7). In addition to the above parameters, the mapping f may require additional parameters Z, M, rational_map, E', or iso_map. When applicable, these MUST be specified. Faz-Hernandez, et al. Expires 14 May 2022 [Page 34] Internet-Draft hash-to-curve November 2021 The below table lists suites RECOMMENDED for some elliptic curves. The corresponding parameters are given in the following subsections. Applications instantiating cryptographic protocols whose security analysis relies on a random oracle that outputs points with a uniform distribution MUST NOT use a nonuniform encoding. Moreover, applications that use a nonuniform encoding SHOULD carefully analyze the security implications of nonuniformity. When the required encoding is not clear, applications SHOULD use a uniform encoding for security. +==============+===================================+=========+ | E | Suites | Section | +==============+===================================+=========+ | NIST P-256 | P256_XMD:SHA-256_SSWU_RO_ | Section | | | P256_XMD:SHA-256_SSWU_NU_ | 8.2 | +--------------+-----------------------------------+---------+ | NIST P-384 | P384_XMD:SHA-384_SSWU_RO_ | Section | | | P384_XMD:SHA-384_SSWU_NU_ | 8.3 | +--------------+-----------------------------------+---------+ | NIST P-521 | P521_XMD:SHA-512_SSWU_RO_ | Section | | | P521_XMD:SHA-512_SSWU_NU_ | 8.4 | +--------------+-----------------------------------+---------+ | curve25519 | curve25519_XMD:SHA-512_ELL2_RO_ | Section | | | curve25519_XMD:SHA-512_ELL2_NU_ | 8.5 | +--------------+-----------------------------------+---------+ | edwards25519 | edwards25519_XMD:SHA-512_ELL2_RO_ | Section | | | edwards25519_XMD:SHA-512_ELL2_NU_ | 8.5 | +--------------+-----------------------------------+---------+ | curve448 | curve448_XMD:SHA-512_ELL2_RO_ | Section | | | curve448_XMD:SHA-512_ELL2_NU_ | 8.6 | +--------------+-----------------------------------+---------+ | edwards448 | edwards448_XMD:SHA-512_ELL2_RO_ | Section | | | edwards448_XMD:SHA-512_ELL2_NU_ | 8.6 | +--------------+-----------------------------------+---------+ | secp256k1 | secp256k1_XMD:SHA-256_SSWU_RO_ | Section | | | secp256k1_XMD:SHA-256_SSWU_NU_ | 8.7 | +--------------+-----------------------------------+---------+ | BLS12-381 G1 | BLS12381G1_XMD:SHA-256_SSWU_RO_ | Section | | | BLS12381G1_XMD:SHA-256_SSWU_NU_ | 8.8 | +--------------+-----------------------------------+---------+ | BLS12-381 G2 | BLS12381G2_XMD:SHA-256_SSWU_RO_ | Section | | | BLS12381G2_XMD:SHA-256_SSWU_NU_ | 8.8 | +--------------+-----------------------------------+---------+ Table 2 Faz-Hernandez, et al. Expires 14 May 2022 [Page 35] Internet-Draft hash-to-curve November 2021 8.1. Implementing a hash-to-curve suite A hash-to-curve suite requires the following functions. Note that some of these require utility functions from Section 4. 1. Base field arithmetic operations for the target elliptic curve, e.g., addition, multiplication, and square root. 2. Elliptic curve point operations for the target curve, e.g., point addition and scalar multiplication. 3. The hash_to_field function; see Section 5. This includes the expand_message variant (Section 5.4) and any constituent hash function or XOF. 4. The suite-specified mapping function; see the corresponding subsection of Section 6. 5. A cofactor clearing function; see Section 7. This may be implemented as scalar multiplication by h_eff or as a faster equivalent method. 6. The desired encoding function; see Section 3. This is either hash_to_curve or encode_to_curve. 8.2. Suites for NIST P-256 This section defines ciphersuites for the NIST P-256 elliptic curve [FIPS186-4]. P256_XMD:SHA-256_SSWU_RO_ is defined as follows: * encoding type: hash_to_curve (Section 3) * E: y^2 = x^3 + A * x + B, where - A = -3 - B = 0x5ac635d8aa3a93e7b3ebbd55769886bc651d06b0cc53b0f63bce3c3e2 7d2604b * p: 2^256 - 2^224 + 2^192 + 2^96 - 1 * m: 1 * k: 128 * expand_message: expand_message_xmd (Section 5.4.1) Faz-Hernandez, et al. Expires 14 May 2022 [Page 36] Internet-Draft hash-to-curve November 2021 * H: SHA-256 * L: 48 * f: Simplified SWU method (Section 6.6.2) * Z: -10 * h_eff: 1 P256_XMD:SHA-256_SSWU_NU_ is identical to P256_XMD:SHA-256_SSWU_RO_, except that the encoding type is encode_to_curve (Section 3). An optimized example implementation of the Simplified SWU mapping to P-256 is given in Appendix F.2. 8.3. Suites for NIST P-384 This section defines ciphersuites for the NIST P-384 elliptic curve [FIPS186-4]. P384_XMD:SHA-384_SSWU_RO_ is defined as follows: * encoding type: hash_to_curve (Section 3) * E: y^2 = x^3 + A * x + B, where - A = -3 - B = 0xb3312fa7e23ee7e4988e056be3f82d19181d9c6efe8141120314088f5 013875ac656398d8a2ed19d2a85c8edd3ec2aef * p: 2^384 - 2^128 - 2^96 + 2^32 - 1 * m: 1 * k: 192 * expand_message: expand_message_xmd (Section 5.4.1) * H: SHA-384 * L: 72 * f: Simplified SWU method (Section 6.6.2) * Z: -12 Faz-Hernandez, et al. Expires 14 May 2022 [Page 37] Internet-Draft hash-to-curve November 2021 * h_eff: 1 P384_XMD:SHA-384_SSWU_NU_ is identical to P384_XMD:SHA-384_SSWU_RO_, except that the encoding type is encode_to_curve (Section 3). An optimized example implementation of the Simplified SWU mapping to P-384 is given in Appendix F.2. 8.4. Suites for NIST P-521 This section defines ciphersuites for the NIST P-521 elliptic curve [FIPS186-4]. P521_XMD:SHA-512_SSWU_RO_ is defined as follows: * encoding type: hash_to_curve (Section 3) * E: y^2 = x^3 + A * x + B, where - A = -3 - B = 0x51953eb9618e1c9a1f929a21a0b68540eea2da725b99b315f3b8b4899 18ef109e156193951ec7e937b1652c0bd3bb1bf073573df883d2c34f1ef451f d46b503f00 * p: 2^521 - 1 * m: 1 * k: 256 * expand_message: expand_message_xmd (Section 5.4.1) * H: SHA-512 * L: 98 * f: Simplified SWU method (Section 6.6.2) * Z: -4 * h_eff: 1 P521_XMD:SHA-512_SSWU_NU_ is identical to P521_XMD:SHA-512_SSWU_RO_, except that the encoding type is encode_to_curve (Section 3). An optimized example implementation of the Simplified SWU mapping to P-521 is given in Appendix F.2. Faz-Hernandez, et al. Expires 14 May 2022 [Page 38] Internet-Draft hash-to-curve November 2021 8.5. Suites for curve25519 and edwards25519 This section defines ciphersuites for curve25519 and edwards25519 [RFC7748]. Note that these ciphersuites SHOULD NOT be used when hashing to ristretto255 [I-D.irtf-cfrg-ristretto255-decaf448]. See Appendix B for information on how to hash to that group. curve25519_XMD:SHA-512_ELL2_RO_ is defined as follows: * encoding type: hash_to_curve (Section 3) * E: K * t^2 = s^3 + J * s^2 + s, where - J = 486662 - K = 1 * p: 2^255 - 19 * m: 1 * k: 128 * expand_message: expand_message_xmd (Section 5.4.1) * H: SHA-512 * L: 48 * f: Elligator 2 method (Section 6.7.1) * Z: 2 * h_eff: 8 edwards25519_XMD:SHA-512_ELL2_RO_ is identical to curve25519_XMD:SHA- 512_ELL2_RO_, except for the following parameters: * E: a * v^2 + w^2 = 1 + d * v^2 * w^2, where - a = -1 - d = 0x52036cee2b6ffe738cc740797779e89800700a4d4141d8ab75eb4dca1 35978a3 * f: Twisted Edwards Elligator 2 method (Section 6.8.2) * M: curve25519 defined in [RFC7748], Section 4.1 Faz-Hernandez, et al. Expires 14 May 2022 [Page 39] Internet-Draft hash-to-curve November 2021 * rational_map: the birational map defined in [RFC7748], Section 4.1 curve25519_XMD:SHA-512_ELL2_NU_ is identical to curve25519_XMD:SHA- 512_ELL2_RO_, except that the encoding type is encode_to_curve (Section 3). edwards25519_XMD:SHA-512_ELL2_NU_ is identical to edwards25519_XMD:SHA-512_ELL2_RO_, except that the encoding type is encode_to_curve (Section 3). Optimized example implementations of the above mappings are given in Appendix G.2.1 and Appendix G.2.2. 8.6. Suites for curve448 and edwards448 This section defines ciphersuites for curve448 and edwards448 [RFC7748]. Note that these ciphersuites SHOULD NOT be used when hashing to decaf448 [I-D.irtf-cfrg-ristretto255-decaf448]. See Appendix C for information on how to hash to that group. curve448_XMD:SHA-512_ELL2_RO_ is defined as follows: * encoding type: hash_to_curve (Section 3) * E: K * t^2 = s^3 + J * s^2 + s, where - J = 156326 - K = 1 * p: 2^448 - 2^224 - 1 * m: 1 * k: 224 * expand_message: expand_message_xmd (Section 5.4.1) * H: SHA-512 * L: 84 * f: Elligator 2 method (Section 6.7.1) * Z: -1 * h_eff: 4 Faz-Hernandez, et al. Expires 14 May 2022 [Page 40] Internet-Draft hash-to-curve November 2021 edwards448_XMD:SHA-512_ELL2_RO_ is identical to curve448_XMD:SHA- 512_ELL2_RO_, except for the following parameters: * E: a * v^2 + w^2 = 1 + d * v^2 * w^2, where - a = 1 - d = -39081 * f: Twisted Edwards Elligator 2 method (Section 6.8.2) * M: curve448, defined in [RFC7748], Section 4.2 * rational_map: the 4-isogeny map defined in [RFC7748], Section 4.2 curve448_XMD:SHA-512_ELL2_NU_ is identical to curve448_XMD:SHA- 512_ELL2_RO_, except that the encoding type is encode_to_curve (Section 3). edwards448_XMD:SHA-512_ELL2_NU_ is identical to edwards448_XMD:SHA- 512_ELL2_RO_, except that the encoding type is encode_to_curve (Section 3). Optimized example implementations of the above mappings are given in Appendix G.2.3 and Appendix G.2.4. 8.7. Suites for secp256k1 This section defines ciphersuites for the secp256k1 elliptic curve [SEC2]. secp256k1_XMD:SHA-256_SSWU_RO_ is defined as follows: * encoding type: hash_to_curve (Section 3) * E: y^2 = x^3 + 7 * p: 2^256 - 2^32 - 2^9 - 2^8 - 2^7 - 2^6 - 2^4 - 1 * m: 1 * k: 128 * expand_message: expand_message_xmd (Section 5.4.1) * H: SHA-256 * L: 48 Faz-Hernandez, et al. Expires 14 May 2022 [Page 41] Internet-Draft hash-to-curve November 2021 * f: Simplified SWU for AB == 0 (Section 6.6.3) * Z: -11 * E': y'^2 = x'^3 + A' * x' + B', where - A': 0x3f8731abdd661adca08a5558f0f5d272e953d363cb6f0e5d405447c01 a444533 - B': 1771 * iso_map: the 3-isogeny map from E' to E given in Appendix E.1 * h_eff: 1 secp256k1_XMD:SHA-256_SSWU_NU_ is identical to secp256k1_XMD:SHA- 256_SSWU_RO_, except that the encoding type is encode_to_curve (Section 3). An optimized example implementation of the Simplified SWU mapping to the curve E' isogenous to secp256k1 is given in Appendix F.2. 8.8. Suites for BLS12-381 This section defines ciphersuites for groups G1 and G2 of the BLS12-381 elliptic curve [BLS12-381]. The curve parameters in this section match the ones listed in [I-D.irtf-cfrg-pairing-friendly-curves], Appendix C. 8.8.1. BLS12-381 G1 BLS12381G1_XMD:SHA-256_SSWU_RO_ is defined as follows: * encoding type: hash_to_curve (Section 3) * E: y^2 = x^3 + 4 * p: 0x1a0111ea397fe69a4b1ba7b6434bacd764774b84f38512bf6730d2a0f6b0f 6241eabfffeb153ffffb9feffffffffaaab * m: 1 * k: 128 * expand_message: expand_message_xmd (Section 5.4.1) * H: SHA-256 Faz-Hernandez, et al. Expires 14 May 2022 [Page 42] Internet-Draft hash-to-curve November 2021 * L: 64 * f: Simplified SWU for AB == 0 (Section 6.6.3) * Z: 11 * E': y'^2 = x'^3 + A' * x' + B', where - A' = 0x144698a3b8e9433d693a02c96d4982b0ea985383ee66a8d8e8981aef d881ac98936f8da0e0f97f5cf428082d584c1d - B' = 0x12e2908d11688030018b12e8753eee3b2016c1f0f24f4070a0b9c14f cef35ef55a23215a316ceaa5d1cc48e98e172be0 * iso_map: the 11-isogeny map from E' to E given in Appendix E.2 * h_eff: 0xd201000000010001 BLS12381G1_XMD:SHA-256_SSWU_NU_ is identical to BLS12381G1_XMD:SHA- 256_SSWU_RO_, except that the encoding type is encode_to_curve (Section 3). Note that the h_eff values for these suites are chosen for compatibility with the fast cofactor clearing method described by Scott ([WB19] Section 5). An optimized example implementation of the Simplified SWU mapping to the curve E' isogenous to BLS12-381 G1 is given in Appendix F.2. 8.8.2. BLS12-381 G2 BLS12381G2_XMD:SHA-256_SSWU_RO_ is defined as follows: * encoding type: hash_to_curve (Section 3) * E: y^2 = x^3 + 4 * (1 + I) * base field F is GF(p^m), where - p: 0x1a0111ea397fe69a4b1ba7b6434bacd764774b84f38512bf6730d2a0f6 b0f6241eabfffeb153ffffb9feffffffffaaab - m: 2 - (1, I) is the basis for F, where I^2 + 1 == 0 in F * k: 128 Faz-Hernandez, et al. Expires 14 May 2022 [Page 43] Internet-Draft hash-to-curve November 2021 * expand_message: expand_message_xmd (Section 5.4.1) * H: SHA-256 * L: 64 * f: Simplified SWU for AB == 0 (Section 6.6.3) * Z: -(2 + I) * E': y'^2 = x'^3 + A' * x' + B', where - A' = 240 * I - B' = 1012 * (1 + I) * iso_map: the isogeny map from E' to E given in Appendix E.3 * h_eff: 0xbc69f08f2ee75b3584c6a0ea91b352888e2a8e9145ad7689986ff0315 08ffe1329c2f178731db956d82bf015d1212b02ec0ec69d7477c1ae954cbc06689 f6a359894c0adebbf6b4e8020005aaa95551 BLS12381G2_XMD:SHA-256_SSWU_NU_ is identical to BLS12381G2_XMD:SHA- 256_SSWU_RO_, except that the encoding type is encode_to_curve (Section 3). Note that the h_eff values for these suites are chosen for compatibility with the fast cofactor clearing method described by Budroni and Pintore ([BP17], Section 4.1), and summarized in Appendix G.3. An optimized example implementation of the Simplified SWU mapping to the curve E' isogenous to BLS12-381 G2 is given in Appendix F.2. 8.9. Defining a new hash-to-curve suite The RECOMMENDED way to define a new hash-to-curve suite is: 1. E, F, p, and m are determined by the elliptic curve and its base field. 2. k is an upper bound on the target security level of the suite (Section 10.5). A reasonable choice of k is ceil(log2(r) / 2), where r is the order of the subgroup G of the curve E (Section 2.1). 3. Choose encoding type, either hash_to_curve or encode_to_curve (Section 3). Faz-Hernandez, et al. Expires 14 May 2022 [Page 44] Internet-Draft hash-to-curve November 2021 4. Compute L as described in Section 5.1. 5. Choose an expand_message variant from Section 5.4 plus any underlying cryptographic primitives (e.g., a hash function H). 6. Choose a mapping following the guidelines in Section 6.1, and select any required parameters for that mapping. 7. Choose h_eff to be either the cofactor of E or, if a fast cofactor clearing method is to be used, a value appropriate to that method as discussed in Section 7. 8. Construct a Suite ID following the guidelines in Section 8.10. When hashing to an elliptic curve not listed in this section, corresponding hash-to-curve suites SHOULD be fully specified as described above. 8.10. Suite ID naming conventions Suite IDs MUST be constructed as follows: CURVE_ID || "_" || HASH_ID || "_" || MAP_ID || "_" || ENC_VAR || "_" The fields CURVE_ID, HASH_ID, MAP_ID, and ENC_VAR are ASCII-encoded strings of at most 64 characters each. Fields MUST contain only ASCII characters between 0x21 and 0x7E (inclusive) except that underscore (i.e., 0x5f) is not allowed. As indicated above, each field (including the last) is followed by an underscore ("_", ASCII 0x5f). This helps to ensure that Suite IDs are prefix free. Suite IDs MUST include the final underscore and MUST NOT include any characters after the final underscore. Suite ID fields MUST be chosen as follows: * CURVE_ID: a human-readable representation of the target elliptic curve. * HASH_ID: a human-readable representation of the expand_message function and any underlying hash primitives used in hash_to_field (Section 5). This field MUST be constructed as follows: EXP_TAG || ":" || HASH_NAME EXP_TAG indicates the expand_message variant: - "XMD" for expand_message_xmd (Section 5.4.1). Faz-Hernandez, et al. Expires 14 May 2022 [Page 45] Internet-Draft hash-to-curve November 2021 - "XOF" for expand_message_xof (Section 5.4.2). HASH_NAME is a human-readable name for the underlying hash primitive. As examples: 1. For expand_message_xof (Section 5.4.2) with SHAKE128, HASH_ID is "XOF:SHAKE128". 2. For expand_message_xmd (Section 5.4.1) with SHA3-256, HASH_ID is "XMD:SHA3-256". Suites that use an alternative hash_to_field function that meets the requirements in Section 5.2 MUST indicate this by appending a tag identifying that function to the HASH_ID field, separated by a colon (":", ASCII 0x3A). * MAP_ID: a human-readable representation of the map_to_curve function as defined in Section 6. These are defined as follows: - "SVDW" for or Shallue and van de Woestijne (Section 6.6.1). - "SSWU" for Simplified SWU (Section 6.6.2, Section 6.6.3). - "ELL2" for Elligator 2 (Section 6.7.1, Section 6.8.2). * ENC_VAR: a string indicating the encoding type and other information. The first two characters of this string indicate whether the suite represents a hash_to_curve or an encode_to_curve operation (Section 3), as follows: - If ENC_VAR begins with "RO", the suite uses hash_to_curve. - If ENC_VAR begins with "NU", the suite uses encode_to_curve. - ENC_VAR MUST NOT begin with any other string. ENC_VAR MAY also be used to encode other information used to identify variants, for example, a version number. The RECOMMENDED way to do so is to add one or more subfields separated by colons. For example, "RO:V02" is an appropriate ENC_VAR value for the second version of a uniform encoding suite, while "RO:V02:FOO01:BAR17" might be used to indicate a variant of that suite. 9. IANA considerations This document has no IANA actions. Faz-Hernandez, et al. Expires 14 May 2022 [Page 46] Internet-Draft hash-to-curve November 2021 10. Security considerations Section 3.1 describes considerations related to domain separation. See Section 10.4 for further discussion. Section 5 describes considerations for uniformly hashing to field elements; see Section 10.2 and Section 10.3 for further discussion. Each encoding type (Section 3) accepts an arbitrary byte string and maps it to a point on the curve sampled from a distribution that depends on the encoding type. It is important to note that using a nonuniform encoding or directly evaluating one of the mappings of Section 6 produces an output that is easily distinguished from a uniformly random point. Applications that use a nonuniform encoding SHOULD carefully analyze the security implications of nonuniformity. When the required encoding is not clear, applications SHOULD use a uniform encoding. Both encodings given in Section 3 can output the identity element of the group G. The probability that either encoding function outputs the identity element is roughly 1/r for a random input, which is negligible for cryptographically useful elliptic curves. Further, it is computationally infeasible to find an input to either encoding function whose corresponding output is the identity element. (Both of these properties hold when the encoding functions are instantiated with a hash_to_field function that follows all guidelines in Section 5.) Protocols that use these encoding functions SHOULD NOT add a special case to detect and "fix" the identity element. When the hash_to_curve function (Section 3) is instantiated with a hash_to_field function that is indifferentiable from a random oracle (Section 5), the resulting function is indifferentiable from a random oracle ([MRH04], [BCIMRT10], [FFSTV13], [LBB19], [H20]). In many cases such a function can be safely used in cryptographic protocols whose security analysis assumes a random oracle that outputs uniformly random points on an elliptic curve. As Ristenpart et al. discuss in [RSS11], however, not all security proofs that rely on random oracles continue to hold when those oracles are replaced by indifferentiable functionalities. This limitation should be considered when analyzing the security of protocols relying on the hash_to_curve function. When hashing passwords using any function described in this document, an adversary who learns the output of the hash function (or potentially any intermediate value, e.g., the output of hash_to_field) may be able to carry out a dictionary attack. To mitigate such attacks, it is recommended to first execute a more costly key derivation function (e.g., PBKDF2 [RFC2898], scrypt Faz-Hernandez, et al. Expires 14 May 2022 [Page 47] Internet-Draft hash-to-curve November 2021 [RFC7914], or Argon2 [I-D.irtf-cfrg-argon2]) on the password, then hash the output of that function to the target elliptic curve. For collision resistance, the hash underlying the key derivation function should be chosen according to the guidelines listed in Section 5.4.1. Constant-time implementations of all functions in this document are STRONGLY RECOMMENDED for all uses, to avoid leaking information via side channels. It is especially important to use a constant-time implementation when inputs to an encoding are secret values; in such cases, constant-time implementations are REQUIRED for security against timing attacks (e.g., [VR20]). When constant-time implementations are required, all basic operations and utility functions must be implemented in constant time, as discussed in Section 4. In some applications (e.g., embedded systems), leakage through other side channels (e.g., power or electromagnetic side channels) may be pertinent. Defending against such leakage is outside the scope of this document, because the nature of the leakage and the appropriate defense depend on the application. 10.1. encode_to_curve: output distribution and indifferentiability The encode_to_curve function (Section 3) returns points sampled from a distribution that is statistically far from uniform. This distribution is bounded roughly as follows: first, it includes at least one eighth of the points in G, and second, the probability of points in the distribution varies by at most a factor of four. These bounds hold when encode_to_curve is instantiated with any of the map_to_curve functions in Section 6. The bounds above are derived from several works in the literature. Specifically: * Shallue and van de Woestijne [SW06] and Fouque and Tibouchi [FT12] derive bounds on the Shallue-van de Woestijne mapping (Section 6.6.1). * Fouque and Tibouchi [FT10] and Tibouchi [T14] derive bounds for the Simplified SWU mapping (Section 6.6.2, Section 6.6.3). * Bernstein et al. [BHKL13] derive bounds for the Elligator 2 mapping (Section 6.7.1, Section 6.8.2). Indifferentiability of encode_to_curve follows from an argument similar to the one given by Brier et al. [BCIMRT10]; we briefly sketch. Consider an ideal random oracle Hc() that samples from the distribution induced by the map_to_curve function called by encode_to_curve, and assume for simplicity that the target elliptic curve has cofactor 1 (a similar argument applies for non-unity Faz-Hernandez, et al. Expires 14 May 2022 [Page 48] Internet-Draft hash-to-curve November 2021 cofactors). Indifferentiability holds just if it is possible to efficiently simulate the "inner" random oracle in encode_to_curve, namely, hash_to_field. The simulator works as follows: on a fresh query msg, the simulator queries Hc(msg) and receives a point P in the image of map_to_curve (if msg is the same as a prior query, the simulator just returns the value it gave in response to that query). The simulator then computes the possible preimages of P under map_to_curve, i.e., elements u of F such that map_to_curve(u) == P (Tibouchi [T14] shows that this can be done efficiently for the Shallue-van de Woestijne and Simplified SWU maps, and Bernstein et al. show the same for Elligator 2). The simulator selects one such preimage at random and returns this value as the simulated output of the "inner" random oracle. By hypothesis, Hc() samples from the distribution induced by map_to_curve on a uniformly random input element of F, so this value is uniformly random and induces the correct point P when passed through map_to_curve. 10.2. hash_to_field security The hash_to_field function defined in Section 5 is indifferentiable from a random oracle [MRH04] when expand_message (Section 5.4) is modeled as a random oracle. By composability of indifferentiability proofs, this also holds when expand_message is proved indifferentiable from a random oracle relative to an underlying primitive that is modeled as a random oracle. When following the guidelines in Section 5.4, both variants of expand_message defined in that section meet this requirement (see also Section 10.3). We very briefly sketch the indifferentiability argument for hash_to_field. Notice that each integer mod p that hash_to_field returns (i.e., each element of the vector representation of F) is a member of an equivalence class of roughly 2^k integers of length log2(p) + k bits, all of which are equal modulo p. For each integer mod p that hash_to_field returns, the simulator samples one member of this equivalence class at random and outputs the byte string returned by I2OSP. (Notice that this is essentially the inverse of the hash_to_field procedure.) 10.3. expand_message_xmd security The expand_message_xmd function defined in Section 5.4.1 is indifferentiable from a random oracle [MRH04] when one of the following holds: 1. H is indifferentiable from a random oracle, 2. H is a sponge-based hash function whose inner function is modeled as a random transformation or random permutation [BDPV08], or Faz-Hernandez, et al. Expires 14 May 2022 [Page 49] Internet-Draft hash-to-curve November 2021 3. H is a Merkle-Damgaard hash function whose compression function is modeled as a random oracle [CDMP05]. For cases (1) and (2), the indifferentiability of expand_message_xmd follows directly from the indifferentiability of H. For case (3), i.e., for H a Merkle-Damgaard hash function, indifferentiability follows from [CDMP05], Theorem 3.5. In particular, expand_message_xmd computes b_0 by prefixing the message with one block of 0-bytes plus auxiliary information (length, counter, and DST). Then, each of the output blocks b_i, i >= 1 in expand_message_xmd is the result of invoking H on a unique, prefix- free encoding of b_0. This is true, first, because the length of the input to all such invocations is equal and fixed by the choice of H and DST, and second, because each such input has a unique suffix (because of the inclusion of the counter byte I2OSP(i, 1)). The essential difference between the construction of [CDMP05] and expand_message_xmd is that the latter hashes a counter appended to strxor(b_0, b_(i - 1)) (step 10) rather than to b_0. This approach increases the Hamming distance between inputs to different invocations of H, which reduces the likelihood that nonidealities in H affect the distribution of the b_i values. We note that expand_message_xmd can be used to instantiate a general- purpose indifferentiable functionality with variable-length output based on any hash function meeting one of the above criteria. Applications that use expand_message_xmd outside of hash_to_field should ensure domain separation by picking a distinct value for DST. 10.4. Domain separation recommendations As discussed in Section 2.2.5, the purpose of domain separation is to ensure that security analyses of cryptographic protocols that query multiple independent random oracles remain valid even if all of these random oracles are instantiated based on one underlying function H. The expand_message variants in this document (Section 5.4) ensure domain separation by appending a suffix-free-encoded domain separation tag DST_prime to all strings hashed by H, an underlying hash or extensible output function. (Other expand_message variants that follow the guidelines in Section 5.4.4 are expected to behave similarly, but these should be analyzed on a case-by-case basis.) For security, applications that use the same function H outside of expand_message should enforce domain separation between those uses of H and expand_message, and should separate all of these from uses of H in other applications. Faz-Hernandez, et al. Expires 14 May 2022 [Page 50] Internet-Draft hash-to-curve November 2021 This section suggests four methods for enforcing domain separation from expand_message variants, explains how each method achieves domain separation, and lists the situations in which each is appropriate. These methods share a high-level structure: the application designer fixes a tag DST_ext distinct from DST_prime and augments calls to H with DST_ext. Each method augments calls to H differently, and each may impose additional requirements on DST_ext. These methods can be used to instantiate multiple domain separated functions (e.g., H1 and H2) by selecting distinct DST_ext values for each (e.g., DST_ext1, DST_ext2). 1. (Suffix-only domain separation.) This method is useful when domain separating invocations of H from expand_message_xmd or expand_message_xof. It is not appropriate for domain separating expand_message from HMAC-H [RFC2104]; for that purpose, see method 4. To instantiate a suffix-only domain separated function Hso, compute Hso(msg) = H(msg || DST_ext) DST_ext should be suffix-free encoded (e.g., by appending one byte encoding the length of DST_ext) to make it infeasible to find distinct (msg, DST_ext) pairs that hash to the same value. This method ensures domain separation because all distinct invocations of H have distinct suffixes, since DST_ext is distinct from DST_prime. 2. (Prefix-suffix domain separation.) This method can be used in the same cases as the suffix-only method. To instantiate a prefix-suffix domain separated function Hps, compute Hps(msg) = H(DST_ext || msg || I2OSP(0, 1)) DST_ext should be prefix-free encoded (e.g., by adding a one-byte prefix that encodes the length of DST_ext) to make it infeasible to find distinct (msg, DST_ext) pairs that hash to the same value. Faz-Hernandez, et al. Expires 14 May 2022 [Page 51] Internet-Draft hash-to-curve November 2021 This method ensures domain separation because appending the byte I2OSP(0, 1) ensures that inputs to H inside Hps are distinct from those inside expand_message. Specifically, the final byte of DST_prime encodes the length of DST, which is required to be nonzero (Section 3.1, requirement 2), and DST_prime is always appended to invocations of H inside expand_message. 3. (Prefix-only domain separation.) This method is only useful for domain separating invocations of H from expand_message_xmd. It does not give domain separation for expand_message_xof or HMAC-H. To instantiate a prefix-only domain separated function Hpo, compute Hpo(msg) = H(DST_ext || msg) In order for this method to give domain separation, DST_ext should be at least b bits long, where b is the number of bits output by the hash function H. In addition, at least one of the first b bits must be nonzero. Finally, DST_ext should be prefix- free encoded (e.g., by adding a one-byte prefix that encodes the length of DST_ext) to make it infeasible to find distinct (msg, DST_ext) pairs that hash to the same value. This method ensures domain separation as follows. First, since DST_ext contains at least one nonzero bit among its first b bits, it is guaranteed to be distinct from the value Z_pad (Section 5.4.1, step 4), which ensures that all inputs to H are distinct from the input used to generate b_0 in expand_message_xmd. Second, since DST_ext is at least b bits long, it is almost certainly distinct from the values b_0 and strxor(b_0, b_(i - 1)), and therefore all inputs to H are distinct from the inputs used to generate b_i, i >= 1, with high probability. 4. (XMD-HMAC domain separation.) This method is useful for domain separating invocations of H inside HMAC-H (i.e., HMAC [RFC2104] instantiated with hash function H) from expand_message_xmd. It also applies to HKDF-H [RFC5869], as discussed below. Specifically, this method applies when HMAC-H is used with a non- secret key to instantiate a random oracle based on a hash function H (note that expand_message_xmd can also be used for this purpose; see Section 10.3). When using HMAC-H with a high- entropy secret key, domain separation is not necessary; see discussion below. Faz-Hernandez, et al. Expires 14 May 2022 [Page 52] Internet-Draft hash-to-curve November 2021 To choose a non-secret HMAC key DST_key that ensures domain separation from expand_message_xmd, compute DST_key_preimage = "DERIVE-HMAC-KEY-" || DST_ext || I2OSP(0, 1) DST_key = H(DST_key_preimage) Then, to instantiate the random oracle Hro using HMAC-H, compute Hro(msg) = HMAC-H(DST_key, msg) The trailing zero byte in DST_key_preimage ensures that this value is distinct from inputs to H inside expand_message_xmd (because all such inputs have suffix DST_prime, which cannot end with a zero byte as discussed above). This ensures domain separation because, with overwhelming probability, all inputs to H inside of HMAC-H using key DST_key have prefixes that are distinct from the values Z_pad, b_0, and strxor(b_0, b_(i - 1)) inside of expand_message_xmd. For uses of HMAC-H that instantiate a private random oracle by fixing a high-entropy secret key, domain separation from expand_message_xmd is not necessary. This is because, similarly to the case above, all inputs to H inside HMAC-H using this secret key almost certainly have distinct prefixes from all inputs to H inside expand_message_xmd. Finally, this method can be used with HKDF-H [RFC5869] by fixing the salt input to HKDF-Extract to DST_key, computed as above. This ensures domain separation for HKDF-Extract by the same argument as for HMAC-H using DST_key. Moreover, assuming that the IKM input to HKDF-Extract has sufficiently high entropy (say, commensurate with the security parameter), the HKDF-Expand step is domain separated by the same argument as for HMAC-H with a high-entropy secret key (since PRK is exactly that). 10.5. Target security levels Each ciphersuite specifies a target security level (in bits) for the underlying curve. This parameter ensures the corresponding hash_to_field instantiation is conservative and correct. We stress that this parameter is only an upper bound on the security level of the curve, and is neither a guarantee nor endorsement of its suitability for a given application. Mathematical and cryptographic advancements may reduce the effective security level for any curve. Faz-Hernandez, et al. Expires 14 May 2022 [Page 53] Internet-Draft hash-to-curve November 2021 11. Acknowledgements The authors would like to thank Adam Langley for his detailed writeup of Elligator 2 with Curve25519 [L13]; Dan Boneh, Christopher Patton, Benjamin Lipp, and Leonid Reyzin for educational discussions; and David Benjamin, Daniel Bourdrez, Frank Denis, Sean Devlin, Justin Drake, Bjoern Haase, Mike Hamburg, Dan Harkins, Daira Hopwood, Thomas Icart, Andy Polyakov, Thomas Pornin, Mamy Ratsimbazafy, Michael Scott, Filippo Valsorda, and Mathy Vanhoef for helpful reviews and feedback. 12. Contributors * Sharon Goldberg Boston University goldbe@cs.bu.edu * Ela Lee Royal Holloway, University of London Ela.Lee.2010@live.rhul.ac.uk * Michele Orru michele.orru@ens.fr 13. References 13.1. Normative References [EID4730] Langley, A., "RFC 7748, Errata ID 4730", July 2016, . [I-D.irtf-cfrg-pairing-friendly-curves] Sakemi, Y., Kobayashi, T., Saito, T., and R. S. Wahby, "Pairing-Friendly Curves", Work in Progress, Internet- Draft, draft-irtf-cfrg-pairing-friendly-curves-10, 30 July 2021, . [RFC2119] Bradner, S., "Key words for use in RFCs to Indicate Requirement Levels", BCP 14, RFC 2119, DOI 10.17487/RFC2119, March 1997, . Faz-Hernandez, et al. Expires 14 May 2022 [Page 54] Internet-Draft hash-to-curve November 2021 [RFC7748] Langley, A., Hamburg, M., and S. Turner, "Elliptic Curves for Security", RFC 7748, DOI 10.17487/RFC7748, January 2016, . [RFC8017] Moriarty, K., Ed., Kaliski, B., Jonsson, J., and A. Rusch, "PKCS #1: RSA Cryptography Specifications Version 2.2", RFC 8017, DOI 10.17487/RFC8017, November 2016, . [RFC8174] Leiba, B., "Ambiguity of Uppercase vs Lowercase in RFC 2119 Key Words", BCP 14, RFC 8174, DOI 10.17487/RFC8174, May 2017, . 13.2. Informative References [AFQTZ14] Aranha, D.F., Fouque, P.A., Qian, C., Tibouchi, M., and J.C. Zapalowicz, "Binary Elligator squared", DOI 10.1007/978-3-319-13051-4_2, pages 20-37, In Selected Areas in Cryptography - SAC 2014, 2014, . [AR13] Adj, G. and F. Rodriguez-Henriquez, "Square Root Computation over Even Extension Fields", DOI 10.1109/TC.2013.145, pages 2829-2841, In IEEE Transactions on Computers. vol 63 issue 11, November 2014, . [BBJLP08] Bernstein, D.J., Birkner, P., Joye, M., Lange, T., and C. Peters, "Twisted Edwards curves", DOI 10.1007/978-3-540-68164-9_26, pages 389-405, In AFRICACRYPT 2008, 2008, . [BCIMRT10] Brier, E., Coron, J-S., Icart, T., Madore, D., Randriam, H., and M. Tibouchi, "Efficient Indifferentiable Hashing into Ordinary Elliptic Curves", DOI 10.1007/978-3-642-14623-7_13, pages 237-254, In Advances in Cryptology - CRYPTO 2010, 2010, . [BDPV08] Bertoni,, G., Daemen, J., Peeters, M., and G. Van Assche, "On the Indifferentiability of the Sponge Construction", DOI 10.1007/978-3-540-78967-3_11, pages 181-197, In Advances in Cryptology - EUROCRYPT 2008, 2008, . Faz-Hernandez, et al. Expires 14 May 2022 [Page 55] Internet-Draft hash-to-curve November 2021 [BF01] Boneh, D. and M. Franklin, "Identity-based encryption from the Weil pairing", DOI 10.1007/3-540-44647-8_13, pages 213-229, In Advances in Cryptology - CRYPTO 2001, August 2001, . [BHKL13] Bernstein, D.J., Hamburg, M., Krasnova, A., and T. Lange, "Elligator: elliptic-curve points indistinguishable from uniform random strings", DOI 10.1145/2508859.2516734, pages 967-980, In Proceedings of the 2013 ACM SIGSAC Conference on Computer and Communications Security, November 2013, . [BLAKE2X] Aumasson, J-P., Neves, S., Wilcox-O'Hearn, Z., and C. Winnerlein, "BLAKE2X", December 2016, . [BLMP19] Bernstein, D.J., Lange, T., Martindale, C., and L. Panny, "Quantum circuits for the CSIDH: optimizing quantum evaluation of isogenies", DOI 10.1007/978-3-030-17656-3, In Advances in Cryptology - EUROCRYPT 2019, 2019, . [BLS01] Boneh, D., Lynn, B., and H. Shacham, "Short signatures from the Weil pairing", DOI 10.1007/s00145-004-0314-9, pages 297-319, In Journal of Cryptology, vol 17, July 2004, . [BLS03] Barreto, P., Lynn, B., and M. Scott, "Constructing Elliptic Curves with Prescribed Embedding Degrees", DOI 10.1007/3-540-36413-7_19, pages 257-267, In Security in Communication Networks, 2003, . [BLS12-381] Bowe, S., "BLS12-381: New zk-SNARK Elliptic Curve Construction", March 2017, . [BM92] Bellovin, S.M. and M. Merritt, "Encrypted key exchange: Password-based protocols secure against dictionary attacks", DOI 10.1109/RISP.1992.213269, pages 72-84, In IEEE Symposium on Security and Privacy - Oakland 1992, 1992, . Faz-Hernandez, et al. Expires 14 May 2022 [Page 56] Internet-Draft hash-to-curve November 2021 [BMP00] Boyko, V., MacKenzie, P.D., and S. Patel, "Provably secure password-authenticated key exchange using Diffie-Hellman", DOI 10.1007/3-540-45539-6_12, pages 156-171, In Advances in Cryptology - EUROCRYPT 2000, May 2000, . [BN05] Barreto, P. and M. Naehrig, "Pairing-Friendly Elliptic Curves of Prime Order", DOI 10.1007/11693383_22, pages 319-331, In Selected Areas in Cryptography 2005, 2006, . [BP17] Budroni, A. and F. Pintore, "Efficient hash maps to G2 on BLS curves", ePrint 2017/419, May 2017, . [BR93] Bellare, M. and P. Rogaway, "Random oracles are practical: a paradigm for designing efficient protocols", DOI 10.1145/168588.168596, pages 62-73, In Proceedings of the 1993 ACM Conference on Computer and Communications Security, December 1993, . [C93] Cohen, H., "A Course in Computational Algebraic Number Theory", ISBN 9783642081422, publisher Springer-Verlag, 1993, . [CDMP05] Coron, J-S., Dodis, Y., Malinaud, C., and P. Puniya, "Merkle-Damgaard Revisited: How to Construct a Hash Function", DOI 10.1007/11535218_26, pages 430-448, In Advances in Cryptology - CRYPTO 2005, 2005, . [CFADLNV05] Cohen, H., Frey, G., Avanzi, R., Doche, C., Lange, T., Nguyen, K., and F. Vercauteren, "Handbook of Elliptic and Hyperelliptic Curve Cryptography", ISBN 9781584885184, publisher Chapman and Hall / CRC, 2005, . [CK11] Couveignes, J. and J. Kammerer, "The geometry of flex tangents to a cubic curve and its parameterizations", DOI 10.1016/j.jsc.2011.11.003, pages 266-281, In Journal of Symbolic Computation, vol 47 issue 3, 2012, . Faz-Hernandez, et al. Expires 14 May 2022 [Page 57] Internet-Draft hash-to-curve November 2021 [F11] Farashahi, R.R., "Hashing into Hessian curves", DOI 10.1007/978-3-642-21969-6_17, pages 278-289, In AFRICACRYPT 2011, 2011, . [FFSTV13] Farashahi, R.R., Fouque, P.A., Shparlinski, I.E., Tibouchi, M., and J.F. Voloch, "Indifferentiable deterministic hashing to elliptic and hyperelliptic curves", DOI 10.1090/S0025-5718-2012-02606-8, pages 491-512, In Math. Comp. vol 82, 2013, . [FIPS180-4] National Institute of Standards and Technology (NIST), "Secure Hash Standard (SHS)", August 2015, . [FIPS186-4] National Institute of Standards and Technology (NIST), "FIPS Publication 186-4: Digital Signature Standard", July 2013, . [FIPS202] National Institute of Standards and Technology (NIST), "SHA-3 Standard: Permutation-Based Hash and Extendable- Output Functions", August 2015, . [FJT13] Fouque, P-A., Joux, A., and M. Tibouchi, "Injective encodings to elliptic curves", DOI 10.1007/978-3-642-39059-3_14, pages 203-218, In ACISP 2013, 2013, . [FKR11] Fuentes-Castaneda, L., Knapp, E., and F. Rodriguez- Henriquez, "Fast Hashing to G2 on Pairing-Friendly Curves", DOI 10.1007/978-3-642-28496-0_25, pages 412-430, In Selected Areas in Cryptography, 2011, . [FSV09] Farashahi, R.R., Shparlinski, I.E., and J.F. Voloch, "On hashing into elliptic curves", DOI 10.1515/JMC.2009.022, pages 353-360, In Journal of Mathematical Cryptology, vol 3 no 4, 2009, . Faz-Hernandez, et al. Expires 14 May 2022 [Page 58] Internet-Draft hash-to-curve November 2021 [FT10] Fouque, P-A. and M. Tibouchi, "Estimating the size of the image of deterministic hash functions to elliptic curves.", DOI 10.1007/978-3-642-14712-8_5, pages 81-91, In Progress in Cryptology - LATINCRYPT 2010, 2010, . [FT12] Fouque, P-A. and M. Tibouchi, "Indifferentiable Hashing to Barreto-Naehrig Curves", DOI 10.1007/978-3-642-33481-8_1, pages 1-7, In Progress in Cryptology - LATINCRYPT 2012, 2012, . [H20] Hamburg, M., "Indifferentiable hashing from Elligator 2", 2020, . [hash2curve-repo] "Hashing to Elliptic Curves - GitHub repository", 2019, . [I-D.irtf-cfrg-argon2] Biryukov, A., Dinu, D., Khovratovich, D., and S. Josefsson, "Argon2 Memory-Hard Function for Password Hashing and Proof-of-Work Applications", Work in Progress, Internet-Draft, draft-irtf-cfrg-argon2-13, 11 March 2021, . [I-D.irtf-cfrg-bls-signature] Boneh, D., Gorbunov, S., Wahby, R. S., Wee, H., and Z. Zhang, "BLS Signatures", Work in Progress, Internet-Draft, draft-irtf-cfrg-bls-signature-04, 10 September 2020, . [I-D.irtf-cfrg-ristretto255-decaf448] Valence, H. D., Grigg, J., Tankersley, G., Valsorda, F., Lovecruft, I., and M. Hamburg, "The ristretto255 and decaf448 Groups", Work in Progress, Internet-Draft, draft- irtf-cfrg-ristretto255-decaf448-01, 4 August 2021, . [I-D.irtf-cfrg-voprf] Davidson, A., Faz-Hernandez, A., Sullivan, N., and C. A. Wood, "Oblivious Pseudorandom Functions (OPRFs) using Prime-Order Groups", Work in Progress, Internet-Draft, draft-irtf-cfrg-voprf-08, 25 October 2021, . Faz-Hernandez, et al. Expires 14 May 2022 [Page 59] Internet-Draft hash-to-curve November 2021 [I-D.irtf-cfrg-vrf] Goldberg, S., Reyzin, L., Papadopoulos, D., and J. Vcelak, "Verifiable Random Functions (VRFs)", Work in Progress, Internet-Draft, draft-irtf-cfrg-vrf-09, 17 May 2021, . [Icart09] Icart, T., "How to Hash into Elliptic Curves", DOI 10.1007/978-3-642-03356-8_18, pages 303-316, In Advances in Cryptology - CRYPTO 2009, 2009, . [J96] Jablon, D.P., "Strong password-only authenticated key exchange", DOI 10.1145/242896.242897, pages 5-26, In SIGCOMM Computer Communication Review, vol 26 issue 5, 1996, . [jubjub-fq] "zkcrypto/jubjub - fq.rs", 2019, . [KLR10] Kammerer, J., Lercier, R., and G. Renault, "Encoding points on hyperelliptic curves over finite fields in deterministic polynomial time", DOI 10.1007/978-3-642-17455-1_18, pages 278-297, In PAIRING 2010, 2010, . [L13] Langley, A., "Implementing Elligator for Curve25519", 2013, . [LBB19] Lipp, B., Blanchet, B., and K. Bhargavan, "A Mechanised Proof of the WireGuard Virtual Private Network Protocol", In INRIA Research Report No. 9269, April 2019, . [MOV96] Menezes, A.J., van Oorschot, P.C., and S.A. Vanstone, "Handbook of Applied Cryptography", ISBN 9780849385230, publisher CRC Press, 1996, . Faz-Hernandez, et al. Expires 14 May 2022 [Page 60] Internet-Draft hash-to-curve November 2021 [MRH04] Maurer, U., Renner, R., and C. Holenstein, "Indifferentiability, impossibility results on reductions, and applications to the random oracle methodology", DOI 10.1007/978-3-540-24638-1_2, pages 21-39, In TCC 2004: Theory of Cryptography, February 2004, . [MRV99] Micali, S., Rabin, M., and S. Vadhan, "Verifiable Random Functions", DOI 10.1109/SFFCS.1999.814584, In Symposium on the Foundations of Computer Science, October 1999, . [MT98] Matsumoto, M. and T. Nishimura, "Mersenne twister: A 623-dimensionally equidistributed uniform pseudo-random number generator", DOI 10.1145/272991.272995, pages 3-30, In ACM Transactions on Modeling and Computer Simulation (TOMACS), Volume 8, Issue 1, January 1998, . [NR97] Naor, M. and O. Reingold, "Number-theoretic constructions of efficient pseudo-random functions", DOI 10.1109/SFCS.1997.646134, In Symposium on the Foundations of Computer Science, October 1997, . [p1363.2] IEEE Computer Society, "IEEE Standard Specification for Password-Based Public-Key Cryptography Techniques", September 2008, . [p1363a] IEEE Computer Society, "IEEE Standard Specifications for Public-Key Cryptography---Amendment 1: Additional Techniques", March 2004, . [P20] Pornin, T., "Efficient Elliptic Curve Operations On Microcontrollers With Finite Field Extensions", 2020, . [RCB16] Renes, J., Costello, C., and L. Batina, "Complete addition formulas for prime order elliptic curves", DOI 10.1007/978-3-662-49890-3_16, pages 403-428, In Advances in Cryptology - EUROCRYPT 2016, May 2016, . Faz-Hernandez, et al. Expires 14 May 2022 [Page 61] Internet-Draft hash-to-curve November 2021 [RFC2104] Krawczyk, H., Bellare, M., and R. Canetti, "HMAC: Keyed- Hashing for Message Authentication", RFC 2104, DOI 10.17487/RFC2104, February 1997, . [RFC2898] Kaliski, B., "PKCS #5: Password-Based Cryptography Specification Version 2.0", RFC 2898, DOI 10.17487/RFC2898, September 2000, . [RFC5869] Krawczyk, H. and P. Eronen, "HMAC-based Extract-and-Expand Key Derivation Function (HKDF)", RFC 5869, DOI 10.17487/RFC5869, May 2010, . [RFC7693] Saarinen, M-J., Ed. and J-P. Aumasson, "The BLAKE2 Cryptographic Hash and Message Authentication Code (MAC)", RFC 7693, DOI 10.17487/RFC7693, November 2015, . [RFC7914] Percival, C. and S. Josefsson, "The scrypt Password-Based Key Derivation Function", RFC 7914, DOI 10.17487/RFC7914, August 2016, . [RSS11] Ristenpart, T., Shacham, H., and T. Shrimpton, "Careful with Composition: Limitations of the Indifferentiability Framework", DOI 10.1007/978-3-642-20465-4_27, pages 487-506, In Advances in Cryptology - EUROCRYPT 2011, May 2011, . [S05] Skalba, M., "Points on elliptic curves over finite fields", DOI 10.4064/aa117-3-7, pages 293-301, In Acta Arithmetica, vol 117 no 3, 2005, . [S85] Schoof, R., "Elliptic Curves Over Finite Fields and the Computation of Square Roots mod p", DOI 10.1090/S0025-5718-1985-0777280-6, pages 483-494, In Mathematics of Computation vol 44 issue 170, April 1985, . [SAGE] The Sage Developers, "SageMath, the Sage Mathematics Software System", 2019, . Faz-Hernandez, et al. Expires 14 May 2022 [Page 62] Internet-Draft hash-to-curve November 2021 [SBCDK09] Scott, M., Benger, N., Charlemagne, M., Dominguez Perez, L.J., and E.J. Kachisa, "Fast Hashing to G2 on Pairing- Friendly Curves", DOI 10.1007/978-3-642-03298-1_8, pages 102-113, In Pairing-Based Cryptography - Pairing 2009, 2009, . [SEC1] Standards for Efficient Cryptography Group (SECG), "SEC 1: Elliptic Curve Cryptography", May 2009, . [SEC2] Standards for Efficient Cryptography Group (SECG), "SEC 2: Recommended Elliptic Curve Domain Parameters", January 2010, . [SS04] Schinzel, A. and M. Skalba, "On equations y^2 = x^n + k in a finite field.", DOI 10.4064/ba52-3-1, pages 223-226, In Bulletin Polish Acad. Sci. Math. vol 52, no 3, 2004, . [SW06] Shallue, A. and C. van de Woestijne, "Construction of rational points on elliptic curves over finite fields", DOI 10.1007/11792086_36, pages 510-524, In Algorithmic Number Theory. ANTS 2006., 2006, . [T14] Tibouchi, M., "Elligator squared: Uniform points on elliptic curves of prime order as uniform random strings", DOI 10.1007/978-3-662-45472-5_10, pages 139-156, In Financial Cryptography and Data Security - FC 2014, 2014, . [TK17] Tibouchi, M. and T. Kim, "Improved elliptic curve hashing and point representation", DOI 10.1007/s10623-016-0288-2, pages 161-177, In Designs, Codes, and Cryptography, vol 82, 2017, . [U07] Ulas, M., "Rational points on certain hyperelliptic curves over finite fields", DOI 10.4064/ba55-2-1, pages 97-104, In Bulletin Polish Acad. Sci. Math. vol 55, no 2, 2007, . [VR20] Vanhoef, M. and E. Ronen, "Dragonblood: Analyzing the Dragonfly Handshake of WPA3 and EAP-pwd", In IEEE Symposium on Security & Privacy (SP), 2020, . Faz-Hernandez, et al. Expires 14 May 2022 [Page 63] Internet-Draft hash-to-curve November 2021 [W08] Washington, L.C., "Elliptic curves: Number theory and cryptography", ISBN 9781420071467, publisher Chapman and Hall / CRC, edition 2nd, 2008, . [W19] Wahby, R.S., "An explicit, generic parameterization for the Shallue--van de Woestijne map", 2019, . [WB19] Wahby, R.S. and D. Boneh, "Fast and simple constant-time hashing to the BLS12-381 elliptic curve", DOI 10.13154/tches.v2019.i4.154-179, ePrint 2019/403, issue 4, volume 2019, In IACR Trans. CHES, August 2019, . Appendix A. Related work The problem of mapping arbitrary bit strings to elliptic curve points has been the subject of both practical and theoretical research. This section briefly describes the background and research results that underly the recommendations in this document. This section is provided for informational purposes only. A naive but generally insecure method of mapping a string msg to a point on an elliptic curve E having n points is to first fix a point P that generates the elliptic curve group, and a hash function Hn from bit strings to integers less than n; then compute Hn(msg) * P, where the * operator represents scalar multiplication. The reason this approach is insecure is that the resulting point has a known discrete log relationship to P. Thus, except in cases where this method is specified by the protocol, it must not be used; doing so risks catastrophic security failures. Boneh et al. [BLS01] describe an encoding method they call MapToGroup, which works roughly as follows: first, use the input string to initialize a pseudorandom number generator, then use the generator to produce a value x in F. If x is the x-coordinate of a point on the elliptic curve, output that point. Otherwise, generate a new value x in F and try again. Since a random value x in F has probability about 1/2 of corresponding to a point on the curve, the expected number of tries is just two. However, the running time of this method depends on the input string, which means that it is not safe to use in protocols sensitive to timing side channels. Schinzel and Skalba [SS04] introduce a method of constructing elliptic curve points deterministically, for a restricted class of curves and a very small number of points. Skalba [S05] generalizes Faz-Hernandez, et al. Expires 14 May 2022 [Page 64] Internet-Draft hash-to-curve November 2021 this construction to more curves and more points on those curves. Shallue and van de Woestijne [SW06] further generalize and simplify Skalba's construction, yielding concretely efficient maps to a constant fraction of the points on almost any curve. Fouque and Tibouchi [FT12] give a parameterization of this mapping for Barreto- Naehrig pairing-friendly curves [BN05]. Ulas [U07] describes a simpler version of the Shallue-van de Woestijne map, and Brier et al. [BCIMRT10] give a further simplification, which the authors call the "simplified SWU" map. That simplified map applies only to fields of characteristic p = 3 (mod 4); Wahby and Boneh [WB19] generalize to fields of any characteristic, and give further optimizations. Boneh and Franklin give a deterministic algorithm mapping to certain supersingular curves over fields of characteristic p = 2 (mod 3) [BF01]. Icart gives another deterministic algorithm which maps to any curve over a field of characteristic p = 2 (mod 3) [Icart09]. Several extensions and generalizations follow this work, including [FSV09], [FT10], [KLR10], [F11], and [CK11]. Following the work of Farashahi [F11], Fouque et al. [FJT13] describe a mapping to curves over fields of characteristic p = 3 (mod 4) having a number of points divisible by 4. Bernstein et al. [BHKL13] optimize this mapping and describe a related mapping that they call "Elligator 2," which applies to any curve over a field of odd characteristic having a point of order 2. This includes Curve25519 and Curve448, both of which are CFRG-recommended curves [RFC7748]. Bernstein et al. [BLMP19] extend the Elligator 2 map to a class of supersingular curves over fields of characteristic p = 3 (mod 4). An important caveat regarding all of the above deterministic mapping functions is that none of them map to the entire curve, but rather to some fraction of the points. This means that they cannot be used directly to construct a random oracle that outputs points on the curve. Faz-Hernandez, et al. Expires 14 May 2022 [Page 65] Internet-Draft hash-to-curve November 2021 Brier et al. [BCIMRT10] give two solutions to this problem. The first, which Brier et al. prove applies to Icart's method, computes f(H0(msg)) + f(H1(msg)) for two distinct hash functions H0 and H1 from bit strings to F and a mapping f from F to the elliptic curve E. The second, which applies to essentially all deterministic mappings but is more costly, computes f(H0(msg)) + H2(msg) * P, for P a generator of the elliptic curve group and H2 a hash from bit strings to integers modulo r, the order of the elliptic curve group. Farashahi et al. [FFSTV13] improve the analysis of the first method, showing that it applies to essentially all deterministic mappings. Tibouchi and Kim [TK17] further refine the analysis and describe additional optimizations. Complementary to the problem of mapping from bit strings to elliptic curve points, Bernstein et al. [BHKL13] study the problem of mapping from elliptic curve points to uniformly random bit strings, giving solutions for a class of curves including Montgomery and twisted Edwards curves. Tibouchi [T14] and Aranha et al. [AFQTZ14] generalize these results. This document does not deal with this complementary problem. Appendix B. Hashing to ristretto255 ristretto255 [I-D.irtf-cfrg-ristretto255-decaf448] provides a prime- order group based on Curve25519 [RFC7748]. This section describes hash_to_ristretto255, which implements a random-oracle encoding to this group that has a uniform output distribution (Section 2.2.3) and the same security properties and interface as the hash_to_curve function (Section 3). The ristretto255 API defines a one-way map ([I-D.irtf-cfrg-ristretto255-decaf448], Section 4.3.4); this section refers to that map as ristretto255_map. The hash_to_ristretto255 function MUST be instantiated with an expand_message function that conforms to the requirements given in Section 5.4. In addition, it MUST use a domain separation tag constructed as described in Section 3.1, and all domain separation recommendations given in Section 10.4 apply when implementing protocols that use hash_to_ristretto255. Faz-Hernandez, et al. Expires 14 May 2022 [Page 66] Internet-Draft hash-to-curve November 2021 hash_to_ristretto255(msg) Parameters: - DST, a domain separation tag (see discussion above). - expand_message, a function that expands a byte string and domain separation tag into a uniformly random byte string (see discussion above). - ristretto255_map, the one-way map from the ristretto255 API. Input: msg, an arbitrary-length byte string. Output: P, an element of the ristretto255 group. Steps: 1. uniform_bytes = expand_message(msg, DST, 64) 2. P = ristretto255_map(uniform_bytes) 3. return P Since hash_to_ristretto255 is not a hash-to-curve suite, it does not have a Suite ID. If a similar identifier is needed, it MUST be constructed following the guidelines in Section 8.10, with the following parameters: * CURVE_ID: "ristretto255" * HASH_ID: as described in Section 8.10 * MAP_ID: "R255MAP" * ENC_VAR: "RO" For example, if expand_message is expand_message_xmd using SHA-512, the REQUIRED identifier is: ristretto255_XMD:SHA-512_R255MAP_RO_ Appendix C. Hashing to decaf448 Similar to ristretto255, decaf448 [I-D.irtf-cfrg-ristretto255-decaf448] provides a prime-order group based on Curve448 [RFC7748]. This section describes hash_to_decaf448, which implements a random-oracle encoding to this group that has a uniform output distribution (Section 2.2.3) and the same security properties and interface as the hash_to_curve function (Section 3). The decaf448 API defines a one-way map ([I-D.irtf-cfrg-ristretto255-decaf448], Section 5.3.4); this section refers to that map as decaf448_map. Faz-Hernandez, et al. Expires 14 May 2022 [Page 67] Internet-Draft hash-to-curve November 2021 The hash_to_decaf448 function MUST be instantiated with an expand_message function that conforms to the requirements given in Section 5.4. In addition, it MUST use a domain separation tag constructed as described in Section 3.1, and all domain separation recommendations given in Section 10.4 apply when implementing protocols that use hash_to_decaf448. hash_to_decaf448(msg) Parameters: - DST, a domain separation tag (see discussion above). - expand_message, a function that expands a byte string and domain separation tag into a uniformly random byte string (see discussion above). - decaf448_map, the one-way map from the decaf448 API. Input: msg, an arbitrary-length byte string. Output: P, an element of the decaf448 group. Steps: 1. uniform_bytes = expand_message(msg, DST, 112) 2. P = decaf448_map(uniform_bytes) 3. return P Since hash_to_decaf448 is not a hash-to-curve suite, it does not have a Suite ID. If a similar identifier is needed, it MUST be constructed following the guidelines in Section 8.10, with the following parameters: * CURVE_ID: "decaf448" * HASH_ID: as described in Section 8.10 * MAP_ID: "D448MAP" * ENC_VAR: "RO" For example, if expand_message is expand_message_xof using SHAKE256, the REQUIRED identifier is: decaf448_XOF:SHAKE256_D448MAP_RO_ Appendix D. Rational maps This section gives rational maps that can be used when hashing to twisted Edwards or Montgomery curves. Faz-Hernandez, et al. Expires 14 May 2022 [Page 68] Internet-Draft hash-to-curve November 2021 Given a twisted Edwards curve, Appendix D.1 shows how to derive a corresponding Montgomery curve and how to map from that curve to the twisted Edwards curve. This mapping may be used when hashing to twisted Edwards curves as described in Section 6.8. Given a Montgomery curve, Appendix D.2 shows how to derive a corresponding Weierstrass curve and how to map from that curve to the Montgomery curve. This mapping can be used to hash to Montgomery or twisted Edwards curves via the Shallue-van de Woestijne (Section 6.6.1) or Simplified SWU (Section 6.6.2) method, as follows: * For Montgomery curves, first map to the Weierstrass curve, then convert to Montgomery coordinates via the mapping. * For twisted Edwards curves, compose the Weierstrass to Montgomery mapping with the Montgomery to twisted Edwards mapping (Appendix D.1) to obtain a Weierstrass curve and a mapping to the target twisted Edwards curve. Map to this Weierstrass curve, then convert to Edwards coordinates via the mapping. D.1. Generic Montgomery to twisted Edwards map This section gives a generic birational map between twisted Edwards and Montgomery curves. The map in this section is a simplified version of the map given in [BBJLP08], Theorem 3.2. Specifically, this section's map handles exceptional cases in a simplified way that is geared towards hashing to a twisted Edwards curve's prime-order subgroup. The twisted Edwards curve a * v^2 + w^2 = 1 + d * v^2 * w^2 is birationally equivalent to the Montgomery curve K * t^2 = s^3 + J * s^2 + s which has the form required by the Elligator 2 mapping of Section 6.7.1. The coefficients of the Montgomery curve are * J = 2 * (a + d) / (a - d) * K = 4 / (a - d) The rational map from the point (s, t) on the above Montgomery curve to the point (v, w) on the twisted Edwards curve is given by Faz-Hernandez, et al. Expires 14 May 2022 [Page 69] Internet-Draft hash-to-curve November 2021 * v = s / t * w = (s - 1) / (s + 1) This mapping is undefined when t == 0 or s == -1, i.e., when the denominator of either of the above rational functions is zero. Implementations MUST detect exceptional cases and return the value (v, w) = (0, 1), which is the identity point on all twisted Edwards curves. The following straight-line implementation of the above rational map handles the exceptional cases. edw_to_monty_generic(s, t) Input: (s, t), a point on the curve K * t^2 = s^3 + J * s^2 + s. Output: (v, w), a point on an equivalent twisted Edwards curve. 1. tv1 = s + 1 2. tv2 = tv1 * t # (s + 1) * t 3. tv2 = inv0(tv2) # 1 / ((s + 1) * t) 4. v = tv2 * tv1 # 1 / t 5. v = v * s # s / t 6. w = tv2 * t # 1 / (s + 1) 7. tv1 = s - 1 8. w = w * tv1 # (s - 1) / (s + 1) 9. e = tv2 == 0 10. w = CMOV(w, 1, e) # handle exceptional case 11. return (v, w) For completeness, we also give the inverse relations. (Note that this map is not required when hashing to twisted Edwards curves.) The coefficients of the twisted Edwards curve corresponding to the above Montgomery curve are * a = (J + 2) / K * d = (J - 2) / K The rational map from the point (v, w) on the twisted Edwards curve to the point (s, t) on the Montgomery curve is given by * s = (1 + w) / (1 - w) * t = (1 + w) / (v * (1 - w)) Faz-Hernandez, et al. Expires 14 May 2022 [Page 70] Internet-Draft hash-to-curve November 2021 The mapping is undefined when v == 0 or w == 1. When the goal is to map into the prime-order subgroup of the Montgomery curve, it suffices to return the identity point on the Montgomery curve in the exceptional cases. D.2. Weierstrass to Montgomery map The rational map from the point (s, t) on the Montgomery curve K * t^2 = s^3 + J * s^2 + s to the point (x, y) on the equivalent Weierstrass curve y^2 = x^3 + A * x + B is given by: * A = (3 - J^2) / (3 * K^2) * B = (2 * J^3 - 9 * J) / (27 * K^3) * x = (3 * s + J) / (3 * K) * y = t / K The inverse map, from the point (x, y) to the point (s, t), is given by * s = (3 * K * x - J) / 3 * t = y * K This mapping can be used to apply the Shallue-van de Woestijne (Section 6.6.1) or Simplified SWU (Section 6.6.2) method to Montgomery curves. Appendix E. Isogeny maps for suites This section specifies the isogeny maps for the secp256k1 and BLS12-381 suites listed in Section 8. These maps are given in terms of affine coordinates. Wahby and Boneh ([WB19], Section 4.3) show how to evaluate these maps in a projective coordinate system (Appendix G.1), which avoids modular inversions. Refer to the draft repository [hash2curve-repo] for a Sage [SAGE] script that constructs these isogenies. Faz-Hernandez, et al. Expires 14 May 2022 [Page 71] Internet-Draft hash-to-curve November 2021 E.1. 3-isogeny map for secp256k1 This section specifies the isogeny map for the secp256k1 suite listed in Section 8.7. The 3-isogeny map from (x', y') on E' to (x, y) on E is given by the following rational functions: * x = x_num / x_den, where - x_num = k_(1,3) * x'^3 + k_(1,2) * x'^2 + k_(1,1) * x' + k_(1,0) - x_den = x'^2 + k_(2,1) * x' + k_(2,0) * y = y' * y_num / y_den, where - y_num = k_(3,3) * x'^3 + k_(3,2) * x'^2 + k_(3,1) * x' + k_(3,0) - y_den = x'^3 + k_(4,2) * x'^2 + k_(4,1) * x' + k_(4,0) The constants used to compute x_num are as follows: * k_(1,0) = 0x8e38e38e38e38e38e38e38e38e38e38e38e38e38e38e38e38e38e38daaaaa8c7 * k_(1,1) = 0x7d3d4c80bc321d5b9f315cea7fd44c5d595d2fc0bf63b92dfff1044f17c6581 * k_(1,2) = 0x534c328d23f234e6e2a413deca25caece4506144037c40314ecbd0b53d9dd262 * k_(1,3) = 0x8e38e38e38e38e38e38e38e38e38e38e38e38e38e38e38e38e38e38daaaaa88c The constants used to compute x_den are as follows: * k_(2,0) = 0xd35771193d94918a9ca34ccbb7b640dd86cd409542f8487d9fe6b745781eb49b * k_(2,1) = 0xedadc6f64383dc1df7c4b2d51b54225406d36b641f5e41bbc52a56612a8c6d14 The constants used to compute y_num are as follows: * k_(3,0) = 0x4bda12f684bda12f684bda12f684bda12f684bda12f684bda12f684b8e38e23c Faz-Hernandez, et al. Expires 14 May 2022 [Page 72] Internet-Draft hash-to-curve November 2021 * k_(3,1) = 0xc75e0c32d5cb7c0fa9d0a54b12a0a6d5647ab046d686da6fdffc90fc201d71a3 * k_(3,2) = 0x29a6194691f91a73715209ef6512e576722830a201be2018a765e85a9ecee931 * k_(3,3) = 0x2f684bda12f684bda12f684bda12f684bda12f684bda12f684bda12f38e38d84 The constants used to compute y_den are as follows: * k_(4,0) = 0xfffffffffffffffffffffffffffffffffffffffffffffffffffffffefffff93b * k_(4,1) = 0x7a06534bb8bdb49fd5e9e6632722c2989467c1bfc8e8d978dfb425d2685c2573 * k_(4,2) = 0x6484aa716545ca2cf3a70c3fa8fe337e0a3d21162f0d6299a7bf8192bfd2a76f E.2. 11-isogeny map for BLS12-381 G1 The 11-isogeny map from (x', y') on E' to (x, y) on E is given by the following rational functions: * x = x_num / x_den, where - x_num = k_(1,11) * x'^11 + k_(1,10) * x'^10 + k_(1,9) * x'^9 + ... + k_(1,0) - x_den = x'^10 + k_(2,9) * x'^9 + k_(2,8) * x'^8 + ... + k_(2,0) * y = y' * y_num / y_den, where - y_num = k_(3,15) * x'^15 + k_(3,14) * x'^14 + k_(3,13) * x'^13 + ... + k_(3,0) - y_den = x'^15 + k_(4,14) * x'^14 + k_(4,13) * x'^13 + ... + k_(4,0) The constants used to compute x_num are as follows: * k_(1,0) = 0x11a05f2b1e833340b809101dd99815856b303e88a2d7005ff2627b 56cdb4e2c85610c2d5f2e62d6eaeac1662734649b7 * k_(1,1) = 0x17294ed3e943ab2f0588bab22147a81c7c17e75b2f6a8417f565e3 3c70d1e86b4838f2a6f318c356e834eef1b3cb83bb Faz-Hernandez, et al. Expires 14 May 2022 [Page 73] Internet-Draft hash-to-curve November 2021 * k_(1,2) = 0xd54005db97678ec1d1048c5d10a9a1bce032473295983e56878e50 1ec68e25c958c3e3d2a09729fe0179f9dac9edcb0 * k_(1,3) = 0x1778e7166fcc6db74e0609d307e55412d7f5e4656a8dbf25f1b332 89f1b330835336e25ce3107193c5b388641d9b6861 * k_(1,4) = 0xe99726a3199f4436642b4b3e4118e5499db995a1257fb3f086eeb6 5982fac18985a286f301e77c451154ce9ac8895d9 * k_(1,5) = 0x1630c3250d7313ff01d1201bf7a74ab5db3cb17dd952799b9ed3ab 9097e68f90a0870d2dcae73d19cd13c1c66f652983 * k_(1,6) = 0xd6ed6553fe44d296a3726c38ae652bfb11586264f0f8ce19008e21 8f9c86b2a8da25128c1052ecaddd7f225a139ed84 * k_(1,7) = 0x17b81e7701abdbe2e8743884d1117e53356de5ab275b4db1a682c6 2ef0f2753339b7c8f8c8f475af9ccb5618e3f0c88e * k_(1,8) = 0x80d3cf1f9a78fc47b90b33563be990dc43b756ce79f5574a2c596c 928c5d1de4fa295f296b74e956d71986a8497e317 * k_(1,9) = 0x169b1f8e1bcfa7c42e0c37515d138f22dd2ecb803a0c5c99676314 baf4bb1b7fa3190b2edc0327797f241067be390c9e * k_(1,10) = 0x10321da079ce07e272d8ec09d2565b0dfa7dccdde6787f96d50af 36003b14866f69b771f8c285decca67df3f1605fb7b * k_(1,11) = 0x6e08c248e260e70bd1e962381edee3d31d79d7e22c837bc23c0bf 1bc24c6b68c24b1b80b64d391fa9c8ba2e8ba2d229 The constants used to compute x_den are as follows: * k_(2,0) = 0x8ca8d548cff19ae18b2e62f4bd3fa6f01d5ef4ba35b48ba9c95886 17fc8ac62b558d681be343df8993cf9fa40d21b1c * k_(2,1) = 0x12561a5deb559c4348b4711298e536367041e8ca0cf0800c0126c2 588c48bf5713daa8846cb026e9e5c8276ec82b3bff * k_(2,2) = 0xb2962fe57a3225e8137e629bff2991f6f89416f5a718cd1fca64e0 0b11aceacd6a3d0967c94fedcfcc239ba5cb83e19 * k_(2,3) = 0x3425581a58ae2fec83aafef7c40eb545b08243f16b1655154cca8a bc28d6fd04976d5243eecf5c4130de8938dc62cd8 * k_(2,4) = 0x13a8e162022914a80a6f1d5f43e7a07dffdfc759a12062bb8d6b44 e833b306da9bd29ba81f35781d539d395b3532a21e Faz-Hernandez, et al. Expires 14 May 2022 [Page 74] Internet-Draft hash-to-curve November 2021 * k_(2,5) = 0xe7355f8e4e667b955390f7f0506c6e9395735e9ce9cad4d0a43bce f24b8982f7400d24bc4228f11c02df9a29f6304a5 * k_(2,6) = 0x772caacf16936190f3e0c63e0596721570f5799af53a1894e2e073 062aede9cea73b3538f0de06cec2574496ee84a3a * k_(2,7) = 0x14a7ac2a9d64a8b230b3f5b074cf01996e7f63c21bca68a81996e1 cdf9822c580fa5b9489d11e2d311f7d99bbdcc5a5e * k_(2,8) = 0xa10ecf6ada54f825e920b3dafc7a3cce07f8d1d7161366b74100da 67f39883503826692abba43704776ec3a79a1d641 * k_(2,9) = 0x95fc13ab9e92ad4476d6e3eb3a56680f682b4ee96f7d03776df533 978f31c1593174e4b4b7865002d6384d168ecdd0a The constants used to compute y_num are as follows: * k_(3,0) = 0x90d97c81ba24ee0259d1f094980dcfa11ad138e48a869522b52af6 c956543d3cd0c7aee9b3ba3c2be9845719707bb33 * k_(3,1) = 0x134996a104ee5811d51036d776fb46831223e96c254f383d0f9063 43eb67ad34d6c56711962fa8bfe097e75a2e41c696 * k_(3,2) = 0xcc786baa966e66f4a384c86a3b49942552e2d658a31ce2c344be4b 91400da7d26d521628b00523b8dfe240c72de1f6 * k_(3,3) = 0x1f86376e8981c217898751ad8746757d42aa7b90eeb791c09e4a3e c03251cf9de405aba9ec61deca6355c77b0e5f4cb * k_(3,4) = 0x8cc03fdefe0ff135caf4fe2a21529c4195536fbe3ce50b879833fd 221351adc2ee7f8dc099040a841b6daecf2e8fedb * k_(3,5) = 0x16603fca40634b6a2211e11db8f0a6a074a7d0d4afadb7bd76505c 3d3ad5544e203f6326c95a807299b23ab13633a5f0 * k_(3,6) = 0x4ab0b9bcfac1bbcb2c977d027796b3ce75bb8ca2be184cb5231413 c4d634f3747a87ac2460f415ec961f8855fe9d6f2 * k_(3,7) = 0x987c8d5333ab86fde9926bd2ca6c674170a05bfe3bdd81ffd038da 6c26c842642f64550fedfe935a15e4ca31870fb29 * k_(3,8) = 0x9fc4018bd96684be88c9e221e4da1bb8f3abd16679dc26c1e8b6e6 a1f20cabe69d65201c78607a360370e577bdba587 * k_(3,9) = 0xe1bba7a1186bdb5223abde7ada14a23c42a0ca7915af6fe06985e7 ed1e4d43b9b3f7055dd4eba6f2bafaaebca731c30 Faz-Hernandez, et al. Expires 14 May 2022 [Page 75] Internet-Draft hash-to-curve November 2021 * k_(3,10) = 0x19713e47937cd1be0dfd0b8f1d43fb93cd2fcbcb6caf493fd1183 e416389e61031bf3a5cce3fbafce813711ad011c132 * k_(3,11) = 0x18b46a908f36f6deb918c143fed2edcc523559b8aaf0c2462e6bf e7f911f643249d9cdf41b44d606ce07c8a4d0074d8e * k_(3,12) = 0xb182cac101b9399d155096004f53f447aa7b12a3426b08ec02710 e807b4633f06c851c1919211f20d4c04f00b971ef8 * k_(3,13) = 0x245a394ad1eca9b72fc00ae7be315dc757b3b080d4c158013e663 2d3c40659cc6cf90ad1c232a6442d9d3f5db980133 * k_(3,14) = 0x5c129645e44cf1102a159f748c4a3fc5e673d81d7e86568d9ab0f 5d396a7ce46ba1049b6579afb7866b1e715475224b * k_(3,15) = 0x15e6be4e990f03ce4ea50b3b42df2eb5cb181d8f84965a3957add 4fa95af01b2b665027efec01c7704b456be69c8b604 The constants used to compute y_den are as follows: * k_(4,0) = 0x16112c4c3a9c98b252181140fad0eae9601a6de578980be6eec323 2b5be72e7a07f3688ef60c206d01479253b03663c1 * k_(4,1) = 0x1962d75c2381201e1a0cbd6c43c348b885c84ff731c4d59ca4a103 56f453e01f78a4260763529e3532f6102c2e49a03d * k_(4,2) = 0x58df3306640da276faaae7d6e8eb15778c4855551ae7f310c35a5d d279cd2eca6757cd636f96f891e2538b53dbf67f2 * k_(4,3) = 0x16b7d288798e5395f20d23bf89edb4d1d115c5dbddbcd30e123da4 89e726af41727364f2c28297ada8d26d98445f5416 * k_(4,4) = 0xbe0e079545f43e4b00cc912f8228ddcc6d19c9f0f69bbb0542eda0 fc9dec916a20b15dc0fd2ededda39142311a5001d * k_(4,5) = 0x8d9e5297186db2d9fb266eaac783182b70152c65550d881c5ecd87 b6f0f5a6449f38db9dfa9cce202c6477faaf9b7ac * k_(4,6) = 0x166007c08a99db2fc3ba8734ace9824b5eecfdfa8d0cf8ef5dd365 bc400a0051d5fa9c01a58b1fb93d1a1399126a775c * k_(4,7) = 0x16a3ef08be3ea7ea03bcddfabba6ff6ee5a4375efa1f4fd7feb34f d206357132b920f5b00801dee460ee415a15812ed9 * k_(4,8) = 0x1866c8ed336c61231a1be54fd1d74cc4f9fb0ce4c6af5920abc575 0c4bf39b4852cfe2f7bb9248836b233d9d55535d4a Faz-Hernandez, et al. Expires 14 May 2022 [Page 76] Internet-Draft hash-to-curve November 2021 * k_(4,9) = 0x167a55cda70a6e1cea820597d94a84903216f763e13d87bb530859 2e7ea7d4fbc7385ea3d529b35e346ef48bb8913f55 * k_(4,10) = 0x4d2f259eea405bd48f010a01ad2911d9c6dd039bb61a6290e591b 36e636a5c871a5c29f4f83060400f8b49cba8f6aa8 * k_(4,11) = 0xaccbb67481d033ff5852c1e48c50c477f94ff8aefce42d28c0f9a 88cea7913516f968986f7ebbea9684b529e2561092 * k_(4,12) = 0xad6b9514c767fe3c3613144b45f1496543346d98adf02267d5cee f9a00d9b8693000763e3b90ac11e99b138573345cc * k_(4,13) = 0x2660400eb2e4f3b628bdd0d53cd76f2bf565b94e72927c1cb748d f27942480e420517bd8714cc80d1fadc1326ed06f7 * k_(4,14) = 0xe0fa1d816ddc03e6b24255e0d7819c171c40f65e273b853324efc d6356caa205ca2f570f13497804415473a1d634b8f E.3. 3-isogeny map for BLS12-381 G2 The 3-isogeny map from (x', y') on E' to (x, y) on E is given by the following rational functions: * x = x_num / x_den, where - x_num = k_(1,3) * x'^3 + k_(1,2) * x'^2 + k_(1,1) * x' + k_(1,0) - x_den = x'^2 + k_(2,1) * x' + k_(2,0) * y = y' * y_num / y_den, where - y_num = k_(3,3) * x'^3 + k_(3,2) * x'^2 + k_(3,1) * x' + k_(3,0) - y_den = x'^3 + k_(4,2) * x'^2 + k_(4,1) * x' + k_(4,0) The constants used to compute x_num are as follows: * k_(1,0) = 0x5c759507e8e333ebb5b7a9a47d7ed8532c52d39fd3a042a88b5842 3c50ae15d5c2638e343d9c71c6238aaaaaaaa97d6 + 0x5c759507e8e333ebb5b7 a9a47d7ed8532c52d39fd3a042a88b58423c50ae15d5c2638e343d9c71c6238aaa aaaaa97d6 * I * k_(1,1) = 0x11560bf17baa99bc32126fced787c88f984f87adf7ae0c7f9a208c 6b4f20a4181472aaa9cb8d555526a9ffffffffc71a * I Faz-Hernandez, et al. Expires 14 May 2022 [Page 77] Internet-Draft hash-to-curve November 2021 * k_(1,2) = 0x11560bf17baa99bc32126fced787c88f984f87adf7ae0c7f9a208c 6b4f20a4181472aaa9cb8d555526a9ffffffffc71e + 0x8ab05f8bdd54cde1909 37e76bc3e447cc27c3d6fbd7063fcd104635a790520c0a395554e5c6aaaa9354ff ffffffe38d * I * k_(1,3) = 0x171d6541fa38ccfaed6dea691f5fb614cb14b4e7f4e810aa22d610 8f142b85757098e38d0f671c7188e2aaaaaaaa5ed1 The constants used to compute x_den are as follows: * k_(2,0) = 0x1a0111ea397fe69a4b1ba7b6434bacd764774b84f38512bf6730d2 a0f6b0f6241eabfffeb153ffffb9feffffffffaa63 * I * k_(2,1) = 0xc + 0x1a0111ea397fe69a4b1ba7b6434bacd764774b84f38512bf 6730d2a0f6b0f6241eabfffeb153ffffb9feffffffffaa9f * I The constants used to compute y_num are as follows: * k_(3,0) = 0x1530477c7ab4113b59a4c18b076d11930f7da5d4a07f649bf54439 d87d27e500fc8c25ebf8c92f6812cfc71c71c6d706 + 0x1530477c7ab4113b59a 4c18b076d11930f7da5d4a07f649bf54439d87d27e500fc8c25ebf8c92f6812cfc 71c71c6d706 * I * k_(3,1) = 0x5c759507e8e333ebb5b7a9a47d7ed8532c52d39fd3a042a88b5842 3c50ae15d5c2638e343d9c71c6238aaaaaaaa97be * I * k_(3,2) = 0x11560bf17baa99bc32126fced787c88f984f87adf7ae0c7f9a208c 6b4f20a4181472aaa9cb8d555526a9ffffffffc71c + 0x8ab05f8bdd54cde1909 37e76bc3e447cc27c3d6fbd7063fcd104635a790520c0a395554e5c6aaaa9354ff ffffffe38f * I * k_(3,3) = 0x124c9ad43b6cf79bfbf7043de3811ad0761b0f37a1e26286b0e977 c69aa274524e79097a56dc4bd9e1b371c71c718b10 The constants used to compute y_den are as follows: * k_(4,0) = 0x1a0111ea397fe69a4b1ba7b6434bacd764774b84f38512bf6730d2 a0f6b0f6241eabfffeb153ffffb9feffffffffa8fb + 0x1a0111ea397fe69a4b1 ba7b6434bacd764774b84f38512bf6730d2a0f6b0f6241eabfffeb153ffffb9fef fffffffa8fb * I * k_(4,1) = 0x1a0111ea397fe69a4b1ba7b6434bacd764774b84f38512bf6730d2 a0f6b0f6241eabfffeb153ffffb9feffffffffa9d3 * I * k_(4,2) = 0x12 + 0x1a0111ea397fe69a4b1ba7b6434bacd764774b84f38512b f6730d2a0f6b0f6241eabfffeb153ffffb9feffffffffaa99 * I Faz-Hernandez, et al. Expires 14 May 2022 [Page 78] Internet-Draft hash-to-curve November 2021 Appendix F. Straight-line implementations of deterministic mappings This section gives straight-line implementations of the mappings of Section 6. These implementations are generic, i.e., they are defined for any curve and field. Appendix G gives further implementations that are optimized for specific classes of curves and fields. F.1. Shallue-van de Woestijne method This section gives a straight-line implementation of the Shallue and van de Woestijne method for any Weierstrass curve of the form given in Section 6.6. See Section 6.6.1 for information on the constants used in this mapping. Note that the constant c3 below MUST be chosen such that sgn0(c3) = 0. In other words, if the square-root computation returns a value cx such that sgn0(cx) = 1, set c3 = -cx; otherwise, set c3 = cx. map_to_curve_svdw(u) Input: u, an element of F. Output: (x, y), a point on E. Constants: 1. c1 = g(Z) 2. c2 = -Z / 2 3. c3 = sqrt(-g(Z) * (3 * Z^2 + 4 * A)) # sgn0(c3) MUST equal 0 4. c4 = -4 * g(Z) / (3 * Z^2 + 4 * A) Steps: 1. tv1 = u^2 2. tv1 = tv1 * c1 3. tv2 = 1 + tv1 4. tv1 = 1 - tv1 5. tv3 = tv1 * tv2 6. tv3 = inv0(tv3) 7. tv4 = u * tv1 8. tv4 = tv4 * tv3 9. tv4 = tv4 * c3 10. x1 = c2 - tv4 11. gx1 = x1^2 12. gx1 = gx1 + A 13. gx1 = gx1 * x1 14. gx1 = gx1 + B 15. e1 = is_square(gx1) 16. x2 = c2 + tv4 17. gx2 = x2^2 18. gx2 = gx2 + A Faz-Hernandez, et al. Expires 14 May 2022 [Page 79] Internet-Draft hash-to-curve November 2021 19. gx2 = gx2 * x2 20. gx2 = gx2 + B 21. e2 = is_square(gx2) AND NOT e1 # Avoid short-circuit logic ops 22. x3 = tv2^2 23. x3 = x3 * tv3 24. x3 = x3^2 25. x3 = x3 * c4 26. x3 = x3 + Z 27. x = CMOV(x3, x1, e1) # x = x1 if gx1 is square, else x = x3 28. x = CMOV(x, x2, e2) # x = x2 if gx2 is square and gx1 is not 29. gx = x^2 30. gx = gx + A 31. gx = gx * x 32. gx = gx + B 33. y = sqrt(gx) 34. e3 = sgn0(u) == sgn0(y) 35. y = CMOV(-y, y, e3) # Select correct sign of y 36. return (x, y) F.2. Simplified SWU method This section gives a straight-line implementation of the simplified SWU method for any Weierstrass curve of the form given in Section 6.6. See Section 6.6.2 for information on the constants used in this mapping. This optimized, straight-line procedure applies to any base field. The sqrt_ratio subroutine is defined in Appendix F.2.1. Faz-Hernandez, et al. Expires 14 May 2022 [Page 80] Internet-Draft hash-to-curve November 2021 map_to_curve_simple_swu(u) Input: u, an element of F. Output: (x, y), a point on E. Steps: 1. tv1 = u^2 2. tv1 = Z * tv1 3. tv2 = tv1^2 4. tv2 = tv2 + tv1 5. tv3 = tv2 + 1 6. tv3 = B * tv3 7. tv4 = CMOV(Z, -tv2, tv2 != 0) 8. tv4 = A * tv4 9. tv2 = tv3^2 10. tv6 = tv4^2 11. tv5 = A * tv6 12. tv2 = tv2 + tv5 13. tv2 = tv2 * tv3 14. tv6 = tv6 * tv4 15. tv5 = B * tv6 16. tv2 = tv2 + tv5 17. x = tv1 * tv3 18. (is_gx1_square, y1) = sqrt_ratio(tv2, tv6) 19. y = tv1 * u 20. y = y * y1 21. x = CMOV(x, tv3, is_gx1_square) 22. y = CMOV(y, y1, is_gx1_square) 23. e1 = sgn0(u) == sgn0(y) 24. y = CMOV(-y, y, e1) 25. x = x / tv4 26. return (x, y) F.2.1. sqrt_ratio subroutines This section defines three variants of the sqrt_ratio subroutine used by the above procedure. The first variant can be used with any field; the others are optimized versions for specific fields. The routines given in this section depend on the constant Z from the simplified SWU map. For correctness, sqrt_ratio and map_to_curve_simple_swu MUST use the same value for Z. F.2.1.1. sqrt_ratio for any field Faz-Hernandez, et al. Expires 14 May 2022 [Page 81] Internet-Draft hash-to-curve November 2021 sqrt_ratio(u, v) Parameters: - F, a finite field of characteristic p and order q = p^m. - Z, the constant from the simplified SWU map. Input: u and v, elements of F, where v != 0. Output: (b, y), where b = True and y = sqrt(u / v) if (u / v) is square in F, and b = False and y = sqrt(Z * (u / v)) otherwise. Constants: 1. c1, the largest integer such that 2^c1 divides q - 1. 2. c2 = (q - 1) / (2^c1) # Integer arithmetic 3. c3 = (c2 - 1) / 2 # Integer arithmetic 4. c4 = 2^c1 - 1 # Integer arithmetic 5. c5 = 2^(c1 - 1) # Integer arithmetic 6. c6 = Z^c2 7. c7 = Z^((c2 + 1) / 2) Procedure: 1. tv1 = c6 2. tv2 = v^c4 3. tv3 = tv2^2 4. tv3 = tv3 * v 5. tv5 = u * tv3 6. tv5 = tv5^c3 7. tv5 = tv5 * tv2 8. tv2 = tv5 * v 9. tv3 = tv5 * u 10. tv4 = tv3 * tv2 11. tv5 = tv4^c5 12. isQR = tv5 == 1 13. tv2 = tv3 * c7 14. tv5 = tv4 * tv1 15. tv3 = CMOV(tv2, tv3, isQR) 16. tv4 = CMOV(tv5, tv4, isQR) 17. for i in (c1, c1 - 1, ..., 2): 18. tv5 = i - 2 19. tv5 = 2^tv5 20. tv5 = tv4^tv5 21. e1 = tv5 == 1 22. tv2 = tv3 * tv1 23. tv1 = tv1 * tv1 24. tv5 = tv4 * tv1 25. tv3 = CMOV(tv2, tv3, e1) 26. tv4 = CMOV(tv5, tv4, e1) 27. return (isQR, tv3) Faz-Hernandez, et al. Expires 14 May 2022 [Page 82] Internet-Draft hash-to-curve November 2021 F.2.1.2. optimized sqrt_ratio for q = 3 mod 4 sqrt_ratio_3mod4(u, v) Parameters: - F, a finite field of characteristic p and order q = p^m, where q = 3 mod 4. - Z, the constant from the simplified SWU map. Input: u and v, elements of F, where v != 0. Output: (b, y), where b = True and y = sqrt(u / v) if (u / v) is square in F, and b = False and y = sqrt(Z * (u / v)) otherwise. Constants: 1. c1 = (q - 3) / 4 # Integer arithmetic 2. c2 = sqrt(-Z) Procedure: 1. tv1 = v^2 2. tv2 = u * v 3. tv1 = tv1 * tv2 4. y1 = tv1^c1 5. y1 = y1 * tv2 6. y2 = y1 * c2 7. tv3 = y1^2 8. tv3 = tv3 * v 9. isQR = tv3 == u 10. y = CMOV(y2, y1, isQR) 11. return (isQR, y) F.2.1.3. optimized sqrt_ratio for q = 5 mod 8 Faz-Hernandez, et al. Expires 14 May 2022 [Page 83] Internet-Draft hash-to-curve November 2021 sqrt_ratio_5mod8(u, v) Parameters: - F, a finite field of characteristic p and order q = p^m, where q = 5 mod 8. - Z, the constant from the simplified SWU map. Input: u and v, elements of F, where v != 0. Output: (b, y), where b = True and y = sqrt(u / v) if (u / v) is square in F, and b = False and y = sqrt(Z * (u / v)) otherwise. Constants: 1. c1 = (q - 5) / 8 2. c2 = sqrt(-1) 3. c3 = sqrt(Z / c2) Steps: 1. tv1 = v^2 2. tv2 = tv1 * v 3. tv1 = tv1^2 4. tv2 = tv2 * u 5. tv1 = tv1 * tv2 6. y1 = tv1^c1 7. y1 = y1 * tv2 8. tv1 = y1 * c2 9. tv2 = tv1^2 10. tv2 = tv2 * v 11. e1 = tv2 == u 12. y1 = CMOV(y1, tv1, e1) 13. tv2 = y1^2 14. tv2 = tv2 * v 15. isQR = tv2 == u 16. y2 = y1 * c3 17. tv1 = y2 * c2 18. tv2 = tv1^2 19. tv2 = tv2 * v 20. tv3 = Z * u 21. e2 = tv2 == tv3 22. y2 = CMOV(y2, tv1, e2) 23. y = CMOV(y2, y1, isQR) 24. return (isQR, y) Faz-Hernandez, et al. Expires 14 May 2022 [Page 84] Internet-Draft hash-to-curve November 2021 F.3. Elligator 2 method This section gives a straight-line implementation of the Elligator 2 method for any Montgomery curve of the form given in Section 6.7. See Section 6.7.1 for information on the constants used in this mapping. Appendix G.2 gives optimized straight-line procedures that apply to specific classes of curves and base fields, including curve25519 and curve448 [RFC7748]. map_to_curve_elligator2(u) Input: u, an element of F. Output: (s, t), a point on M. Constants: 1. c1 = J / K 2. c2 = 1 / K^2 Steps: 1. tv1 = u^2 2. tv1 = Z * tv1 # Z * u^2 3. e1 = tv1 == -1 # exceptional case: Z * u^2 == -1 4. tv1 = CMOV(tv1, 0, e1) # if tv1 == -1, set tv1 = 0 5. x1 = tv1 + 1 6. x1 = inv0(x1) 7. x1 = -c1 * x1 # x1 = -(J / K) / (1 + Z * u^2) 8. gx1 = x1 + c1 9. gx1 = gx1 * x1 10. gx1 = gx1 + c2 11. gx1 = gx1 * x1 # gx1 = x1^3 + (J / K) * x1^2 + x1 / K^2 12. x2 = -x1 - c1 13. gx2 = tv1 * gx1 14. e2 = is_square(gx1) # If is_square(gx1) 15. x = CMOV(x2, x1, e2) # then x = x1, else x = x2 16. y2 = CMOV(gx2, gx1, e2) # then y2 = gx1, else y2 = gx2 17. y = sqrt(y2) 18. e3 = sgn0(y) == 1 19. y = CMOV(y, -y, e2 XOR e3) # fix sign of y 20. s = x * K 21. t = y * K 22. return (s, t) Faz-Hernandez, et al. Expires 14 May 2022 [Page 85] Internet-Draft hash-to-curve November 2021 Appendix G. Curve-specific optimized sample code This section gives sample implementations optimized for some of the elliptic curves listed in Section 8. Sample Sage [SAGE] code for each algorithm can also be found in the draft repository [hash2curve-repo]. G.1. Interface and projective coordinate systems The sample code in this section uses a different interface than the mappings of Section 6. Specifically, each mapping function in this section has the following signature: (xn, xd, yn, yd) = map_to_curve(u) The resulting affine point (x, y) is given by (xn / xd, yn / yd). The reason for this modified interface is that it enables further optimizations when working with points in a projective coordinate system. This is desirable, for example, when the resulting point will be immediately multiplied by a scalar, since most scalar multiplication algorithms operate on projective points. Projective coordinates are also useful when implementing random oracle encodings (Section 3). One reason is that, in general, point addition is faster using projective coordinates. Another reason is that, for Weierstrass curves, projective coordinates allow using complete addition formulas [RCB16]. This is especially convenient when implementing a constant-time encoding, because it eliminates the need for a special case when Q0 == Q1, which incomplete addition formulas usually do not handle. The following are two commonly used projective coordinate systems and the corresponding conversions: * A point (X, Y, Z) in homogeneous projective coordinates corresponds to the affine point (x, y) = (X / Z, Y / Z); the inverse conversion is given by (X, Y, Z) = (x, y, 1). To convert (xn, xd, yn, yd) to homogeneous projective coordinates, compute (X, Y, Z) = (xn * yd, yn * xd, xd * yd). * A point (X', Y', Z') in Jacobian projective coordinates corresponds to the affine point (x, y) = (X' / Z'^2, Y' / Z'^3); the inverse conversion is given by (X', Y', Z') = (x, y, 1). To convert (xn, xd, yn, yd) to Jacobian projective coordinates, compute (X', Y', Z') = (xn * xd * yd^2, yn * yd^2 * xd^3, xd * yd). Faz-Hernandez, et al. Expires 14 May 2022 [Page 86] Internet-Draft hash-to-curve November 2021 G.2. Elligator 2 G.2.1. curve25519 (q = 5 (mod 8), K = 1) The following is a straight-line implementation of Elligator 2 for curve25519 [RFC7748] as specified in Section 8.5. This implementation can also be used for any Montgomery curve with K = 1 over GF(q) where q = 5 (mod 8). map_to_curve_elligator2_curve25519(u) Input: u, an element of F. Output: (xn, xd, yn, yd) such that (xn / xd, yn / yd) is a point on curve25519. Constants: 1. c1 = (q + 3) / 8 # Integer arithmetic 2. c2 = 2^c1 3. c3 = sqrt(-1) 4. c4 = (q - 5) / 8 # Integer arithmetic Steps: 1. tv1 = u^2 2. tv1 = 2 * tv1 3. xd = tv1 + 1 # Nonzero: -1 is square (mod p), tv1 is not 4. x1n = -J # x1 = x1n / xd = -J / (1 + 2 * u^2) 5. tv2 = xd^2 6. gxd = tv2 * xd # gxd = xd^3 7. gx1 = J * tv1 # x1n + J * xd 8. gx1 = gx1 * x1n # x1n^2 + J * x1n * xd 9. gx1 = gx1 + tv2 # x1n^2 + J * x1n * xd + xd^2 10. gx1 = gx1 * x1n # x1n^3 + J * x1n^2 * xd + x1n * xd^2 11. tv3 = gxd^2 12. tv2 = tv3^2 # gxd^4 13. tv3 = tv3 * gxd # gxd^3 14. tv3 = tv3 * gx1 # gx1 * gxd^3 15. tv2 = tv2 * tv3 # gx1 * gxd^7 16. y11 = tv2^c4 # (gx1 * gxd^7)^((p - 5) / 8) 17. y11 = y11 * tv3 # gx1 * gxd^3 * (gx1 * gxd^7)^((p - 5) / 8) 18. y12 = y11 * c3 19. tv2 = y11^2 20. tv2 = tv2 * gxd 21. e1 = tv2 == gx1 22. y1 = CMOV(y12, y11, e1) # If g(x1) is square, this is its sqrt 23. x2n = x1n * tv1 # x2 = x2n / xd = 2 * u^2 * x1n / xd 24. y21 = y11 * u 25. y21 = y21 * c2 Faz-Hernandez, et al. Expires 14 May 2022 [Page 87] Internet-Draft hash-to-curve November 2021 26. y22 = y21 * c3 27. gx2 = gx1 * tv1 # g(x2) = gx2 / gxd = 2 * u^2 * g(x1) 28. tv2 = y21^2 29. tv2 = tv2 * gxd 30. e2 = tv2 == gx2 31. y2 = CMOV(y22, y21, e2) # If g(x2) is square, this is its sqrt 32. tv2 = y1^2 33. tv2 = tv2 * gxd 34. e3 = tv2 == gx1 35. xn = CMOV(x2n, x1n, e3) # If e3, x = x1, else x = x2 36. y = CMOV(y2, y1, e3) # If e3, y = y1, else y = y2 37. e4 = sgn0(y) == 1 # Fix sign of y 38. y = CMOV(y, -y, e3 XOR e4) 39. return (xn, xd, y, 1) G.2.2. edwards25519 The following is a straight-line implementation of Elligator 2 for edwards25519 [RFC7748] as specified in Section 8.5. The subroutine map_to_curve_elligator2_curve25519 is defined in Appendix G.2.1. Note that the sign of the constant c1 below is chosen as specified in Section 6.8.1, i.e., applying the rational map to the edwards25519 base point yields the curve25519 base point (see erratum [EID4730]). map_to_curve_elligator2_edwards25519(u) Input: u, an element of F. Output: (xn, xd, yn, yd) such that (xn / xd, yn / yd) is a point on edwards25519. Constants: 1. c1 = sqrt(-486664) # sgn0(c1) MUST equal 0 Steps: 1. (xMn, xMd, yMn, yMd) = map_to_curve_elligator2_curve25519(u) 2. xn = xMn * yMd 3. xn = xn * c1 4. xd = xMd * yMn # xn / xd = c1 * xM / yM 5. yn = xMn - xMd 6. yd = xMn + xMd # (n / d - 1) / (n / d + 1) = (n - d) / (n + d) 7. tv1 = xd * yd 8. e = tv1 == 0 9. xn = CMOV(xn, 0, e) 10. xd = CMOV(xd, 1, e) 11. yn = CMOV(yn, 1, e) 12. yd = CMOV(yd, 1, e) 13. return (xn, xd, yn, yd) Faz-Hernandez, et al. Expires 14 May 2022 [Page 88] Internet-Draft hash-to-curve November 2021 G.2.3. curve448 (q = 3 (mod 4), K = 1) The following is a straight-line implementation of Elligator 2 for curve448 [RFC7748] as specified in Section 8.6. This implementation can also be used for any Montgomery curve with K = 1 over GF(q) where q = 3 (mod 4). map_to_curve_elligator2_curve448(u) Input: u, an element of F. Output: (xn, xd, yn, yd) such that (xn / xd, yn / yd) is a point on curve448. Constants: 1. c1 = (q - 3) / 4 # Integer arithmetic Steps: 1. tv1 = u^2 2. e1 = tv1 == 1 3. tv1 = CMOV(tv1, 0, e1) # If Z * u^2 == -1, set tv1 = 0 4. xd = 1 - tv1 5. x1n = -J 6. tv2 = xd^2 7. gxd = tv2 * xd # gxd = xd^3 8. gx1 = -J * tv1 # x1n + J * xd 9. gx1 = gx1 * x1n # x1n^2 + J * x1n * xd 10. gx1 = gx1 + tv2 # x1n^2 + J * x1n * xd + xd^2 11. gx1 = gx1 * x1n # x1n^3 + J * x1n^2 * xd + x1n * xd^2 12. tv3 = gxd^2 13. tv2 = gx1 * gxd # gx1 * gxd 14. tv3 = tv3 * tv2 # gx1 * gxd^3 15. y1 = tv3^c1 # (gx1 * gxd^3)^((p - 3) / 4) 16. y1 = y1 * tv2 # gx1 * gxd * (gx1 * gxd^3)^((p - 3) / 4) 17. x2n = -tv1 * x1n # x2 = x2n / xd = -1 * u^2 * x1n / xd 18. y2 = y1 * u 19. y2 = CMOV(y2, 0, e1) 20. tv2 = y1^2 21. tv2 = tv2 * gxd 22. e2 = tv2 == gx1 23. xn = CMOV(x2n, x1n, e2) # If e2, x = x1, else x = x2 24. y = CMOV(y2, y1, e2) # If e2, y = y1, else y = y2 25. e3 = sgn0(y) == 1 # Fix sign of y 26. y = CMOV(y, -y, e2 XOR e3) 27. return (xn, xd, y, 1) Faz-Hernandez, et al. Expires 14 May 2022 [Page 89] Internet-Draft hash-to-curve November 2021 G.2.4. edwards448 The following is a straight-line implementation of Elligator 2 for edwards448 [RFC7748] as specified in Section 8.6. The subroutine map_to_curve_elligator2_curve448 is defined in Appendix G.2.3. Faz-Hernandez, et al. Expires 14 May 2022 [Page 90] Internet-Draft hash-to-curve November 2021 map_to_curve_elligator2_edwards448(u) Input: u, an element of F. Output: (xn, xd, yn, yd) such that (xn / xd, yn / yd) is a point on edwards448. Steps: 1. (xn, xd, yn, yd) = map_to_curve_elligator2_curve448(u) 2. xn2 = xn^2 3. xd2 = xd^2 4. xd4 = xd2^2 5. yn2 = yn^2 6. yd2 = yd^2 7. xEn = xn2 - xd2 8. tv2 = xEn - xd2 9. xEn = xEn * xd2 10. xEn = xEn * yd 11. xEn = xEn * yn 12. xEn = xEn * 4 13. tv2 = tv2 * xn2 14. tv2 = tv2 * yd2 15. tv3 = 4 * yn2 16. tv1 = tv3 + yd2 17. tv1 = tv1 * xd4 18. xEd = tv1 + tv2 19. tv2 = tv2 * xn 20. tv4 = xn * xd4 21. yEn = tv3 - yd2 22. yEn = yEn * tv4 23. yEn = yEn - tv2 24. tv1 = xn2 + xd2 25. tv1 = tv1 * xd2 26. tv1 = tv1 * xd 27. tv1 = tv1 * yn2 28. tv1 = -2 * tv1 29. yEd = tv2 + tv1 30. tv4 = tv4 * yd2 31. yEd = yEd + tv4 32. tv1 = xEd * yEd 33. e = tv1 == 0 34. xEn = CMOV(xEn, 0, e) 35. xEd = CMOV(xEd, 1, e) 36. yEn = CMOV(yEn, 1, e) 37. yEd = CMOV(yEd, 1, e) 38. return (xEn, xEd, yEn, yEd) Faz-Hernandez, et al. Expires 14 May 2022 [Page 91] Internet-Draft hash-to-curve November 2021 G.2.5. q = 3 (mod 4) The following is a straight-line implementation of Elligator 2 that applies to any curve over GF(q) where q = 3 (mod 4). For curves where K = 1, the implementation given in Appendix G.2.3 gives identical results with slightly reduced cost. Faz-Hernandez, et al. Expires 14 May 2022 [Page 92] Internet-Draft hash-to-curve November 2021 map_to_curve_elligator2_3mod4(u) Input: u, an element of F. Output: (xn, xd, yn, yd) such that (xn / xd, yn / yd) is a point on the target curve. Constants: 1. c1 = (q - 3) / 4 # Integer arithmetic 2. c2 = K^2 Steps: 1. tv1 = u^2 2. e1 = tv1 == 1 3. tv1 = CMOV(tv1, 0, e1) # If Z * u^2 == -1, set tv1 = 0 4. xd = 1 - tv1 5. xd = xd * K 6. x1n = -J # x1 = x1n / xd = -J / (K * (1 + 2 * u^2)) 7. tv2 = xd^2 8. gxd = tv2 * xd 9. gxd = gxd * c2 # gxd = xd^3 * K^2 10. gx1 = x1n * K 11. tv3 = xd * J 12. tv3 = gx1 + tv3 # x1n * K + xd * J 13. gx1 = gx1 * tv3 # K^2 * x1n^2 + J * K * x1n * xd 14. gx1 = gx1 + tv2 # K^2 * x1n^2 + J * K * x1n * xd + xd^2 15. gx1 = gx1 * x1n # K^2 * x1n^3 + J * K * x1n^2 * xd + x1n * xd^2 16. tv3 = gxd^2 17. tv2 = gx1 * gxd # gx1 * gxd 18. tv3 = tv3 * tv2 # gx1 * gxd^3 19. y1 = tv3^c1 # (gx1 * gxd^3)^((q - 3) / 4) 20. y1 = y1 * tv2 # gx1 * gxd * (gx1 * gxd^3)^((q - 3) / 4) 21. x2n = -tv1 * x1n # x2 = x2n / xd = -1 * u^2 * x1n / xd 22. y2 = y1 * u 23. y2 = CMOV(y2, 0, e1) 24. tv2 = y1^2 25. tv2 = tv2 * gxd 26. e2 = tv2 == gx1 27. xn = CMOV(x2n, x1n, e2) # If e2, x = x1, else x = x2 28. xn = xn * K 29. y = CMOV(y2, y1, e2) # If e2, y = y1, else y = y2 30. e3 = sgn0(y) == 1 # Fix sign of y 31. y = CMOV(y, -y, e2 XOR e3) 32. y = y * K 33. return (xn, xd, y, 1) Faz-Hernandez, et al. Expires 14 May 2022 [Page 93] Internet-Draft hash-to-curve November 2021 G.2.6. q = 5 (mod 8) The following is a straight-line implementation of Elligator 2 that applies to any curve over GF(q) where q = 5 (mod 8). For curves where K = 1, the implementation given in Appendix G.2.1 gives identical results with slightly reduced cost. map_to_curve_elligator2_5mod8(u) Input: u, an element of F. Output: (xn, xd, yn, yd) such that (xn / xd, yn / yd) is a point on the target curve. Constants: 1. c1 = (q + 3) / 8 # Integer arithmetic 2. c2 = 2^c1 3. c3 = sqrt(-1) 4. c4 = (q - 5) / 8 # Integer arithmetic 5. c5 = K^2 Steps: 1. tv1 = u^2 2. tv1 = 2 * tv1 3. xd = tv1 + 1 # Nonzero: -1 is square (mod p), tv1 is not 4. xd = xd * K 5. x1n = -J # x1 = x1n / xd = -J / (K * (1 + 2 * u^2)) 6. tv2 = xd^2 7. gxd = tv2 * xd 8. gxd = gxd * c5 # gxd = xd^3 * K^2 9. gx1 = x1n * K 10. tv3 = xd * J 11. tv3 = gx1 + tv3 # x1n * K + xd * J 12. gx1 = gx1 * tv3 # K^2 * x1n^2 + J * K * x1n * xd 13. gx1 = gx1 + tv2 # K^2 * x1n^2 + J * K * x1n * xd + xd^2 14. gx1 = gx1 * x1n # K^2 * x1n^3 + J * K * x1n^2 * xd + x1n * xd^2 15. tv3 = gxd^2 16. tv2 = tv3^2 # gxd^4 17. tv3 = tv3 * gxd # gxd^3 18. tv3 = tv3 * gx1 # gx1 * gxd^3 19. tv2 = tv2 * tv3 # gx1 * gxd^7 20. y11 = tv2^c4 # (gx1 * gxd^7)^((q - 5) / 8) 21. y11 = y11 * tv3 # gx1 * gxd^3 * (gx1 * gxd^7)^((q - 5) / 8) 22. y12 = y11 * c3 23. tv2 = y11^2 24. tv2 = tv2 * gxd 25. e1 = tv2 == gx1 26. y1 = CMOV(y12, y11, e1) # If g(x1) is square, this is its sqrt Faz-Hernandez, et al. Expires 14 May 2022 [Page 94] Internet-Draft hash-to-curve November 2021 27. x2n = x1n * tv1 # x2 = x2n / xd = 2 * u^2 * x1n / xd 28. y21 = y11 * u 29. y21 = y21 * c2 30. y22 = y21 * c3 31. gx2 = gx1 * tv1 # g(x2) = gx2 / gxd = 2 * u^2 * g(x1) 32. tv2 = y21^2 33. tv2 = tv2 * gxd 34. e2 = tv2 == gx2 35. y2 = CMOV(y22, y21, e2) # If g(x2) is square, this is its sqrt 36. tv2 = y1^2 37. tv2 = tv2 * gxd 38. e3 = tv2 == gx1 39. xn = CMOV(x2n, x1n, e3) # If e3, x = x1, else x = x2 40. xn = xn * K 41. y = CMOV(y2, y1, e3) # If e3, y = y1, else y = y2 42. e4 = sgn0(y) == 1 # Fix sign of y 43. y = CMOV(y, -y, e3 XOR e4) 44. y = y * K 45. return (xn, xd, y, 1) G.3. Cofactor clearing for BLS12-381 G2 The curve BLS12-381, whose parameters are defined in Section 8.8.2, admits an efficiently-computable endomorphism psi that can be used to speed up cofactor clearing for G2 [SBCDK09] [FKR11] [BP17] (see also Section 7). This section implements the endomorphism psi and a fast cofactor clearing method described by Budroni and Pintore [BP17]. The functions in this section operate on points whose coordinates are represented as ratios, i.e., (xn, xd, yn, yd) corresponds to the point (xn / xd, yn / yd); see Appendix G.1 for further discussion of projective coordinates. When points are represented in affine coordinates, one can simply ignore the denominators (xd == 1 and yd == 1). The following function computes the Frobenius endomorphism for an element of F = GF(p^2) with basis (1, I), where I^2 + 1 == 0 in F. (This is the base field of the elliptic curve E defined in Section 8.8.2.) Faz-Hernandez, et al. Expires 14 May 2022 [Page 95] Internet-Draft hash-to-curve November 2021 frobenius(x) Input: x, an element of GF(p^2). Output: a, an element of GF(p^2). Notation: x = x0 + I * x1, where x0 and x1 are elements of GF(p). Steps: 1. a = x0 - I * x1 2. return a The following function computes the endomorphism psi for points on the elliptic curve E defined in Section 8.8.2. psi(xn, xd, yn, yd) Input: P, the point (xn / xd, yn / yd) on the curve E (see above). Output: Q, a point on the same curve. Constants: 1. c1 = 1 / (1 + I)^((p - 1) / 3) # in GF(p^2) 2. c2 = 1 / (1 + I)^((p - 1) / 2) # in GF(p^2) Steps: 1. qxn = c1 * frobenius(xn) 2. qxd = frobenius(xd) 3. qyn = c2 * frobenius(yn) 4. qyd = frobenius(yd) 5. return (qxn, qxd, qyn, qyd) The following function efficiently computes psi(psi(P)). psi2(xn, xd, yn, yd) Input: P, the point (xn / xd, yn / yd) on the curve E (see above). Output: Q, a point on the same curve. Constants: 1. c1 = 1 / 2^((p - 1) / 3) # in GF(p^2) Steps: 1. qxn = c1 * xn 2. qyn = -yn 3. return (qxn, xd, qyn, yd) Faz-Hernandez, et al. Expires 14 May 2022 [Page 96] Internet-Draft hash-to-curve November 2021 The following function maps any point on the elliptic curve E (Section 8.8.2) into the prime-order subgroup G2. This function returns a point equal to h_eff * G2, where h_eff is the parameter given in Section 8.8.2. clear_cofactor_bls12381_g2(P) Input: P, the point (xn / xd, yn / yd) on the curve E (see above). Output: Q, a point in the subgroup G2 of BLS12-381. Constants: 1. c1 = -15132376222941642752 # the BLS parameter for BLS12-381 # i.e., -0xd201000000010000 Notation: in this procedure, + and - represent elliptic curve point addition and subtraction, respectively, and * represents scalar multiplication. Steps: 1. t1 = c1 * P 2. t2 = psi(P) 3. t3 = 2 * P 4. t3 = psi2(t3) 5. t3 = t3 - t2 6. t2 = t1 + t2 7. t2 = c1 * t2 8. t3 = t3 + t2 9. t3 = t3 - t1 10. Q = t3 - P 11. return Q Appendix H. Scripts for parameter generation This section gives Sage [SAGE] scripts used to generate parameters for the mappings of Section 6. H.1. Finding Z for the Shallue-van de Woestijne map The below function outputs an appropriate Z for the Shallue and van de Woestijne map (Section 6.6.1). Faz-Hernandez, et al. Expires 14 May 2022 [Page 97] Internet-Draft hash-to-curve November 2021 # Arguments: # - F, a field object, e.g., F = GF(2^521 - 1) # - A and B, the coefficients of the curve y^2 = x^3 + A * x + B def find_z_svdw(F, A, B, init_ctr=1): g = lambda x: F(x)^3 + F(A) * F(x) + F(B) h = lambda Z: -(F(3) * Z^2 + F(4) * A) / (F(4) * g(Z)) # NOTE: if init_ctr=1 fails to find Z, try setting it to F.gen() ctr = init_ctr while True: for Z_cand in (F(ctr), F(-ctr)): # Criterion 1: # g(Z) != 0 in F. if g(Z_cand) == F(0): continue # Criterion 2: # -(3 * Z^2 + 4 * A) / (4 * g(Z)) != 0 in F. if h(Z_cand) == F(0): continue # Criterion 3: # -(3 * Z^2 + 4 * A) / (4 * g(Z)) is square in F. if not is_square(h(Z_cand)): continue # Criterion 4: # At least one of g(Z) and g(-Z / 2) is square in F. if is_square(g(Z_cand)) or is_square(g(-Z_cand / F(2))): return Z_cand ctr += 1 H.2. Finding Z for Simplified SWU The below function outputs an appropriate Z for the Simplified SWU map (Section 6.6.2). Faz-Hernandez, et al. Expires 14 May 2022 [Page 98] Internet-Draft hash-to-curve November 2021 # Arguments: # - F, a field object, e.g., F = GF(2^521 - 1) # - A and B, the coefficients of the curve y^2 = x^3 + A * x + B def find_z_sswu(F, A, B): R. = F[] # Polynomial ring over F g = xx^3 + F(A) * xx + F(B) # y^2 = g(x) = x^3 + A * x + B ctr = F.gen() while True: for Z_cand in (F(ctr), F(-ctr)): # Criterion 1: Z is non-square in F. if is_square(Z_cand): continue # Criterion 2: Z != -1 in F. if Z_cand == F(-1): continue # Criterion 3: g(x) - Z is irreducible over F. if not (g - Z_cand).is_irreducible(): continue # Criterion 4: g(B / (Z * A)) is square in F. if is_square(g(B / (Z_cand * A))): return Z_cand ctr += 1 H.3. Finding Z for Elligator 2 The below function outputs an appropriate Z for the Elligator 2 map (Section 6.7.1). # Argument: # - F, a field object, e.g., F = GF(2^255 - 19) def find_z_ell2(F): ctr = F.gen() while True: for Z_cand in (F(ctr), F(-ctr)): # Z must be a non-square in F. if is_square(Z_cand): continue return Z_cand ctr += 1 Appendix I. sqrt and is_square functions This section defines special-purpose sqrt functions for the three most common cases, q = 3 (mod 4), q = 5 (mod 8), and q = 9 (mod 16), plus a generic constant-time algorithm that works for any prime modulus. In addition, it gives an optimized is_square method for GF(p^2). Faz-Hernandez, et al. Expires 14 May 2022 [Page 99] Internet-Draft hash-to-curve November 2021 I.1. q = 3 (mod 4) sqrt_3mod4(x) Parameters: - F, a finite field of characteristic p and order q = p^m. Input: x, an element of F. Output: z, an element of F such that (z^2) == x, if x is square in F. Constants: 1. c1 = (q + 1) / 4 # Integer arithmetic Procedure: 1. return x^c1 I.2. q = 5 (mod 8) sqrt_5mod8(x) Parameters: - F, a finite field of characteristic p and order q = p^m. Input: x, an element of F. Output: z, an element of F such that (z^2) == x, if x is square in F. Constants: 1. c1 = sqrt(-1) in F, i.e., (c1^2) == -1 in F 2. c2 = (q + 3) / 8 # Integer arithmetic Procedure: 1. tv1 = x^c2 2. tv2 = tv1 * c1 3. e = (tv1^2) == x 4. z = CMOV(tv2, tv1, e) 5. return z I.3. q = 9 (mod 16) Faz-Hernandez, et al. Expires 14 May 2022 [Page 100] Internet-Draft hash-to-curve November 2021 sqrt_9mod16(x) Parameters: - F, a finite field of characteristic p and order q = p^m. Input: x, an element of F. Output: z, an element of F such that (z^2) == x, if x is square in F. Constants: 1. c1 = sqrt(-1) in F, i.e., (c1^2) == -1 in F 2. c2 = sqrt(c1) in F, i.e., (c2^2) == c1 in F 3. c3 = sqrt(-c1) in F, i.e., (c3^2) == -c1 in F 4. c4 = (q + 7) / 16 # Integer arithmetic Procedure: 1. tv1 = x^c4 2. tv2 = c1 * tv1 3. tv3 = c2 * tv1 4. tv4 = c3 * tv1 5. e1 = (tv2^2) == x 6. e2 = (tv3^2) == x 7. tv1 = CMOV(tv1, tv2, e1) # Select tv2 if (tv2^2) == x 8. tv2 = CMOV(tv4, tv3, e2) # Select tv3 if (tv3^2) == x 9. e3 = (tv2^2) == x 10. z = CMOV(tv1, tv2, e3) # Select the sqrt from tv1 and tv2 11. return z I.4. Constant-time Tonelli-Shanks algorithm This algorithm is a constant-time version of the classic Tonelli- Shanks algorithm ([C93], Algorithm 1.5.1) due to Sean Bowe, Jack Grigg, and Eirik Ogilvie-Wigley [jubjub-fq], adapted and optimized by Michael Scott. This algorithm applies to GF(p) for any p. Note, however, that the special-purpose algorithms given in the prior sections are faster, when they apply. Faz-Hernandez, et al. Expires 14 May 2022 [Page 101] Internet-Draft hash-to-curve November 2021 sqrt_ts_ct(x) Parameters: - F, a finite field of characteristic p and order q = p^m. Input x, an element of F. Output: z, an element of F such that z^2 == x, if x is square in F. Constants: 1. c1, the largest integer such that 2^c1 divides q - 1. 2. c2 = (q - 1) / (2^c1) # Integer arithmetic 3. c3 = (c2 - 1) / 2 # Integer arithmetic 4. c4, a non-square value in F 5. c5 = c4^c2 in F Procedure: 1. z = x^c3 2. t = z * z 3. t = t * x 4. z = z * x 5. b = t 6. c = c5 7. for i in (c1, c1 - 1, ..., 2): 8. for j in (1, 2, ..., i - 2): 9. b = b * b 10. e = b == 1 11. zt = z * c 12. z = CMOV(zt, z, e) 13. c = c * c 14. tt = t * c 15. t = CMOV(tt, t, e) 16. b = t 17. return z I.5. is_square for F = GF(p^2) The following is_square method applies to any field F = GF(p^2) with basis (1, I) represented as described in Section 2.1, i.e., an element x = (x_1, x_2) = x_1 + x_2 * I. Other optimizations of this type are possible in other extension fields; see, e.g., [AR13] for more information. Faz-Hernandez, et al. Expires 14 May 2022 [Page 102] Internet-Draft hash-to-curve November 2021 is_square(x) Parameters: - F, an extension field of characteristic p and order q = p^2 with basis (1, I). Input: x, an element of F. Output: True if x is square in F, and False otherwise. Constants: 1. c1 = (p - 1) / 2 # Integer arithmetic Procedure: 1. tv1 = x_1^2 2. tv2 = I * x_2 3. tv2 = tv2^2 4. tv1 = tv1 - tv2 5. tv1 = tv1^c1 6. e1 = tv1 != -1 # Note: -1 in F 7. return e1 Appendix J. Suite test vectors This section gives test vectors for each suite defined in Section 8. The test vectors in this section were generated using code that is available from [hash2curve-repo]. Each test vector in this section lists values computed by the appropriate encoding function, with variable names defined as in Section 3. For example, for a suite whose encoding type is random oracle, the test vector gives the value for msg, u, Q0, Q1, and the output point P. J.1. NIST P-256 J.1.1. P256_XMD:SHA-256_SSWU_RO_ suite = P256_XMD:SHA-256_SSWU_RO_ dst = QUUX-V01-CS02-with-P256_XMD:SHA-256_SSWU_RO_ msg = P.x = 2c15230b26dbc6fc9a37051158c95b79656e17a1a920b11394ca91 c44247d3e4 P.y = 8a7a74985cc5c776cdfe4b1f19884970453912e9d31528c060be9a b5c43e8415 u[0] = ad5342c66a6dd0ff080df1da0ea1c04b96e0330dd89406465eeba1 1582515009 u[1] = 8c0f1d43204bd6f6ea70ae8013070a1518b43873bcd850aafa0a9e Faz-Hernandez, et al. Expires 14 May 2022 [Page 103] Internet-Draft hash-to-curve November 2021 220e2eea5a Q0.x = ab640a12220d3ff283510ff3f4b1953d09fad35795140b1c5d64f3 13967934d5 Q0.y = dccb558863804a881d4fff3455716c836cef230e5209594ddd33d8 5c565b19b1 Q1.x = 51cce63c50d972a6e51c61334f0f4875c9ac1cd2d3238412f84e31 da7d980ef5 Q1.y = b45d1a36d00ad90e5ec7840a60a4de411917fbe7c82c3949a6e699 e5a1b66aac msg = abc P.x = 0bb8b87485551aa43ed54f009230450b492fead5f1cc91658775da c4a3388a0f P.y = 5c41b3d0731a27a7b14bc0bf0ccded2d8751f83493404c84a88e71 ffd424212e u[0] = afe47f2ea2b10465cc26ac403194dfb68b7f5ee865cda61e9f3e07 a537220af1 u[1] = 379a27833b0bfe6f7bdca08e1e83c760bf9a338ab335542704edcd 69ce9e46e0 Q0.x = 5219ad0ddef3cc49b714145e91b2f7de6ce0a7a7dc7406c7726c7e 373c58cb48 Q0.y = 7950144e52d30acbec7b624c203b1996c99617d0b61c2442354301 b191d93ecf Q1.x = 019b7cb4efcfeaf39f738fe638e31d375ad6837f58a852d032ff60 c69ee3875f Q1.y = 589a62d2b22357fed5449bc38065b760095ebe6aeac84b01156ee4 252715446e msg = abcdef0123456789 P.x = 65038ac8f2b1def042a5df0b33b1f4eca6bff7cb0f9c6c15268118 64e544ed80 P.y = cad44d40a656e7aff4002a8de287abc8ae0482b5ae825822bb870d 6df9b56ca3 u[0] = 0fad9d125a9477d55cf9357105b0eb3a5c4259809bf87180aa01d6 51f53d312c u[1] = b68597377392cd3419d8fcc7d7660948c8403b19ea78bbca4b133c 9d2196c0fb Q0.x = a17bdf2965eb88074bc01157e644ed409dac97cfcf0c61c998ed0f a45e79e4a2 Q0.y = 4f1bc80c70d411a3cc1d67aeae6e726f0f311639fee560c7f5a664 554e3c9c2e Q1.x = 7da48bb67225c1a17d452c983798113f47e438e4202219dd0715f8 419b274d66 Q1.y = b765696b2913e36db3016c47edb99e24b1da30e761a8a3215dc0ec 4d8f96e6f9 msg = q128_qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq Faz-Hernandez, et al. Expires 14 May 2022 [Page 104] Internet-Draft hash-to-curve November 2021 qqqqqqqqqqqqqqqqqqqqqqqqq P.x = 4be61ee205094282ba8a2042bcb48d88dfbb609301c49aa8b07853 3dc65a0b5d P.y = 98f8df449a072c4721d241a3b1236d3caccba603f916ca680f4539 d2bfb3c29e u[0] = 3bbc30446f39a7befad080f4d5f32ed116b9534626993d2cc5033f 6f8d805919 u[1] = 76bb02db019ca9d3c1e02f0c17f8baf617bbdae5c393a81d9ce11e 3be1bf1d33 Q0.x = c76aaa823aeadeb3f356909cb08f97eee46ecb157c1f56699b5efe bddf0e6398 Q0.y = 776a6f45f528a0e8d289a4be12c4fab80762386ec644abf2bffb9b 627e4352b1 Q1.x = 418ac3d85a5ccc4ea8dec14f750a3a9ec8b85176c95a7022f39182 6794eb5a75 Q1.y = fd6604f69e9d9d2b74b072d14ea13050db72c932815523305cb9e8 07cc900aff msg = a512_aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa P.x = 457ae2981f70ca85d8e24c308b14db22f3e3862c5ea0f652ca38b5 e49cd64bc5 P.y = ecb9f0eadc9aeed232dabc53235368c1394c78de05dd96893eefa6 2b0f4757dc u[0] = 4ebc95a6e839b1ae3c63b847798e85cb3c12d3817ec6ebc10af6ee 51adb29fec u[1] = 4e21af88e22ea80156aff790750121035b3eefaa96b425a8716e0d 20b4e269ee Q0.x = d88b989ee9d1295df413d4456c5c850b8b2fb0f5402cc5c4c7e815 412e926db8 Q0.y = bb4a1edeff506cf16def96afff41b16fc74f6dbd55c2210e5b8f01 1ba32f4f40 Q1.x = a281e34e628f3a4d2a53fa87ff973537d68ad4fbc28d3be5e8d9f6 a2571c5a4b Q1.y = f6ed88a7aab56a488100e6f1174fa9810b47db13e86be999644922 961206e184 J.1.2. P256_XMD:SHA-256_SSWU_NU_ Faz-Hernandez, et al. Expires 14 May 2022 [Page 105] Internet-Draft hash-to-curve November 2021 suite = P256_XMD:SHA-256_SSWU_NU_ dst = QUUX-V01-CS02-with-P256_XMD:SHA-256_SSWU_NU_ msg = P.x = f871caad25ea3b59c16cf87c1894902f7e7b2c822c3d3f73596c5a ce8ddd14d1 P.y = 87b9ae23335bee057b99bac1e68588b18b5691af476234b8971bc4 f011ddc99b u[0] = b22d487045f80e9edcb0ecc8d4bf77833e2bf1f3a54004d7df1d57 f4802d311f Q.x = f871caad25ea3b59c16cf87c1894902f7e7b2c822c3d3f73596c5a ce8ddd14d1 Q.y = 87b9ae23335bee057b99bac1e68588b18b5691af476234b8971bc4 f011ddc99b msg = abc P.x = fc3f5d734e8dce41ddac49f47dd2b8a57257522a865c124ed02b92 b5237befa4 P.y = fe4d197ecf5a62645b9690599e1d80e82c500b22ac705a0b421fac 7b47157866 u[0] = c7f96eadac763e176629b09ed0c11992225b3a5ae99479760601cb d69c221e58 Q.x = fc3f5d734e8dce41ddac49f47dd2b8a57257522a865c124ed02b92 b5237befa4 Q.y = fe4d197ecf5a62645b9690599e1d80e82c500b22ac705a0b421fac 7b47157866 msg = abcdef0123456789 P.x = f164c6674a02207e414c257ce759d35eddc7f55be6d7f415e2cc17 7e5d8faa84 P.y = 3aa274881d30db70485368c0467e97da0e73c18c1d00f34775d012 b6fcee7f97 u[0] = 314e8585fa92068b3ea2c3bab452d4257b38be1c097d58a2189045 6c2929614d Q.x = f164c6674a02207e414c257ce759d35eddc7f55be6d7f415e2cc17 7e5d8faa84 Q.y = 3aa274881d30db70485368c0467e97da0e73c18c1d00f34775d012 b6fcee7f97 msg = q128_qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqq P.x = 324532006312be4f162614076460315f7a54a6f85544da773dc659 aca0311853 P.y = 8d8197374bcd52de2acfefc8a54fe2c8d8bebd2a39f16be9b710e4 b1af6ef883 u[0] = 752d8eaa38cd785a799a31d63d99c2ae4261823b4a367b133b2c66 27f48858ab Faz-Hernandez, et al. Expires 14 May 2022 [Page 106] Internet-Draft hash-to-curve November 2021 Q.x = 324532006312be4f162614076460315f7a54a6f85544da773dc659 aca0311853 Q.y = 8d8197374bcd52de2acfefc8a54fe2c8d8bebd2a39f16be9b710e4 b1af6ef883 msg = a512_aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa P.x = 5c4bad52f81f39c8e8de1260e9a06d72b8b00a0829a8ea004a610b 0691bea5d9 P.y = c801e7c0782af1f74f24fc385a8555da0582032a3ce038de637ccd cb16f7ef7b u[0] = 0e1527840b9df2dfbef966678ff167140f2b27c4dccd884c25014d ce0e41dfa3 Q.x = 5c4bad52f81f39c8e8de1260e9a06d72b8b00a0829a8ea004a610b 0691bea5d9 Q.y = c801e7c0782af1f74f24fc385a8555da0582032a3ce038de637ccd cb16f7ef7b J.2. NIST P-384 J.2.1. P384_XMD:SHA-384_SSWU_RO_ suite = P384_XMD:SHA-384_SSWU_RO_ dst = QUUX-V01-CS02-with-P384_XMD:SHA-384_SSWU_RO_ msg = P.x = eb9fe1b4f4e14e7140803c1d99d0a93cd823d2b024040f9c067a8e ca1f5a2eeac9ad604973527a356f3fa3aeff0e4d83 P.y = 0c21708cff382b7f4643c07b105c2eaec2cead93a917d825601e63 c8f21f6abd9abc22c93c2bed6f235954b25048bb1a u[0] = 25c8d7dc1acd4ee617766693f7f8829396065d1b447eedb155871f effd9c6653279ac7e5c46edb7010a0e4ff64c9f3b4 u[1] = 59428be4ed69131df59a0c6a8e188d2d4ece3f1b2a3a02602962b4 7efa4d7905945b1e2cc80b36aa35c99451073521ac Q0.x = e4717e29eef38d862bee4902a7d21b44efb58c464e3e1f0d03894d 94de310f8ffc6de86786dd3e15a1541b18d4eb2846 Q0.y = 6b95a6e639822312298a47526bb77d9cd7bcf76244c991c8cd7007 5e2ee6e8b9a135c4a37e3c0768c7ca871c0ceb53d4 Q1.x = 509527cfc0750eedc53147e6d5f78596c8a3b7360e0608e2fab056 3a1670d58d8ae107c9f04bcf90e89489ace5650efd Faz-Hernandez, et al. Expires 14 May 2022 [Page 107] Internet-Draft hash-to-curve November 2021 Q1.y = 33337b13cb35e173fdea4cb9e8cce915d836ff57803dbbeb7998aa 49d17df2ff09b67031773039d09fbd9305a1566bc4 msg = abc P.x = e02fc1a5f44a7519419dd314e29863f30df55a514da2d655775a81 d413003c4d4e7fd59af0826dfaad4200ac6f60abe1 P.y = 01f638d04d98677d65bef99aef1a12a70a4cbb9270ec55248c0453 0d8bc1f8f90f8a6a859a7c1f1ddccedf8f96d675f6 u[0] = 53350214cb6bef0b51abb791b1c4209a2b4c16a0c67e1ab1401017 fad774cd3b3f9a8bcdf7f6229dd8dd5a075cb149a0 u[1] = c0473083898f63e03f26f14877a2407bd60c75ad491e7d26cbc6cc 5ce815654075ec6b6898c7a41d74ceaf720a10c02e Q0.x = fc853b69437aee9a19d5acf96a4ee4c5e04cf7b53406dfaa2afbdd 7ad2351b7f554e4bbc6f5db4177d4d44f933a8f6ee Q0.y = 7e042547e01834c9043b10f3a8221c4a879cb156f04f72bfccab0c 047a304e30f2aa8b2e260d34c4592c0c33dd0c6482 Q1.x = 57912293709b3556b43a2dfb137a315d256d573b82ded120ef8c78 2d607c05d930d958e50cb6dc1cc480b9afc38c45f1 Q1.y = de9387dab0eef0bda219c6f168a92645a84665c4f2137c14270fb4 24b7532ff84843c3da383ceea24c47fa343c227bb8 msg = abcdef0123456789 P.x = bdecc1c1d870624965f19505be50459d363c71a699a496ab672f9a 5d6b78676400926fbceee6fcd1780fe86e62b2aa89 P.y = 57cf1f99b5ee00f3c201139b3bfe4dd30a653193778d89a0accc5e 0f47e46e4e4b85a0595da29c9494c1814acafe183c u[0] = aab7fb87238cf6b2ab56cdcca7e028959bb2ea599d34f68484139d de85ec6548a6e48771d17956421bdb7790598ea52e u[1] = 26e8d833552d7844d167833ca5a87c35bcfaa5a0d86023479fb28e 5cd6075c18b168bf1f5d2a0ea146d057971336d8d1 Q0.x = 0ceece45b73f89844671df962ad2932122e878ad2259e650626924 e4e7f132589341dec1480ebcbbbe3509d11fb570b7 Q0.y = fafd71a3115298f6be4ae5c6dfc96c400cfb55760f185b7b03f3fa 45f3f91eb65d27628b3c705cafd0466fafa54883ce Q1.x = dea1be8d3f9be4cbf4fab9d71d549dde76875b5d9b876832313a08 3ec81e528cbc2a0a1d0596b3bcb0ba77866b129776 Q1.y = eb15fe71662214fb03b65541f40d3eb0f4cf5c3b559f647da138c9 f9b7484c48a08760e02c16f1992762cb7298fa52cf msg = q128_qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqq P.x = 03c3a9f401b78c6c36a52f07eeee0ec1289f178adf78448f43a385 0e0456f5dd7f7633dd31676d990eda32882ab486c0 P.y = cc183d0d7bdfd0a3af05f50e16a3f2de4abbc523215bf57c848d5e a662482b8c1f43dc453a93b94a8026db58f3f5d878 u[0] = 04c00051b0de6e726d228c85bf243bf5f4789efb512b22b498cde3 821db9da667199b74bd5a09a79583c6d353a3bb41c Faz-Hernandez, et al. Expires 14 May 2022 [Page 108] Internet-Draft hash-to-curve November 2021 u[1] = 97580f218255f899f9204db64cd15e6a312cb4d8182375d1e5157c 8f80f41d6a1a4b77fb1ded9dce56c32058b8d5202b Q0.x = 051a22105e0817a35d66196338c8d85bd52690d79bba373ead8a86 dd9899411513bb9f75273f6483395a7847fb21edb4 Q0.y = f168295c1bbcff5f8b01248e9dbc885335d6d6a04aea960f7384f7 46ba6502ce477e624151cc1d1392b00df0f5400c06 Q1.x = 6ad7bc8ed8b841efd8ad0765c8a23d0b968ec9aa360a558ff33500 f164faa02bee6c704f5f91507c4c5aad2b0dc5b943 Q1.y = 47313cc0a873ade774048338fc34ca5313f96bbf6ae22ac6ef475d 85f03d24792dc6afba8d0b4a70170c1b4f0f716629 msg = a512_aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa P.x = 7b18d210b1f090ac701f65f606f6ca18fb8d081e3bc6cbd937c560 4325f1cdea4c15c10a54ef303aabf2ea58bd9947a4 P.y = ea857285a33abb516732915c353c75c576bf82ccc96adb63c094dd e580021eddeafd91f8c0bfee6f636528f3d0c47fd2 u[0] = 480cb3ac2c389db7f9dac9c396d2647ae946db844598971c26d1af d53912a1491199c0a5902811e4b809c26fcd37a014 u[1] = d28435eb34680e148bf3908536e42231cba9e1f73ae2c6902a222a 89db5c49c97db2f8fa4d4cd6e424b17ac60bdb9bb6 Q0.x = 42e6666f505e854187186bad3011598d9278b9d6e3e4d2503c3d23 6381a56748dec5d139c223129b324df53fa147c4df Q0.y = 8ee51dbda46413bf621838cc935d18d617881c6f33f3838a79c767 a1e5618e34b22f79142df708d2432f75c7366c8512 Q1.x = 4ff01ceeba60484fa1bc0d825fe1e5e383d8f79f1e5bb78e5fb26b 7a7ef758153e31e78b9d60ce75c5e32e43869d4e12 Q1.y = 0f84b978fac8ceda7304b47e229d6037d32062e597dc7a9b95bcd9 af441f3c56c619a901d21635f9ec6ab4710b9fcd0e J.2.2. P384_XMD:SHA-384_SSWU_NU_ suite = P384_XMD:SHA-384_SSWU_NU_ dst = QUUX-V01-CS02-with-P384_XMD:SHA-384_SSWU_NU_ msg = P.x = de5a893c83061b2d7ce6a0d8b049f0326f2ada4b966dc7e7292725 6b033ef61058029a3bfb13c1c7ececd6641881ae20 P.y = 63f46da6139785674da315c1947e06e9a0867f5608cf24724eb379 3a1f5b3809ee28eb21a0c64be3be169afc6cdb38ca Faz-Hernandez, et al. Expires 14 May 2022 [Page 109] Internet-Draft hash-to-curve November 2021 u[0] = bc7dc1b2cdc5d588a66de3276b0f24310d4aca4977efda7d6272e1 be25187b001493d267dc53b56183c9e28282368e60 Q.x = de5a893c83061b2d7ce6a0d8b049f0326f2ada4b966dc7e7292725 6b033ef61058029a3bfb13c1c7ececd6641881ae20 Q.y = 63f46da6139785674da315c1947e06e9a0867f5608cf24724eb379 3a1f5b3809ee28eb21a0c64be3be169afc6cdb38ca msg = abc P.x = 1f08108b87e703c86c872ab3eb198a19f2b708237ac4be53d7929f b4bd5194583f40d052f32df66afe5249c9915d139b P.y = 1369dc8d5bf038032336b989994874a2270adadb67a7fcc32f0f88 24bc5118613f0ac8de04a1041d90ff8a5ad555f96c u[0] = 9de6cf41e6e41c03e4a7784ac5c885b4d1e49d6de390b3cdd5a1ac 5dd8c40afb3dfd7bb2686923bab644134483fc1926 Q.x = 1f08108b87e703c86c872ab3eb198a19f2b708237ac4be53d7929f b4bd5194583f40d052f32df66afe5249c9915d139b Q.y = 1369dc8d5bf038032336b989994874a2270adadb67a7fcc32f0f88 24bc5118613f0ac8de04a1041d90ff8a5ad555f96c msg = abcdef0123456789 P.x = 4dac31ec8a82ee3c02ba2d7c9fa431f1e59ffe65bf977b948c59e1 d813c2d7963c7be81aa6db39e78ff315a10115c0d0 P.y = 845333cdb5702ad5c525e603f302904d6fc84879f0ef2ee2014a6b 13edd39131bfd66f7bd7cdc2d9ccf778f0c8892c3f u[0] = 84e2d430a5e2543573e58e368af41821ca3ccc97baba7e9aab51a8 4543d5a0298638a22ceee6090d9d642921112af5b7 Q.x = 4dac31ec8a82ee3c02ba2d7c9fa431f1e59ffe65bf977b948c59e1 d813c2d7963c7be81aa6db39e78ff315a10115c0d0 Q.y = 845333cdb5702ad5c525e603f302904d6fc84879f0ef2ee2014a6b 13edd39131bfd66f7bd7cdc2d9ccf778f0c8892c3f msg = q128_qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqq P.x = 13c1f8c52a492183f7c28e379b0475486718a7e3ac1dfef39283b9 ce5fb02b73f70c6c1f3dfe0c286b03e2af1af12d1d P.y = 57e101887e73e40eab8963324ed16c177d55eb89f804ec9df06801 579820420b5546b579008df2145fd770f584a1a54c u[0] = 504e4d5a529333b9205acaa283107bd1bffde753898f7744161f7d d19ba57fbb6a64214a2e00ddd2613d76cd508ddb30 Q.x = 13c1f8c52a492183f7c28e379b0475486718a7e3ac1dfef39283b9 ce5fb02b73f70c6c1f3dfe0c286b03e2af1af12d1d Q.y = 57e101887e73e40eab8963324ed16c177d55eb89f804ec9df06801 579820420b5546b579008df2145fd770f584a1a54c msg = a512_aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa Faz-Hernandez, et al. Expires 14 May 2022 [Page 110] Internet-Draft hash-to-curve November 2021 aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa P.x = af129727a4207a8cb9e9dce656d88f79fce25edbcea350499d65e9 bf1204537bdde73c7cefb752a6ed5ebcd44e183302 P.y = ce68a3d5e161b2e6a968e4ddaa9e51504ad1516ec170c7eef3ca6b 5327943eca95d90b23b009ba45f58b72906f2a99e2 u[0] = 7b01ce9b8c5a60d9fbc202d6dde92822e46915d8c17e03fcb92ece 1ed6074d01e149fc9236def40d673de903c1d4c166 Q.x = af129727a4207a8cb9e9dce656d88f79fce25edbcea350499d65e9 bf1204537bdde73c7cefb752a6ed5ebcd44e183302 Q.y = ce68a3d5e161b2e6a968e4ddaa9e51504ad1516ec170c7eef3ca6b 5327943eca95d90b23b009ba45f58b72906f2a99e2 J.3. NIST P-521 J.3.1. P521_XMD:SHA-512_SSWU_RO_ suite = P521_XMD:SHA-512_SSWU_RO_ dst = QUUX-V01-CS02-with-P521_XMD:SHA-512_SSWU_RO_ msg = P.x = 00fd767cebb2452030358d0e9cf907f525f50920c8f607889a6a35 680727f64f4d66b161fafeb2654bea0d35086bec0a10b30b14adef 3556ed9f7f1bc23cecc9c088 P.y = 0169ba78d8d851e930680322596e39c78f4fe31b97e57629ef6460 ddd68f8763fd7bd767a4e94a80d3d21a3c2ee98347e024fc73ee1c 27166dc3fe5eeef782be411d u[0] = 01e5f09974e5724f25286763f00ce76238c7a6e03dc396600350ee 2c4135fb17dc555be99a4a4bae0fd303d4f66d984ed7b6a3ba3860 93752a855d26d559d69e7e9e u[1] = 00ae593b42ca2ef93ac488e9e09a5fe5a2f6fb330d18913734ff60 2f2a761fcaaf5f596e790bcc572c9140ec03f6cccc38f767f1c197 5a0b4d70b392d95a0c7278aa Q0.x = 00b70ae99b6339fffac19cb9bfde2098b84f75e50ac1e80d6acb95 4e4534af5f0e9c4a5b8a9c10317b8e6421574bae2b133b4f2b8c6c e4b3063da1d91d34fa2b3a3c Q0.y = 007f368d98a4ddbf381fb354de40e44b19e43bb11a1278759f4ea7 b485e1b6db33e750507c071250e3e443c1aaed61f2c28541bb54b1 b456843eda1eb15ec2a9b36e Q1.x = 01143d0e9cddcdacd6a9aafe1bcf8d218c0afc45d4451239e821f5 d2a56df92be942660b532b2aa59a9c635ae6b30e803c45a6ac8714 32452e685d661cd41cf67214 Q1.y = 00ff75515df265e996d702a5380defffab1a6d2bc232234c7bcffa Faz-Hernandez, et al. Expires 14 May 2022 [Page 111] Internet-Draft hash-to-curve November 2021 433cd8aa791fbc8dcf667f08818bffa739ae25773b32073213cae9 a0f2a917a0b1301a242dda0c msg = abc P.x = 002f89a1677b28054b50d15e1f81ed6669b5a2158211118ebdef8a 6efc77f8ccaa528f698214e4340155abc1fa08f8f613ef14a04371 7503d57e267d57155cf784a4 P.y = 010e0be5dc8e753da8ce51091908b72396d3deed14ae166f66d8eb f0a4e7059ead169ea4bead0232e9b700dd380b316e9361cfdba55a 08c73545563a80966ecbb86d u[0] = 003d00c37e95f19f358adeeaa47288ec39998039c3256e13c2a4c0 0a7cb61a34c8969472960150a27276f2390eb5e53e47ab193351c2 d2d9f164a85c6a5696d94fe8 u[1] = 01f3cbd3df3893a45a2f1fecdac4d525eb16f345b03e2820d69bc5 80f5cbe9cb89196fdf720ef933c4c0361fcfe29940fd0db0a5da6b afb0bee8876b589c41365f15 Q0.x = 01b254e1c99c835836f0aceebba7d77750c48366ecb07fb658e4f5 b76e229ae6ca5d271bb0006ffcc42324e15a6d3daae587f9049de2 dbb0494378ffb60279406f56 Q0.y = 01845f4af72fc2b1a5a2fe966f6a97298614288b456cfc385a425b 686048b25c952fbb5674057e1eb055d04568c0679a8e2dda3158dc 16ac598dbb1d006f5ad915b0 Q1.x = 007f08e813c620e527c961b717ffc74aac7afccb9158cebc347d57 15d5c2214f952c97e194f11d114d80d3481ed766ac0a3dba3eb73f 6ff9ccb9304ad10bbd7b4a36 Q1.y = 0022468f92041f9970a7cc025d71d5b647f822784d29ca7b3bc3b0 829d6bb8581e745f8d0cc9dc6279d0450e779ac2275c4c3608064a d6779108a7828ebd9954caeb msg = abcdef0123456789 P.x = 006e200e276a4a81760099677814d7f8794a4a5f3658442de63c18 d2244dcc957c645e94cb0754f95fcf103b2aeaf94411847c24187b 89fb7462ad3679066337cbc4 P.y = 001dd8dfa9775b60b1614f6f169089d8140d4b3e4012949b52f98d b2deff3e1d97bf73a1fa4d437d1dcdf39b6360cc518d8ebcc0f899 018206fded7617b654f6b168 u[0] = 00183ee1a9bbdc37181b09ec336bcaa34095f91ef14b66b1485c16 6720523dfb81d5c470d44afcb52a87b704dbc5c9bc9d0ef524dec2 9884a4795f55c1359945baf3 u[1] = 00504064fd137f06c81a7cf0f84aa7e92b6b3d56c2368f0a08f447 76aa8930480da1582d01d7f52df31dca35ee0a7876500ece3d8fe0 293cd285f790c9881c998d5e Q0.x = 0021482e8622aac14da60e656043f79a6a110cbae5012268a62dd6 a152c41594549f373910ebed170ade892dd5a19f5d687fae7095a4 61d583f8c4295f7aaf8cd7da Q0.y = 0177e2d8c6356b7de06e0b5712d8387d529b848748e54a8bc0ef5f 1475aa569f8f492fa85c3ad1c5edc51faf7911f11359bfa2a12d2e f0bd73df9cb5abd1b101c8b1 Faz-Hernandez, et al. Expires 14 May 2022 [Page 112] Internet-Draft hash-to-curve November 2021 Q1.x = 00abeafb16fdbb5eb95095678d5a65c1f293291dfd20a3751dbe05 d0a9bfe2d2eef19449fe59ec32cdd4a4adc3411177c0f2dffd0159 438706159a1bbd0567d9b3d0 Q1.y = 007cc657f847db9db651d91c801741060d63dab4056d0a1d3524e2 eb0e819954d8f677aa353bd056244a88f00017e00c3ce8beeedb43 82d83d74418bd48930c6c182 msg = q128_qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqq P.x = 01b264a630bd6555be537b000b99a06761a9325c53322b65bdc41b f196711f9708d58d34b3b90faf12640c27b91c70a507998e559406 48caa8e71098bf2bc8d24664 P.y = 01ea9f445bee198b3ee4c812dcf7b0f91e0881f0251aab272a1220 1fd89b1a95733fd2a699c162b639e9acdcc54fdc2f6536129b6beb 0432be01aa8da02df5e59aaa u[0] = 0159871e222689aad7694dc4c3480a49807b1eedd9c8cb4ae1b219 d5ba51655ea5b38e2e4f56b36bf3e3da44a7b139849d28f598c816 fe1bc7ed15893b22f63363c3 u[1] = 004ef0cffd475152f3858c0a8ccbdf7902d8261da92744e98df9b7 fadb0a5502f29c5086e76e2cf498f47321434a40b1504911552ce4 4ad7356a04e08729ad9411f5 Q0.x = 0005eac7b0b81e38727efcab1e375f6779aea949c3e409b53a1d37 aa2acbac87a7e6ad24aafbf3c52f82f7f0e21b872e88c55e17b7fa 21ce08a94ea2121c42c2eb73 Q0.y = 00a173b6a53a7420dbd61d4a21a7c0a52de7a5c6ce05f31403bef7 47d16cc8604a039a73bdd6e114340e55dacd6bea8e217ffbadfb8c 292afa3e1b2afc839a6ce7bb Q1.x = 01881e3c193a69e4d88d8180a6879b74782a0bc7e529233e9f84bf 7f17d2f319c36920ffba26f9e57a1e045cc7822c834c239593b6e1 42a694aa00c757b0db79e5e8 Q1.y = 01558b16d396d866e476e001f2dd0758927655450b84e12f154032 c7c2a6db837942cd9f44b814f79b4d729996ced61eec61d85c6751 39cbffe3fbf071d2c21cfecb msg = a512_aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa P.x = 00c12bc3e28db07b6b4d2a2b1167ab9e26fc2fa85c7b0498a17b03 47edf52392856d7e28b8fa7a2dd004611159505835b687ecf1a764 857e27e9745848c436ef3925 Faz-Hernandez, et al. Expires 14 May 2022 [Page 113] Internet-Draft hash-to-curve November 2021 P.y = 01cd287df9a50c22a9231beb452346720bb163344a41c5f5a24e83 35b6ccc595fd436aea89737b1281aecb411eb835f0b939073fdd1d d4d5a2492e91ef4a3c55bcbd u[0] = 0033d06d17bc3b9a3efc081a05d65805a14a3050a0dd4dfb488461 8eb5c73980a59c5a246b18f58ad022dd3630faa22889fbb8ba1593 466515e6ab4aeb7381c26334 u[1] = 0092290ab99c3fea1a5b8fb2ca49f859994a04faee3301cefab312 d34227f6a2d0c3322cf76861c6a3683bdaa2dd2a6daa5d6906c663 e065338b2344d20e313f1114 Q0.x = 00041f6eb92af8777260718e4c22328a7d74203350c6c8f5794d99 d5789766698f459b83d5068276716f01429934e40af3d1111a2278 0b1e07e72238d2207e5386be Q0.y = 001c712f0182813942b87cab8e72337db017126f52ed797dd23458 4ac9ae7e80dfe7abea11db02cf1855312eae1447dbaecc9d7e8c88 0a5e76a39f6258074e1bc2e0 Q1.x = 0125c0b69bcf55eab49280b14f707883405028e05c927cd7625d4e 04115bd0e0e6323b12f5d43d0d6d2eff16dbcf244542f84ec05891 1260dc3bb6512ab5db285fbd Q1.y = 008bddfb803b3f4c761458eb5f8a0aee3e1f7f68e9d7424405fa69 172919899317fb6ac1d6903a432d967d14e0f80af63e7035aaae0c 123e56862ce969456f99f102 J.3.2. P521_XMD:SHA-512_SSWU_NU_ suite = P521_XMD:SHA-512_SSWU_NU_ dst = QUUX-V01-CS02-with-P521_XMD:SHA-512_SSWU_NU_ msg = P.x = 01ec604b4e1e3e4c7449b7a41e366e876655538acf51fd40d08b97 be066f7d020634e906b1b6942f9174b417027c953d75fb6ec64b8c ee2a3672d4f1987d13974705 P.y = 00944fc439b4aad2463e5c9cfa0b0707af3c9a42e37c5a57bb4ecd 12fef9fb21508568aedcdd8d2490472df4bbafd79081c81e99f4da 3286eddf19be47e9c4cf0e91 u[0] = 01e4947fe62a4e47792cee2798912f672fff820b2556282d9843b4 b465940d7683a986f93ccb0e9a191fbc09a6e770a564490d2a4ae5 1b287ca39f69c3d910ba6a4f Q.x = 01ec604b4e1e3e4c7449b7a41e366e876655538acf51fd40d08b97 be066f7d020634e906b1b6942f9174b417027c953d75fb6ec64b8c ee2a3672d4f1987d13974705 Q.y = 00944fc439b4aad2463e5c9cfa0b0707af3c9a42e37c5a57bb4ecd 12fef9fb21508568aedcdd8d2490472df4bbafd79081c81e99f4da 3286eddf19be47e9c4cf0e91 msg = abc P.x = 00c720ab56aa5a7a4c07a7732a0a4e1b909e32d063ae1b58db5f0e b5e09f08a9884bff55a2bef4668f715788e692c18c1915cd034a6b 998311fcf46924ce66a2be9a Faz-Hernandez, et al. Expires 14 May 2022 [Page 114] Internet-Draft hash-to-curve November 2021 P.y = 003570e87f91a4f3c7a56be2cb2a078ffc153862a53d5e03e5dad5 bccc6c529b8bab0b7dbb157499e1949e4edab21cf5d10b782bc1e9 45e13d7421ad8121dbc72b1d u[0] = 0019b85ef78596efc84783d42799e80d787591fe7432dee1d9fa2b 7651891321be732ddf653fa8fefa34d86fb728db569d36b5b6ed39 83945854b2fc2dc6a75aa25b Q.x = 00c720ab56aa5a7a4c07a7732a0a4e1b909e32d063ae1b58db5f0e b5e09f08a9884bff55a2bef4668f715788e692c18c1915cd034a6b 998311fcf46924ce66a2be9a Q.y = 003570e87f91a4f3c7a56be2cb2a078ffc153862a53d5e03e5dad5 bccc6c529b8bab0b7dbb157499e1949e4edab21cf5d10b782bc1e9 45e13d7421ad8121dbc72b1d msg = abcdef0123456789 P.x = 00bcaf32a968ff7971b3bbd9ce8edfbee1309e2019d7ff373c3838 7a782b005dce6ceffccfeda5c6511c8f7f312f343f3a891029c585 8f45ee0bf370aba25fc990cc P.y = 00923517e767532d82cb8a0b59705eec2b7779ce05f9181c7d5d5e 25694ef8ebd4696343f0bc27006834d2517215ecf79482a84111f5 0c1bae25044fe1dd77744bbd u[0] = 01dba0d7fa26a562ee8a9014ebc2cca4d66fd9de036176aca8fc11 ef254cd1bc208847ab7701dbca7af328b3f601b11a1737a899575a 5c14f4dca5aaca45e9935e07 Q.x = 00bcaf32a968ff7971b3bbd9ce8edfbee1309e2019d7ff373c3838 7a782b005dce6ceffccfeda5c6511c8f7f312f343f3a891029c585 8f45ee0bf370aba25fc990cc Q.y = 00923517e767532d82cb8a0b59705eec2b7779ce05f9181c7d5d5e 25694ef8ebd4696343f0bc27006834d2517215ecf79482a84111f5 0c1bae25044fe1dd77744bbd msg = q128_qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqq P.x = 001ac69014869b6c4ad7aa8c443c255439d36b0e48a0f57b03d6fe 9c40a66b4e2eaed2a93390679a5cc44b3a91862b34b673f0e92c83 187da02bf3db967d867ce748 P.y = 00d5603d530e4d62b30fccfa1d90c2206654d74291c1db1c25b86a 051ee3fffc294e5d56f2e776853406bd09206c63d40f37ad882952 4cf89ad70b5d6e0b4a3b7341 u[0] = 00844da980675e1244cb209dcf3ea0aabec23bd54b2cda69fff86e b3acc318bf3d01bae96e9cd6f4c5ceb5539df9a7ad7fcc5e9d5469 6081ba9782f3a0f6d14987e3 Q.x = 001ac69014869b6c4ad7aa8c443c255439d36b0e48a0f57b03d6fe 9c40a66b4e2eaed2a93390679a5cc44b3a91862b34b673f0e92c83 187da02bf3db967d867ce748 Q.y = 00d5603d530e4d62b30fccfa1d90c2206654d74291c1db1c25b86a 051ee3fffc294e5d56f2e776853406bd09206c63d40f37ad882952 4cf89ad70b5d6e0b4a3b7341 Faz-Hernandez, et al. Expires 14 May 2022 [Page 115] Internet-Draft hash-to-curve November 2021 msg = a512_aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa P.x = 01801de044c517a80443d2bd4f503a9e6866750d2f94a22970f62d 721f96e4310e4a828206d9cdeaa8f2d476705cc3bbc490a6165c68 7668f15ec178a17e3d27349b P.y = 0068889ea2e1442245fe42bfda9e58266828c0263119f35a61631a 3358330f3bb84443fcb54fcd53a1d097fccbe310489b74ee143fc2 938959a83a1f7dd4a6fd395b u[0] = 01aab1fb7e5cd44ba4d9f32353a383cb1bb9eb763ed40b32bdd5f6 66988970205998c0e44af6e2b5f6f8e48e969b3f649cae3c6ab463 e1b274d968d91c02f00cce91 Q.x = 01801de044c517a80443d2bd4f503a9e6866750d2f94a22970f62d 721f96e4310e4a828206d9cdeaa8f2d476705cc3bbc490a6165c68 7668f15ec178a17e3d27349b Q.y = 0068889ea2e1442245fe42bfda9e58266828c0263119f35a61631a 3358330f3bb84443fcb54fcd53a1d097fccbe310489b74ee143fc2 938959a83a1f7dd4a6fd395b J.4. curve25519 J.4.1. curve25519_XMD:SHA-512_ELL2_RO_ suite = curve25519_XMD:SHA-512_ELL2_RO_ dst = QUUX-V01-CS02-with-curve25519_XMD:SHA-512_ELL2_RO_ msg = P.x = 2de3780abb67e861289f5749d16d3e217ffa722192d16bbd9d1bfb 9d112b98c0 P.y = 3b5dc2a498941a1033d176567d457845637554a2fe7a3507d21abd 1c1bd6e878 u[0] = 005fe8a7b8fef0a16c105e6cadf5a6740b3365e18692a9c05bfbb4 d97f645a6a u[1] = 1347edbec6a2b5d8c02e058819819bee177077c9d10a4ce165aab0 fd0252261a Q0.x = 36b4df0c864c64707cbf6cf36e9ee2c09a6cb93b28313c169be295 61bb904f98 Q0.y = 6cd59d664fb58c66c892883cd0eb792e52055284dac3907dd756b4 5d15c3983d Q1.x = 3fa114783a505c0b2b2fbeef0102853c0b494e7757f2a089d0daae 7ed9a0db2b Faz-Hernandez, et al. Expires 14 May 2022 [Page 116] Internet-Draft hash-to-curve November 2021 Q1.y = 76c0fe7fec932aaafb8eefb42d9cbb32eb931158f469ff3050af15 cfdbbeff94 msg = abc P.x = 2b4419f1f2d48f5872de692b0aca72cc7b0a60915dd70bde432e82 6b6abc526d P.y = 1b8235f255a268f0a6fa8763e97eb3d22d149343d495da1160eff9 703f2d07dd u[0] = 49bed021c7a3748f09fa8cdfcac044089f7829d3531066ac9e74e0 994e05bc7d u[1] = 5c36525b663e63389d886105cee7ed712325d5a97e60e140aba7e2 ce5ae851b6 Q0.x = 16b3d86e056b7970fa00165f6f48d90b619ad618791661b7b5e1ec 78be10eac1 Q0.y = 4ab256422d84c5120b278cbdfc4e1facc5baadffeccecf8ee9bf39 46106d50ca Q1.x = 7ec29ddbf34539c40adfa98fcb39ec36368f47f30e8f888cc7e86f 4d46e0c264 Q1.y = 10d1abc1cae2d34c06e247f2141ba897657fb39f1080d54f09ce0a f128067c74 msg = abcdef0123456789 P.x = 68ca1ea5a6acf4e9956daa101709b1eee6c1bb0df1de3b90d46023 82a104c036 P.y = 2a375b656207123d10766e68b938b1812a4a6625ff83cb8d5e86f5 8a4be08353 u[0] = 6412b7485ba26d3d1b6c290a8e1435b2959f03721874939b21782d f17323d160 u[1] = 24c7b46c1c6d9a21d32f5707be1380ab82db1054fde82865d5c9e3 d968f287b2 Q0.x = 71de3dadfe268872326c35ac512164850860567aea0e7325e6b91a 98f86533ad Q0.y = 26a08b6e9a18084c56f2147bf515414b9b63f1522e1b6c5649f7d4 b0324296ec Q1.x = 5704069021f61e41779e2ba6b932268316d6d2a6f064f997a22fef 16d1eaeaca Q1.y = 50483c7540f64fb4497619c050f2c7fe55454ec0f0e79870bb4430 2e34232210 msg = q128_qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqq P.x = 096e9c8bae6c06b554c1ee69383bb0e82267e064236b3a30608d4e d20b73ac5a P.y = 1eb5a62612cafb32b16c3329794645b5b948d9f8ffe501d4e26b07 3fef6de355 u[0] = 5e123990f11bbb5586613ffabdb58d47f64bb5f2fa115f8ea8df01 88e0c9e1b5 Faz-Hernandez, et al. Expires 14 May 2022 [Page 117] Internet-Draft hash-to-curve November 2021 u[1] = 5e8553eb00438a0bb1e7faa59dec6d8087f9c8011e5fb8ed9df31c b6c0d4ac19 Q0.x = 7a94d45a198fb5daa381f45f2619ab279744efdd8bd8ed587fc5b6 5d6cea1df0 Q0.y = 67d44f85d376e64bb7d713585230cdbfafc8e2676f7568e0b6ee59 361116a6e1 Q1.x = 30506fb7a32136694abd61b6113770270debe593027a968a01f271 e146e60c18 Q1.y = 7eeee0e706b40c6b5174e551426a67f975ad5a977ee2f01e8e20a6 d612458c3b msg = a512_aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa P.x = 1bc61845a138e912f047b5e70ba9606ba2a447a4dade024c8ef3dd 42b7bbc5fe P.y = 623d05e47b70e25f7f1d51dda6d7c23c9a18ce015fe3548df596ea 9e38c69bf1 u[0] = 20f481e85da7a3bf60ac0fb11ed1d0558fc6f941b3ac5469aa8b56 ec883d6d7d u[1] = 017d57fd257e9a78913999a23b52ca988157a81b09c5442501d07f ed20869465 Q0.x = 02d606e2699b918ee36f2818f2bc5013e437e673c9f9b9cdc15fd0 c5ee913970 Q0.y = 29e9dc92297231ef211245db9e31767996c5625dfbf92e1c8107ef 887365de1e Q1.x = 38920e9b988d1ab7449c0fa9a6058192c0c797bb3d42ac34572434 1a1aa98745 Q1.y = 24dcc1be7c4d591d307e89049fd2ed30aae8911245a9d8554bf603 2e5aa40d3d J.4.2. curve25519_XMD:SHA-512_ELL2_NU_ suite = curve25519_XMD:SHA-512_ELL2_NU_ dst = QUUX-V01-CS02-with-curve25519_XMD:SHA-512_ELL2_NU_ msg = P.x = 1bb913f0c9daefa0b3375378ffa534bda5526c97391952a7789eb9 76edfe4d08 P.y = 4548368f4f983243e747b62a600840ae7c1dab5c723991f85d3a97 68479f3ec4 Faz-Hernandez, et al. Expires 14 May 2022 [Page 118] Internet-Draft hash-to-curve November 2021 u[0] = 608d892b641f0328523802a6603427c26e55e6f27e71a91a478148 d45b5093cd Q.x = 51125222da5e763d97f3c10fcc92ea6860b9ccbbd2eb1285728f56 6721c1e65b Q.y = 343d2204f812d3dfc5304a5808c6c0d81a903a5d228b342442aa3c 9ba5520a3d msg = abc P.x = 7c22950b7d900fa866334262fcaea47a441a578df43b894b4625c9 b450f9a026 P.y = 5547bc00e4c09685dcbc6cb6765288b386d8bdcb595fa5a6e3969e 08097f0541 u[0] = 46f5b22494bfeaa7f232cc8d054be68561af50230234d7d1d63d1d 9abeca8da5 Q.x = 7d56d1e08cb0ccb92baf069c18c49bb5a0dcd927eff8dcf75ca921 ef7f3e6eeb Q.y = 404d9a7dc25c9c05c44ab9a94590e7c3fe2dcec74533a0b24b188a 5d5dacf429 msg = abcdef0123456789 P.x = 31ad08a8b0deeb2a4d8b0206ca25f567ab4e042746f792f4b7973f 3ae2096c52 P.y = 405070c28e78b4fa269427c82827261991b9718bd6c6e95d627d70 1a53c30db1 u[0] = 235fe40c443766ce7e18111c33862d66c3b33267efa50d50f9e8e5 d252a40aaa Q.x = 3fbe66b9c9883d79e8407150e7c2a1c8680bee496c62fabe4619a7 2b3cabe90f Q.y = 08ec476147c9a0a3ff312d303dbbd076abb7551e5fce82b48ab14b 433f8d0a7b msg = q128_qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqq P.x = 027877759d155b1997d0d84683a313eb78bdb493271d935b622900 459d52ceaa P.y = 54d691731a53baa30707f4a87121d5169fb5d587d70fb0292b5830 dedbec4c18 u[0] = 001e92a544463bda9bd04ddbe3d6eed248f82de32f522669efc5dd ce95f46f5b Q.x = 227e0bb89de700385d19ec40e857db6e6a3e634b1c32962f370d26 f84ff19683 Q.y = 5f86ff3851d262727326a32c1bf7655a03665830fa7f1b8b1e5a09 d85bc66e4a msg = a512_aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa Faz-Hernandez, et al. Expires 14 May 2022 [Page 119] Internet-Draft hash-to-curve November 2021 aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa P.x = 5fd892c0958d1a75f54c3182a18d286efab784e774d1e017ba2fb2 52998b5dc1 P.y = 750af3c66101737423a4519ac792fb93337bd74ee751f19da4cf1e 94f4d6d0b8 u[0] = 1a68a1af9f663592291af987203393f707305c7bac9c8d63d6a729 bdc553dc19 Q.x = 3bcd651ee54d5f7b6013898aab251ee8ecc0688166fce6e9548d38 472f6bd196 Q.y = 1bb36ad9197299f111b4ef21271c41f4b7ecf5543db8bb5931307e bdb2eaa465 J.5. edwards25519 J.5.1. edwards25519_XMD:SHA-512_ELL2_RO_ suite = edwards25519_XMD:SHA-512_ELL2_RO_ dst = QUUX-V01-CS02-with-edwards25519_XMD:SHA-512_ELL2_RO_ msg = P.x = 3c3da6925a3c3c268448dcabb47ccde5439559d9599646a8260e47 b1e4822fc6 P.y = 09a6c8561a0b22bef63124c588ce4c62ea83a3c899763af26d7953 02e115dc21 u[0] = 03fef4813c8cb5f98c6eef88fae174e6e7d5380de2b007799ac7ee 712d203f3a u[1] = 780bdddd137290c8f589dc687795aafae35f6b674668d92bf92ae7 93e6a60c75 Q0.x = 6549118f65bb617b9e8b438decedc73c496eaed496806d3b2eb9ee 60b88e09a7 Q0.y = 7315bcc8cf47ed68048d22bad602c6680b3382a08c7c5d3f439a97 3fb4cf9feb Q1.x = 31dcfc5c58aa1bee6e760bf78cbe71c2bead8cebb2e397ece0f37a 3da19c9ed2 Q1.y = 7876d81474828d8a5928b50c82420b2bd0898d819e9550c5c82c39 fc9bafa196 msg = abc P.x = 608040b42285cc0d72cbb3985c6b04c935370c7361f4b7fbdb1ae7 f8c1a8ecad P.y = 1a8395b88338f22e435bbd301183e7f20a5f9de643f11882fb237f 88268a5531 Faz-Hernandez, et al. Expires 14 May 2022 [Page 120] Internet-Draft hash-to-curve November 2021 u[0] = 5081955c4141e4e7d02ec0e36becffaa1934df4d7a270f70679c78 f9bd57c227 u[1] = 005bdc17a9b378b6272573a31b04361f21c371b256252ae5463119 aa0b925b76 Q0.x = 5c1525bd5d4b4e034512949d187c39d48e8cd84242aa4758956e4a dc7d445573 Q0.y = 2bf426cf7122d1a90abc7f2d108befc2ef415ce8c2d09695a74072 40faa01f29 Q1.x = 37b03bba828860c6b459ddad476c83e0f9285787a269df2156219b 7e5c86210c Q1.y = 285ebf5412f84d0ad7bb4e136729a9ffd2195d5b8e73c0dc85110c e06958f432 msg = abcdef0123456789 P.x = 6d7fabf47a2dc03fe7d47f7dddd21082c5fb8f86743cd020f3fb14 7d57161472 P.y = 53060a3d140e7fbcda641ed3cf42c88a75411e648a1add71217f70 ea8ec561a6 u[0] = 285ebaa3be701b79871bcb6e225ecc9b0b32dff2d60424b4c50642 636a78d5b3 u[1] = 2e253e6a0ef658fedb8e4bd6a62d1544fd6547922acb3598ec6b36 9760b81b31 Q0.x = 3ac463dd7fddb773b069c5b2b01c0f6b340638f54ee3bd92d452fc ec3015b52d Q0.y = 7b03ba1e8db9ec0b390d5c90168a6a0b7107156c994c674b61fe69 6cbeb46baf Q1.x = 0757e7e904f5e86d2d2f4acf7e01c63827fde2d363985aa7432106 f1b3a444ec Q1.y = 50026c96930a24961e9d86aa91ea1465398ff8e42015e2ec1fa397 d416f6a1c0 msg = q128_qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqq P.x = 5fb0b92acedd16f3bcb0ef83f5c7b7a9466b5f1e0d8d217421878e a3686f8524 P.y = 2eca15e355fcfa39d2982f67ddb0eea138e2994f5956ed37b7f72e ea5e89d2f7 u[0] = 4fedd25431c41f2a606952e2945ef5e3ac905a42cf64b8b4d4a83c 533bf321af u[1] = 02f20716a5801b843987097a8276b6d869295b2e11253751ca72c1 09d37485a9 Q0.x = 703e69787ea7524541933edf41f94010a201cc841c1cce60205ec3 8513458872 Q0.y = 32bb192c4f89106466f0874f5fd56a0d6b6f101cb714777983336c 159a9bec75 Q1.x = 0c9077c5c31720ed9413abe59bf49ce768506128d810cb882435aa 90f713ef6b Faz-Hernandez, et al. Expires 14 May 2022 [Page 121] Internet-Draft hash-to-curve November 2021 Q1.y = 7d5aec5210db638c53f050597964b74d6dda4be5b54fa73041bf90 9ccb3826cb msg = a512_aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa P.x = 0efcfde5898a839b00997fbe40d2ebe950bc81181afbd5cd6b9618 aa336c1e8c P.y = 6dc2fc04f266c5c27f236a80b14f92ccd051ef1ff027f26a07f8c0 f327d8f995 u[0] = 6e34e04a5106e9bd59f64aba49601bf09d23b27f7b594e56d5de06 df4a4ea33b u[1] = 1c1c2cb59fc053f44b86c5d5eb8c1954b64976d0302d3729ff66e8 4068f5fd96 Q0.x = 21091b2e3f9258c7dfa075e7ae513325a94a3d8a28e1b1cb3b5b6f 5d65675592 Q0.y = 41a33d324c89f570e0682cdf7bdb78852295daf8084c669f2cc969 2896ab5026 Q1.x = 4c07ec48c373e39a23bd7954f9e9b66eeab9e5ee1279b867b3d531 5aa815454f Q1.y = 67ccac7c3cb8d1381242d8d6585c57eabaddbb5dca5243a68a8aeb 5477d94b3a J.5.2. edwards25519_XMD:SHA-512_ELL2_NU_ suite = edwards25519_XMD:SHA-512_ELL2_NU_ dst = QUUX-V01-CS02-with-edwards25519_XMD:SHA-512_ELL2_NU_ msg = P.x = 1ff2b70ecf862799e11b7ae744e3489aa058ce805dd323a936375a 84695e76da P.y = 222e314d04a4d5725e9f2aff9fb2a6b69ef375a1214eb19021ceab 2d687f0f9b u[0] = 7f3e7fb9428103ad7f52db32f9df32505d7b427d894c5093f7a0f0 374a30641d Q.x = 42836f691d05211ebc65ef8fcf01e0fb6328ec9c4737c26050471e 50803022eb Q.y = 22cb4aaa555e23bd460262d2130d6a3c9207aa8bbb85060928beb2 63d6d42a95 msg = abc Faz-Hernandez, et al. Expires 14 May 2022 [Page 122] Internet-Draft hash-to-curve November 2021 P.x = 5f13cc69c891d86927eb37bd4afc6672360007c63f68a33ab423a3 aa040fd2a8 P.y = 67732d50f9a26f73111dd1ed5dba225614e538599db58ba30aaea1 f5c827fa42 u[0] = 09cfa30ad79bd59456594a0f5d3a76f6b71c6787b04de98be5cd20 1a556e253b Q.x = 333e41b61c6dd43af220c1ac34a3663e1cf537f996bab50ab66e33 c4bd8e4e19 Q.y = 51b6f178eb08c4a782c820e306b82c6e273ab22e258d972cd0c511 787b2a3443 msg = abcdef0123456789 P.x = 1dd2fefce934ecfd7aae6ec998de088d7dd03316aa1847198aecf6 99ba6613f1 P.y = 2f8a6c24dd1adde73909cada6a4a137577b0f179d336685c4a955a 0a8e1a86fb u[0] = 475ccff99225ef90d78cc9338e9f6a6bb7b17607c0c4428937de75 d33edba941 Q.x = 55186c242c78e7d0ec5b6c9553f04c6aeef64e69ec2e824472394d a32647cfc6 Q.y = 5b9ea3c265ee42256a8f724f616307ef38496ef7eba391c08f99f3 bea6fa88f0 msg = q128_qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqq P.x = 35fbdc5143e8a97afd3096f2b843e07df72e15bfca2eaf6879bf97 c5d3362f73 P.y = 2af6ff6ef5ebba128b0774f4296cb4c2279a074658b083b8dcca91 f57a603450 u[0] = 049a1c8bd51bcb2aec339f387d1ff51428b88d0763a91bcdf69298 14ac95d03d Q.x = 024b6e1621606dca8071aa97b43dce4040ca78284f2a527dcf5d0f bfac2b07e7 Q.y = 5102353883d739bdc9f8a3af650342b171217167dcce34f8db5720 8ec1dfdbf2 msg = a512_aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa P.x = 6e5e1f37e99345887fc12111575fc1c3e36df4b289b8759d23af14 Faz-Hernandez, et al. Expires 14 May 2022 [Page 123] Internet-Draft hash-to-curve November 2021 d774b66bff P.y = 2c90c3d39eb18ff291d33441b35f3262cdd307162cc97c31bfcc7a 4245891a37 u[0] = 3cb0178a8137cefa5b79a3a57c858d7eeeaa787b2781be4a362a2f 0750d24fa0 Q.x = 3e6368cff6e88a58e250c54bd27d2c989ae9b3acb6067f2651ad28 2ab8c21cd9 Q.y = 38fb39f1566ca118ae6c7af42810c0bb9767ae5960abb5a8ca7925 30bfb9447d J.6. curve448 J.6.1. curve448_XMD:SHA-512_ELL2_RO_ suite = curve448_XMD:SHA-512_ELL2_RO_ dst = QUUX-V01-CS02-with-curve448_XMD:SHA-512_ELL2_RO_ msg = P.x = da2332a516a063fef60267e6d89120bb999247ff7f52b313c8eee2 777e03320f30996a53280b6d8c3847cfb9ea565f46310e582a3733 4f69 P.y = dc7be59148778dbf9fbeaeaf2ce578b9b82f8d72fe8e7073ad92f2 77fb987b34711ba8571b30f89de8049d744ba3107399f2dc3c9d5c c8b2 u[0] = e06d3a0f99597cd9fa6ccb2c3db31d163e50940d2c7504e1bfba16 ac69c2a7cbb52df77f100c4e6908788b50ebfb7c47b2e96586ca59 b47b u[1] = 88267fb8a9a813556844b3ac7861b380ad7597ed0ef030be490274 54b83f441e34aee8682afabdae4f3deafaa894b15de9bd6af5059e f0ff Q0.x = 8219c3ff382cfe2f02a2a20f5ffd54564203edc7336022abc6b397 3ec7e61fc2d458a81846385080febb458695746c0ffc04e080b2fd ecf2 Q0.y = 9712f659ce8ddbd2bc581af3c6c359038d877174805b8772a647b3 b0bc9d66a579f72bc9ada3b836aaf2642d909ed9b96dc686ae668a b5c1 Q1.x = 2730fc1f5ea277c6ee5096eece84901d42fa3f78c018b1174c4685 e0be780f769933d28d29b13b330352353b9e1c98bb5ea6dabdf7e5 8e5a Q1.y = 5cb3a598ff66725b74c0e9f33e23b317a82a8bd6d1be02816688ef 74a5d704c14d09440f123573666e81a01cb19d91e25a4e98bab1f2 4668 msg = abc P.x = 126bdbaa7d8690fbf97447adf5b0ead68a48e3c75fb49d4ee584d9 7f08fadb3fd00d107455bd5a032c682d8a80b4f796960d61fc01e3 9faf P.y = d973b5f9d4babfb95e1e28484068fdd3314b2e334f8bfcbccb9878 Faz-Hernandez, et al. Expires 14 May 2022 [Page 124] Internet-Draft hash-to-curve November 2021 a1b9d0247bb4294c035caf1558c7d5fe140fb440fc32f7c4637f56 2db5 u[0] = c0f1c170cea7276b72c0e744f4b1d6974da6a57b50bf3e0551f208 c500a3797fd2279e9d19a3379fcb82ee31d22654645eb4e1440e0f 012a u[1] = f264aea0654f5055d2fbe15a00635fd8a93c90bde40f22632da6c0 cc2e62403261ebd0d21d08ef90704772b9f381f03d46a0a271fcde 22ee Q0.x = 97f4538634980c078431983bc90dd20bffaa3c7e3d0343742738eb 2f9a6a49357798a8900239e49b384e88acfeeaa4819de34b6be12b b583 Q0.y = cd0a7ca6b2bc2f3483ece0ae77307301fe8de23d31077f792ac7a6 bc6362178b22be3188ac29940e576d33f477be976aa1bd60272dde 8fae Q1.x = 7c03c44df3e7d00f16eb363e25573f1cbe229303deb83c4744df1c d1f8542748d41d8eb004fa7633752c8ca82c71e30de3dda8cbe542 3a9d Q1.y = eb29f9825c4f564a6b94bacf78e0eea3888597ee6a893cea9f5ebe 7ada5edbb1a1601a98124e3c3355ed413f9661089b5a11947685c4 371a msg = abcdef0123456789 P.x = 474e477439d7a9592a13094b38c54e25828c681c7c8b81ba2d54c8 0148a25be684ce2e8e25fc0149c10ed1f601bf0883ae16c364c6d3 fa63 P.y = 1989aff846273a5cc768b4624884c707f25d050b9dc9293cbe109c 019ee084d28de004b83ebb0ba175081792c91b721215df14dabf0d 6a70 u[0] = 5f09c4906a56cd7b4a620fbed243b8c1bffa30e58ca6c709273e63 e14e547f1fde8545e04b63e34d4bb2df6143ed0469172fee016040 3e9d u[1] = 8816db97222c7cf1a728ae635542497683c6959f042e51c62ba270 eb437d7e9cce9080e5c8e05b05e0b0a298823bd661d5272258c7f6 d76e Q0.x = e6d0c0fd0e30e1a27d9bbee310398e2461e9199000476e57d819be 2f343a32165bcc5e524c3feffca3b272bc801367b47032b6f5ae21 b514 Q0.y = 87ecc285bada245d203e67f6a5cf19876abdd4fb994b1d72f2bd21 5b19048721359eed24d2de20d344a06cb859445a726c4c76156160 13a7 Q1.x = feebcf52ae07ffc9c6665912c8e6ab14adf880dba9679d1862af6a 1eedc25b33c641af1cf8385771afec096701e247602e54c1e183ff d151 Q1.y = 47a95cb1361c02caeaac9d036b99bb11e733fa2f140c9831694057 eab0e7878b0b823891b02428ee2cb09627bb6cbf860bf72cad6b44 61c4 msg = q128_qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq Faz-Hernandez, et al. Expires 14 May 2022 [Page 125] Internet-Draft hash-to-curve November 2021 qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqq P.x = 9b12ca7cc0fbb013278d7ff476c64983baf7494ddec6c01e4df643 a3ca5aff2842c5122986ed4bb0ca9f9439d497fa8615747e1dfa5d 9788 P.y = 80c9ba33dc4b549b1e4342c4f7ed5bcb53e5a2caf6b322bf2518c1 3d27cfd7770200ace0af3f8f9c8fa5408b467661fd724cb770e5b1 27dc u[0] = 9f3656bf8698d3da48fe5920145fa6b38e4ac91891ae6fcc36f1dd 057a7e653d39eaade8f0009adbf3d47abfb3b4821d8331f50e8e85 2859 u[1] = 601f6005529f083544cd79bef793c8240fddb35ed35aa6a9d6bce0 3a782f59a020ce4ae7a89a2a1cb3fbbcfbdc8fd5228afba13b1776 3d22 Q0.x = 4c961329e6aae24e793c9579fb9035d36f02e634e5dc7010eb4cfd fdf394adb56a6b09ea63127ea1c14b5ff0eaa5e7d1be81832a9e1e b4ff Q0.y = b9afd57b25c525e193a4305b77bef349368f7e9248851a84db2a3b 0367098b30599fc5d396a93da7dfb1ad67cf2fdc8f9c988f95e422 d1ba Q1.x = 0147be9e6313e84dba7e9b86218e5ed2e280cd84bd2b9ab65f26dd 2010f8fc8ccb904884a8fc9c63ec6bb428c04bdc049cfae7ef920d 4249 Q1.y = 265038ec80101c9cec9fc10fc3c740ff9a0f27bd6969f6cbe1dbaa 11e4fbd86cc2971728bf0672f01daed6c639454c6a5c1e56303aa6 a6cf msg = a512_aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa P.x = 63f7b02ccfdcc3dda4c2f3162b3a29cce131bfec05c937f47fbea6 92da6c35611ea3b02056184530a7a16a266e24aa440201418650a9 b1d6 P.y = e7c55ef24be0109fd3a32e982809aa389e9583cc0861ff42e08b4d 1d261f50ddaa4887642aa80a3a7658a107159d0f60c6809bf052f2 feec u[0] = ec36b3337ec9b94b8a32a192617f278c72d89b0c31f19d184c29c2 a26c7d53ce880de19d980be7ba2c43451966a3e5b9b7b3496834e7 3868 u[1] = 8b5ff5b833a4655293cc8f8feb940bc84e8f2e240370a002457346 628a892cbb8abd96c1140ca044a1bca85181a0a4ffcaf8a42a95ef Faz-Hernandez, et al. Expires 14 May 2022 [Page 126] Internet-Draft hash-to-curve November 2021 f4c9 Q0.x = 3d94294cbb050707f5ad5a0dfc596445848c05a81d2a175a05b34e 409748cbe98970d1a0fd4bcdc030969481d669f0ce8befe72e0e7a 3506 Q0.y = 11718321d9ed5ce14aaa7d1d06f6a91ca3875eb2bef70e96b54aa2 51387629633860898faf3ec18ae47d5a6a0d605536435140d0f8e9 802d Q1.x = b283ab36f5dff4d54dca265e74ca355d751983fb013f458f44dcb6 b00302569788ed0a3567ca93be803e6a5aa883587e3a9da9368626 6ecf Q1.y = 7bebe0e4520198e026127cf8bd4308db737358afaca143788637c6 75812282336699de18e4b239e7c5e95797154b8cd00aa51b830993 9abc J.6.2. curve448_XMD:SHA-512_ELL2_NU_ suite = curve448_XMD:SHA-512_ELL2_NU_ dst = QUUX-V01-CS02-with-curve448_XMD:SHA-512_ELL2_NU_ msg = P.x = ea84fb65c9404271a743b99e734888d7d5170dee33421936180745 66b8c4faf0d751b46bcdcebbce41e44d101f93b098e150836eeb26 3d90 P.y = a8378c8c97c14c4127ebf6b36c9d4a6524be2b85ad76fd195315d3 d6eecf5147c9d96edb7f574935d0e945f6664040dbe0270d4ae24d ab64 u[0] = 89b0674b247b36697f028e39edb34bd9ee6ba968148447c80773ea 54650f5f57e005f69898502ea754f3dd710562cf80f347296b15f2 b040 Q.x = e57024bb58651499c5e87dffb879b0bd3abfbfa9f5af2962c2597c 61cc24e2ad7a2802d5f98bc6265ea54e7b83befb8c59afd0854f5e bc09 Q.y = 0f2a66e25fba03deb43daf0dc694d6265e0f426f041a0bc5970206 871f88a0a09b0463607ff6ac94cb3609ed74d7eb9e7842a7b5f652 89c2 msg = abc P.x = 2e7014413676426069da399013d0a825ea436f6036fc895099838d 0c2e047b69a8c98b2b5e5a5e1d203bc58829141bbcc1bbf66d5d7c eb85 P.y = 843b3542bd5c3175fa8a1160b0f5b3ce54f9650a18b0b8f02b83b6 2f4adfda146b0ab04bf902fb098459d0cf2171c640f003df8d79ee e4db u[0] = 413957bbc65b091215af8af48ad7e24bf048e9f6d9a73aec17e998 a2b51cfc4ccdb4c25693e764db7799619f163532ec1ce5692e1753 0384 Q.x = 8e4ff7f3bf4e42202441a441da4afaa6d4f32f95d0406742172e88 af8ffa304022eb3d2fddc5cffbb0241466daf3f152fddc26184074 Faz-Hernandez, et al. Expires 14 May 2022 [Page 127] Internet-Draft hash-to-curve November 2021 12fb Q.y = 22e7d04d6dabbf15eee6529bf1f6a3a9efaa3a772956de1c08441d 94ef63b163ecea2065aebb004e8cfc12cfd2f946de05277caabba1 4a96 msg = abcdef0123456789 P.x = fec43c1455df411dcb549c6cd3c25915bc73b1bb1655b164b98298 c557dc1bf6f33791a43d167375cd645a51a13e34e645d0f5a05def b6a5 P.y = b9ede648411291433b4ecb333869315db05522a26c34ad87a73d52 3fa34f77b6ae8299d992ad5dc5d0d08f708975d19124168dd7d840 f7e4 u[0] = fe602896e4a559685e0deb1c7c8ff4eec02cda9037bb6bd009d003 31b0b51227b64704e2462fd9cad0ebc24cb8817c5758703f720125 1c94 Q.x = 1f1aefee6f36a913a0c63a69439400e00d6b900554489e3a25eaa6 6d4b68cafd0dc63e5645701b37edd7535b1b38305efb0460d8f103 616a Q.y = 2e8d502126f2ab8c125f8213c7a5fed8ca59857a8c9bd2955f938f 4beb89b57f3edbb823bdc4fa94e15e9bbf9f644137bf865ec7659b deb9 msg = q128_qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqq P.x = b1e9ec646a3d80bbb9fe4893ad6499835d926141892deae28a1099 b7d3c8bf1b973babd5cdc59cbb1740ff3818369573e7005b5d8abe 84af P.y = 73268f982e98e27091ac79d1c20f5cf0a678bf56f990788427306d ef1dd5a6cbf49605614429cb7948b09a19028d09c38fc6ecc28dd5 43fb u[0] = 8fb83575d02983a80e69abf9592bb4fcb2f7d0a15c3899b569d276 798d0dee60d81646dfef632f529770d162e045935e63f73a586157 5366 Q.x = 7c5cd86fe37dc942089a3e4983b71173ceb37353fe1e0f1b5d207b 3183b595348a72860b3343ab20b3b9d6dd5ec5b8a36817e1f0ec1b 6dba Q.y = 2bd01a61674d241f1e5c7dab88028b3055cf849ede4c6781c7c5fc f22ad7e65bbcf0cb2cbee89b1caeb6f616c81346564a00611f4731 f9dc msg = a512_aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa Faz-Hernandez, et al. Expires 14 May 2022 [Page 128] Internet-Draft hash-to-curve November 2021 aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa P.x = 1953f9cb8340b7968c1d820fad943cb58d132ec5db85a3bc22a408 b834f4d14fae0826164d92f285a71977ee1ca21c48fb08ecccf71f b43f P.y = 2a5ebf1f4082bb8fa4ead0a54bb9b9d820018b06c2a9c81be048bc 6cae60fb99099cbcb9daf82a88cb177be328283f96f6e623af362a 6165 u[0] = 264d09a96a80db8aac3b51d54f7f115dfe3a615e85713f2d4d4bf6 2c47ce0e8ebe261fc3a281166e9c25ea689010639f8131ffb6d8c0 d5c9 Q.x = 62d81718b0b327cf3b0dd77885de6cf3202bc2c0a20bbd3af18127 16104fb5a39878dbe92862f40c28e1ae078a0b8c25ba23c9bca5fa ed8a Q.y = 6505d475959b61e33f9e4c2c6d8033b5dbd09432993997c07ae2c2 4a31d7a35c02e15f357aebf0178b7b8525074b2400ec3d1bb8eda1 b9ee J.7. edwards448 J.7.1. edwards448_XMD:SHA-512_ELL2_RO_ suite = edwards448_XMD:SHA-512_ELL2_RO_ dst = QUUX-V01-CS02-with-edwards448_XMD:SHA-512_ELL2_RO_ msg = P.x = 132e90a3f7110a41ad796ef3e83baff6839d5569854f5a11c16b2a 7a26d86e32a6bcdc93e954aaea197768d04091aab18f5779ae4f85 9d18 P.y = 12165b672a52370f21268ce2c261c3ed32adac6a3404fbd50d31a5 51a9111d109b0c691868aa6e9d92fd94dfa2a7397a2c111be9c432 c0cd u[0] = 6369f90cf2806a86841398114dccf2ffd06d2d57c782a449df5297 189de02384b22cf73a0b5f678fe486ad59280d15431f4a65fec93d 0039 u[1] = a6b424f6f6c995dec127ac862ebc93b82ab3604087d70a78189054 b09be6c9e76ab85ee8351870bfd93dca7607fe864dac096e3f333a 145b Q0.x = bb132e877b5fe35b189ee7041c53f03d943d36ea7265dce2c267cc 5500f281a2f8981c393a05e94962f8d8017ec6ad3a6b34cbad3480 6c12 Q0.y = 15e1762b3c85a31ba3a9b7ba52eba6c7f02d5a802068afa8935e4d 20ffa3e0356cf57398b8b9065554c3036a6f17e48b1f2ee0e5615c 5d73 Q1.x = cc44d47637c8fcbe1dc1020b87eb21205f055d6dec872efcc09b8c 50935cdb38342340f389d6284a2b4bc5486f8c05a3fec6c7ef59c0 2e2f Faz-Hernandez, et al. Expires 14 May 2022 [Page 129] Internet-Draft hash-to-curve November 2021 Q1.y = c875e0cb547828c2707097c2cb6e3d7f03f92a715d6c66c370b08d 09be11b69cc07c4798ff3ed30e64b5c1742bed375bb51b61963617 9adb msg = abc P.x = bdb13f312a5d478f57bc852b743acd3ce51dd2de96f181ff88556f 2b41d568ed64c2b210286be54b7c84d23cbc09a01902172a903b9c 8c1e P.y = 6165182fc928064e375777e700f44e56b54b5980a42c72f747ba95 829bd8ff80de5fd159e149a8169397029553bd89b5c28df416e10c b36b u[0] = 4c3b600dba4a986944840b32e154510af806aa095f856fdf8fe320 1221314704aba830579f3da5e30f3300e89814324a0bef26a9cd3a b6a0 u[1] = f0ae99c47274ed37fc582a7edc75aa0ddfdb7ed77e7eebbc293dca 8312a6ff43e7c34c6796f9fe2a21c337deab5523d4825aebde5f14 1869 Q0.x = 46b89d09965812c09af0f484a1262246b14f25e8f68a34678302ca 76e461968c61f6ae1d4d7f32d930bbe153f2a02085a12a15e3e11b 5af8 Q0.y = d1cfb0fb4572b51b0c4e27d574adb4fd5a3276e4c59709c732ef1b 7ce9e59fd87df98a79c65fb709110a3922ffdf8b4593c3ed9caf96 6e9e Q1.x = 94558aaac183cb8f263caa55eb79d1c4d44f681d2c9e2efc23927b 6e272b7e3261f5d178dc166a6724aa2b2b5abfd4a97c6fc38ce423 998f Q1.y = 1ee82f88f4e0b552b1a2bf048de95f8196d2f565de8edc1ae94d9f 8fc8fd1a0d193f9bd58833b3bbab5ab97e85ae758b5fa01eadcb0b 9ced msg = abcdef0123456789 P.x = 01a04ed1b758a20245ea227827350eaedef92ef2860e58e5c5a982 0a1fc6157082b3722d25c9754bc2642b126c4a9188a1fe9c8b2b39 6b53 P.y = 88ac7496b9ebc2b446695324e2f76ad54f4b8b21d0077ac15b69ad 7a4abcd00e881cadbd829db2d2d0f28fc84eac4fc59264b2d14063 e770 u[0] = f3c7984b7cbaf248dacc25599c8ac774782c5a3ed7bd24343fb935 602469b76541f8fd54dd78a7c54b6d991e17ec416742f92a18e8e6 805b u[1] = 98fb498e5f6c40762899444a200e052ab25d336b84571f27670c7b cea54efc78f167a770275179dad42b6df0e3cc477abdf115f0b5f1 cc49 Q0.x = cc2286ee7e052ffccbc10607bfa8bb7ced94924dfc6fee6f34eedd fa559000f8d0d238c14cbe2e3509b39843e5711583982a80f652dd e738 Q0.y = 774c9a9f0f4143ca2779b4bbb4ef397da643db7b74ac8bad1a54fb c86f728571fc401ca5b2a8432a018a5d545b526387f364701b232e Faz-Hernandez, et al. Expires 14 May 2022 [Page 130] Internet-Draft hash-to-curve November 2021 3dbd Q1.x = e0fa0e6ff453760f52ea77ae9d605d00fc9930e9302a7779243fd6 5e29970c3ebc26799a17ce7a24b7ea18f2c5b572fc29d6dbf73a89 672a Q1.y = 6b5b06e9ab686c8377ac404f142444ded5adcd68daafde1bdfa7ef 671708636da22adba4fda2b2472e37cddebec47fe30e38a11dcd06 d10d msg = q128_qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqq P.x = 547a8928866ea7fdd556f9fe228eb414e3337338608bbf5746c1cf d39ad5ffa1a8f7c5c9dbd39043484a820453ee518801c38a7ec83f f69d P.y = d8b9295825b37781ba0503242951154c31fc9457f6b2a67ec5cde8 ea577bd8214bc5525d1df3171202ef43782a7fc72982069c858357 519b u[0] = 2d1aa960178c6b26f0c40648d9c8072640b449182b55b72446980b 7cd78b81c6dd0a4195564546d0633934e74b9d73057954a77820af 5ff7 u[1] = 138d93e220a6c35b7afbc2329eb7e5d027f2e64551c41c0550a378 143ae0a6b49855cf9ce5886cfb8dfc6bff6cb62139161e90868359 b566 Q0.x = 39ffc36783c6db4113f60c6754398c95fec611f4a8e05fe479e7f0 df2f89556f8090bb713b57f51e3e68b02670390054656e7cbf06ea bd1e Q0.y = 1ef77f01c770dae258a5af1d08ed7f72311487e4aa273c8ce08f89 808de5ced5ff83241199265bab4cea11a9d4ea0feed61858b4573f 6916 Q1.x = c7b100d6c0a39411a7d3fa37d0e1a1673d3c19af5cc721db2f1342 607e5d860d96fab2313e0f4987f839baa3293c8a4d6fa2f30e51ba fa7d Q1.y = f6a2294965922a98247d066926b265cfc168c9a66861c07f4ac867 5cab13a1486d1d614ac8a56095531ebc4f8233e9d5942436d6d61e 5a1c msg = a512_aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa P.x = c7c7899dab9813b5d10cadf43aa0cf5f26c21e98e5c0ffbe690059 67dbfd066e5564bdc6c67adeb87f55951c4a077c8d4a6095a23559 Faz-Hernandez, et al. Expires 14 May 2022 [Page 131] Internet-Draft hash-to-curve November 2021 3201 P.y = 547c2293a8b8489270957118e69a241227dc32c5587819269630ff 687bb88284b88376fa8f86e0dbafc09d8dbba49a4a433fbd379d10 3c5c u[0] = 967a9f376c20a33d6a041ed783fd7a6b7fba24ab92029bc69340a2 6441c287d368b8496fb14d19631ff406e057bcf8e8553660e7e73d 561d u[1] = 78d710dd21fedeb0dc93ec424e17e879935a689aa2c14b207a0055 f1e57a44d83929731ca74fcca2fd0cb6bbaab12265fe104dd6fae4 87a5 Q0.x = 5f238a69135b8207607ea4dcebed925425adb75dfdb86e6e7f1279 6f2086cba1add20f36a2e38176dd4feda1ad25aa8657728d390ce4 b818 Q0.y = 225a31f6a260be2b81b735ea1bdaaa65486a50777e3e3b83265e69 a572e2054e63f0519a06537fba0d15076c993c6e19121fd077e8e4 5ee9 Q1.x = a1d4f1d07717adbcc3926caaf2d731532c2065c191dfca7b60a5e1 499a1c237985807b441004b5476c896105806f8157e362ef96f16e 7293 Q1.y = e320a16c1417d41241f03507ca0ea48d0e494cfd8bb067107bc18a 939c69581ecaee71a5f656fcd091688391f41ca1b23b3fad4876e6 8e01 J.7.2. edwards448_XMD:SHA-512_ELL2_NU_ suite = edwards448_XMD:SHA-512_ELL2_NU_ dst = QUUX-V01-CS02-with-edwards448_XMD:SHA-512_ELL2_NU_ msg = P.x = 9edd52909ac5f8d4506149c30e1ea8709eed77c409d3ba2b3834a9 18c4d7bf47cb11c464847fc5edfc4ec5dcb6e2e1c4a4bf2cf93914 44af P.y = 6d73a9acd7c51479d59f7aab60bd0090ec44fe64d82ffea0ccff18 ac5060632be1c44219adef88937e2f28ebab0b4edf16c501b6ae83 7409 u[0] = d244401e5d2f510c21944e96203cff15813d57c3f40b1a15bd73e1 d9ac031966137e39e1111cf46e590ed4726ea9c96616a581a57cb4 6010 Q.x = ce981eeb6c73a7edba5c4c29af37010c398f1dbe39fe00be52100e b7c107f71793ce5928c2a34ce7fc37e054838d2788c46abad5b1f7 009f Q.y = e822cb7fa3f0e97e1215619f13ed7fde59137dc807a37ffbb7a375 7f948f3fb50168c7fd6a5077a1fe7e6f484ba4881c964e5fff2b99 e9af msg = abc P.x = 2df5a5ee45640cc4e297f969c9771b36e4358463d47a530e375fe1 3d442a17cc5f27818365eead72adee48c5911eb4ad7ed4e242f81d Faz-Hernandez, et al. Expires 14 May 2022 [Page 132] Internet-Draft hash-to-curve November 2021 37b8 P.y = a1ba8d111dd6338338f88c552d3338077fc5fe037860f2eb2b7496 6a4a96f7fc59f859518caba87a1cf3fefbcace608e7b651ab5cb9e 1eed u[0] = 2f3bcc6253b74569ea64a9d90ea4ee52c71bd168241eaa88c6f458 93ffe00fe9888f2966a6308ce194cff35059d09cf59c68f3b961d0 fd17 Q.x = 4dc176c8c7cacf550289ae7cba158307880fa606e7bdb607d976dd 398e60088bf1316e193b56784ea4a295884176efee84a500d46003 7e1f Q.y = 711909a05b086efe857ec86729a667a531d958d17f5f7a4b9d9fa1 59d3d7668d74ae4f4dd1b7273e2a26c68029c881255e462d046f89 ebe0 msg = abcdef0123456789 P.x = 7120e8a2d8f4c34743b8afc1ec9f72fee0e9fd60d25f776fabc9a8 bd5c3ea1d5797653b782483cad1af66e41acf04fa893844e0986be 739c P.y = 6098bc3d041ba1d31074e9ca1b788c42c9db695f13ddda8d3e8275 478d54fc4ba754c2b397ddafc021810a4068a2b3f6e6e08cfd7787 ebd6 u[0] = b9e80bb5fc08b259cd3b8e9f9ab722b427becf3797b2316ec94e2b 6ff8286908fd2c70470c1290939ab14bda375185054ea4205e0ea0 23ca Q.x = c7c2691fbb36a69fc3a04970d2c9881dbede0b68a69ba7c92b9160 112293d97103b360cdca664a47ebdc31b7b60c6d5bebb6c5691a46 c1c8 Q.y = 8c9e09b8a17559a59cf3071f48e68c5d1fa1836f66c59bc3b76df0 9a62df5ebb5c0ac1a6f6f71e51f876241b3e7d459f45b142891057 eb63 msg = q128_qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqq P.x = 1fe7630fdd70fcc20ef2463e07ab5ec990c8a60a625de07b8114d4 d960d1ab3c3ad5e8d4e98d991bb35a4c257f31d8e9f7951112172f 5a3b P.y = b81d594af4ed69062aa35c2a3ca5f8fbbcc1ea13a893ff79873958 b48a79dde9b2a9422376d66d1bb1321c6bb90e18ae413c4669e16b 63c7 u[0] = 9849d55e9866d3ffb6f2bcbd45dfba339fc02138f07b30b669b088 be9a8d17bcb639943fcd2e920f6eda62f32b1aca5c871d93013cdf 1d01 Q.x = cd18e60afd20ce97349af7732e50f2d8da587187dbabf767abbffc be08abef5c60f300a502f8d4b841c6076e218a301fad6c58da3b81 61f5 Q.y = 6dac55c4fdde9118fb8f64d02b1268467fe709eac8b84d2cef0db2 c786b51916fdccdc7d0999ab073b1e743603dfba27425d282570bb Faz-Hernandez, et al. Expires 14 May 2022 [Page 133] Internet-Draft hash-to-curve November 2021 6373 msg = a512_aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa P.x = e6856da5667f5083bddc84e447578522373fe36b45af60245d5e40 adbe2c1934918a1d2071a1826dbbaed45bbaef0d888a4ffd513281 26a0 P.y = 3e2108cc4d0a88c5bceb7a0ac983b1f3b224efa288f59178f2cc30 615635d1269be84482cc443035bfae0a01f25a262b3b650c778162 4c1e u[0] = 74e63d0db5942bc43c2505caf4b71eb5a826ce0f3f4f23ba2bbbd0 9948a0b0eee887163a93a90566b456c687bb8671ef2e2fe4e51ebd 8def Q.x = 60d1975f6f5b78a8e6edf580958acead694b229a7b35e79b1f214d e76b2391d3fa39ccb1b0d19d719f37f91fc191c554b97acff501af 6fac Q.y = 5dbd9a91c9865928da7fffcd548ad4da813539d349aa40798fed5d 33c872ca56e6545ca0b851eba4b1b05cf9a710b15c94de700a6d6d d1c8 J.8. secp256k1 J.8.1. secp256k1_XMD:SHA-256_SSWU_RO_ suite = secp256k1_XMD:SHA-256_SSWU_RO_ dst = QUUX-V01-CS02-with-secp256k1_XMD:SHA-256_SSWU_RO_ msg = P.x = c1cae290e291aee617ebaef1be6d73861479c48b841eaba9b7b585 2ddfeb1346 P.y = 64fa678e07ae116126f08b022a94af6de15985c996c3a91b64c406 a960e51067 u[0] = 6b0f9910dd2ba71c78f2ee9f04d73b5f4c5f7fc773a701abea1e57 3cab002fb3 u[1] = 1ae6c212e08fe1a5937f6202f929a2cc8ef4ee5b9782db68b0d579 9fd8f09e16 Q0.x = 74519ef88b32b425a095e4ebcc84d81b64e9e2c2675340a720bb1a 1857b99f1e Q0.y = c174fa322ab7c192e11748beed45b508e9fdb1ce046dee9c2cd3a2 a86b410936 Faz-Hernandez, et al. Expires 14 May 2022 [Page 134] Internet-Draft hash-to-curve November 2021 Q1.x = 44548adb1b399263ded3510554d28b4bead34b8cf9a37b4bd0bd2b a4db87ae63 Q1.y = 96eb8e2faf05e368efe5957c6167001760233e6dd2487516b46ae7 25c4cce0c6 msg = abc P.x = 3377e01eab42db296b512293120c6cee72b6ecf9f9205760bd9ff1 1fb3cb2c4b P.y = 7f95890f33efebd1044d382a01b1bee0900fb6116f94688d487c6c 7b9c8371f6 u[0] = 128aab5d3679a1f7601e3bdf94ced1f43e491f544767e18a4873f3 97b08a2b61 u[1] = 5897b65da3b595a813d0fdcc75c895dc531be76a03518b044daaa0 f2e4689e00 Q0.x = 07dd9432d426845fb19857d1b3a91722436604ccbbbadad8523b8f c38a5322d7 Q0.y = 604588ef5138cffe3277bbd590b8550bcbe0e523bbaf1bed4014a4 67122eb33f Q1.x = e9ef9794d15d4e77dde751e06c182782046b8dac05f8491eb88764 fc65321f78 Q1.y = cb07ce53670d5314bf236ee2c871455c562dd76314aa41f012919f e8e7f717b3 msg = abcdef0123456789 P.x = bac54083f293f1fe08e4a70137260aa90783a5cb84d3f35848b324 d0674b0e3a P.y = 4436476085d4c3c4508b60fcf4389c40176adce756b398bdee27bc a19758d828 u[0] = ea67a7c02f2cd5d8b87715c169d055a22520f74daeb080e6180958 380e2f98b9 u[1] = 7434d0d1a500d38380d1f9615c021857ac8d546925f5f2355319d8 23a478da18 Q0.x = 576d43ab0260275adf11af990d130a5752704f7947862876172080 8862544b5d Q0.y = 643c4a7fb68ae6cff55edd66b809087434bbaff0c07f3f9ec4d49b b3c16623c3 Q1.x = f89d6d261a5e00fe5cf45e827b507643e67c2a947a20fd9ad71039 f8b0e29ff8 Q1.y = b33855e0cc34a9176ead91c6c3acb1aacb1ce936d563bc1cee1dcf fc806caf57 msg = q128_qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqq P.x = e2167bc785333a37aa562f021f1e881defb853839babf52a7f72b1 02e41890e9 P.y = f2401dd95cc35867ffed4f367cd564763719fbc6a53e969fb8496a 1e6685d873 Faz-Hernandez, et al. Expires 14 May 2022 [Page 135] Internet-Draft hash-to-curve November 2021 u[0] = eda89a5024fac0a8207a87e8cc4e85aa3bce10745d501a30deb873 41b05bcdf5 u[1] = dfe78cd116818fc2c16f3837fedbe2639fab012c407eac9dfe9245 bf650ac51d Q0.x = 9c91513ccfe9520c9c645588dff5f9b4e92eaf6ad4ab6f1cd720d1 92eb58247a Q0.y = c7371dcd0134412f221e386f8d68f49e7fa36f9037676e163d4a06 3fbf8a1fb8 Q1.x = 10fee3284d7be6bd5912503b972fc52bf4761f47141a0015f1c6ae 36848d869b Q1.y = 0b163d9b4bf21887364332be3eff3c870fa053cf508732900fc69a 6eb0e1b672 msg = a512_aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa P.x = e3c8d35aaaf0b9b647e88a0a0a7ee5d5bed5ad38238152e4e6fd8c 1f8cb7c998 P.y = 8446eeb6181bf12f56a9d24e262221cc2f0c4725c7e3803024b588 8ee5823aa6 u[0] = 8d862e7e7e23d7843fe16d811d46d7e6480127a6b78838c277bca1 7df6900e9f u[1] = 68071d2530f040f081ba818d3c7188a94c900586761e9115efa47a e9bd847938 Q0.x = b32b0ab55977b936f1e93fdc68cec775e13245e161dbfe556bbb1f 72799b4181 Q0.y = 2f5317098360b722f132d7156a94822641b615c91f8663be691698 70a12af9e8 Q1.x = 148f98780f19388b9fa93e7dc567b5a673e5fca7079cd9cdafd719 82ec4c5e12 Q1.y = 3989645d83a433bc0c001f3dac29af861f33a6fd1e04f4b36873f5 bff497298a J.8.2. secp256k1_XMD:SHA-256_SSWU_NU_ suite = secp256k1_XMD:SHA-256_SSWU_NU_ dst = QUUX-V01-CS02-with-secp256k1_XMD:SHA-256_SSWU_NU_ msg = P.x = a4792346075feae77ac3b30026f99c1441b4ecf666ded19b7522cf 65c4c55c5b Faz-Hernandez, et al. Expires 14 May 2022 [Page 136] Internet-Draft hash-to-curve November 2021 P.y = 62c59e2a6aeed1b23be5883e833912b08ba06be7f57c0e9cdc663f 31639ff3a7 u[0] = 0137fcd23bc3da962e8808f97474d097a6c8aa2881fceef4514173 635872cf3b Q.x = a4792346075feae77ac3b30026f99c1441b4ecf666ded19b7522cf 65c4c55c5b Q.y = 62c59e2a6aeed1b23be5883e833912b08ba06be7f57c0e9cdc663f 31639ff3a7 msg = abc P.x = 3f3b5842033fff837d504bb4ce2a372bfeadbdbd84a1d2b678b6e1 d7ee426b9d P.y = 902910d1fef15d8ae2006fc84f2a5a7bda0e0407dc913062c3a493 c4f5d876a5 u[0] = e03f894b4d7caf1a50d6aa45cac27412c8867a25489e32c5ddeb50 3229f63a2e Q.x = 3f3b5842033fff837d504bb4ce2a372bfeadbdbd84a1d2b678b6e1 d7ee426b9d Q.y = 902910d1fef15d8ae2006fc84f2a5a7bda0e0407dc913062c3a493 c4f5d876a5 msg = abcdef0123456789 P.x = 07644fa6281c694709f53bdd21bed94dab995671e4a8cd1904ec4a a50c59bfdf P.y = c79f8d1dad79b6540426922f7fbc9579c3018dafeffcd4552b1626 b506c21e7b u[0] = e7a6525ae7069ff43498f7f508b41c57f80563c1fe4283510b3224 46f32af41b Q.x = 07644fa6281c694709f53bdd21bed94dab995671e4a8cd1904ec4a a50c59bfdf Q.y = c79f8d1dad79b6540426922f7fbc9579c3018dafeffcd4552b1626 b506c21e7b msg = q128_qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqq P.x = b734f05e9b9709ab631d960fa26d669c4aeaea64ae62004b9d34f4 83aa9acc33 P.y = 03fc8a4a5a78632e2eb4d8460d69ff33c1d72574b79a35e402e801 f2d0b1d6ee u[0] = d97cf3d176a2f26b9614a704d7d434739d194226a706c886c5c3c3 9806bc323c Q.x = b734f05e9b9709ab631d960fa26d669c4aeaea64ae62004b9d34f4 83aa9acc33 Q.y = 03fc8a4a5a78632e2eb4d8460d69ff33c1d72574b79a35e402e801 f2d0b1d6ee msg = a512_aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa Faz-Hernandez, et al. Expires 14 May 2022 [Page 137] Internet-Draft hash-to-curve November 2021 aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa P.x = 17d22b867658977b5002dbe8d0ee70a8cfddec3eec50fb93f36136 070fd9fa6c P.y = e9178ff02f4dab73480f8dd590328aea99856a7b6cc8e5a6cdf289 ecc2a51718 u[0] = a9ffbeee1d6e41ac33c248fb3364612ff591b502386c1bf6ac4aaf 1ea51f8c3b Q.x = 17d22b867658977b5002dbe8d0ee70a8cfddec3eec50fb93f36136 070fd9fa6c Q.y = e9178ff02f4dab73480f8dd590328aea99856a7b6cc8e5a6cdf289 ecc2a51718 J.9. BLS12-381 G1 J.9.1. BLS12381G1_XMD:SHA-256_SSWU_RO_ suite = BLS12381G1_XMD:SHA-256_SSWU_RO_ dst = QUUX-V01-CS02-with-BLS12381G1_XMD:SHA-256_SSWU_RO_ msg = P.x = 052926add2207b76ca4fa57a8734416c8dc95e24501772c8142787 00eed6d1e4e8cf62d9c09db0fac349612b759e79a1 P.y = 08ba738453bfed09cb546dbb0783dbb3a5f1f566ed67bb6be0e8c6 7e2e81a4cc68ee29813bb7994998f3eae0c9c6a265 u[0] = 0ba14bd907ad64a016293ee7c2d276b8eae71f25a4b941eece7b0d 89f17f75cb3ae5438a614fb61d6835ad59f29c564f u[1] = 019b9bd7979f12657976de2884c7cce192b82c177c80e0ec604436 a7f538d231552f0d96d9f7babe5fa3b19b3ff25ac9 Q0.x = 11a3cce7e1d90975990066b2f2643b9540fa40d6137780df4e753a 8054d07580db3b7f1f03396333d4a359d1fe3766fe Q0.y = 0eeaf6d794e479e270da10fdaf768db4c96b650a74518fc67b04b0 3927754bac66f3ac720404f339ecdcc028afa091b7 Q1.x = 160003aaf1632b13396dbad518effa00fff532f604de1a7fc2082f f4cb0afa2d63b2c32da1bef2bf6c5ca62dc6b72f9c Q1.y = 0d8bb2d14e20cf9f6036152ed386d79189415b6d015a20133acb4e 019139b94e9c146aaad5817f866c95d609a361735e msg = abc P.x = 03567bc5ef9c690c2ab2ecdf6a96ef1c139cc0b2f284dca0a9a794 3388a49a3aee664ba5379a7655d3c68900be2f6903 Faz-Hernandez, et al. Expires 14 May 2022 [Page 138] Internet-Draft hash-to-curve November 2021 P.y = 0b9c15f3fe6e5cf4211f346271d7b01c8f3b28be689c8429c85b67 af215533311f0b8dfaaa154fa6b88176c229f2885d u[0] = 0d921c33f2bad966478a03ca35d05719bdf92d347557ea166e5bba 579eea9b83e9afa5c088573c2281410369fbd32951 u[1] = 003574a00b109ada2f26a37a91f9d1e740dffd8d69ec0c35e1e9f4 652c7dba61123e9dd2e76c655d956e2b3462611139 Q0.x = 125435adce8e1cbd1c803e7123f45392dc6e326d292499c2c45c58 65985fd74fe8f042ecdeeec5ecac80680d04317d80 Q0.y = 0e8828948c989126595ee30e4f7c931cbd6f4570735624fd25aef2 fa41d3f79cfb4b4ee7b7e55a8ce013af2a5ba20bf2 Q1.x = 11def93719829ecda3b46aa8c31fc3ac9c34b428982b898369608e 4f042babee6c77ab9218aad5c87ba785481eff8ae4 Q1.y = 0007c9cef122ccf2efd233d6eb9bfc680aa276652b0661f4f820a6 53cec1db7ff69899f8e52b8e92b025a12c822a6ce6 msg = abcdef0123456789 P.x = 11e0b079dea29a68f0383ee94fed1b940995272407e3bb916bbf26 8c263ddd57a6a27200a784cbc248e84f357ce82d98 P.y = 03a87ae2caf14e8ee52e51fa2ed8eefe80f02457004ba4d486d6aa 1f517c0889501dc7413753f9599b099ebcbbd2d709 u[0] = 062d1865eb80ebfa73dcfc45db1ad4266b9f3a93219976a3790ab8 d52d3e5f1e62f3b01795e36834b17b70e7b76246d4 u[1] = 0cdc3e2f271f29c4ff75020857ce6c5d36008c9b48385ea2f2bf6f 96f428a3deb798aa033cd482d1cdc8b30178b08e3a Q0.x = 08834484878c217682f6d09a4b51444802fdba3d7f2df9903a0dda db92130ebbfa807fffa0eabf257d7b48272410afff Q0.y = 0b318f7ecf77f45a0f038e62d7098221d2dbbca2a394164e2e3fe9 53dc714ac2cde412d8f2d7f0c03b259e6795a2508e Q1.x = 158418ed6b27e2549f05531a8281b5822b31c3bf3144277fbb977f 8d6e2694fedceb7011b3c2b192f23e2a44b2bd106e Q1.y = 1879074f344471fac5f839e2b4920789643c075792bec5af4282c7 3f7941cda5aa77b00085eb10e206171b9787c4169f msg = q128_qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqq P.x = 15f68eaa693b95ccb85215dc65fa81038d69629f70aeee0d0f677c f22285e7bf58d7cb86eefe8f2e9bc3f8cb84fac488 P.y = 1807a1d50c29f430b8cafc4f8638dfeeadf51211e1602a5f184443 076715f91bb90a48ba1e370edce6ae1062f5e6dd38 u[0] = 010476f6a060453c0b1ad0b628f3e57c23039ee16eea5e71bb87c3 b5419b1255dc0e5883322e563b84a29543823c0e86 u[1] = 0b1a912064fb0554b180e07af7e787f1f883a0470759c03c1b6509 eb8ce980d1670305ae7b928226bb58fdc0a419f46e Q0.x = 0cbd7f84ad2c99643fea7a7ac8f52d63d66cefa06d9a56148e58b9 84b3dd25e1f41ff47154543343949c64f88d48a710 Q0.y = 052c00e4ed52d000d94881a5638ae9274d3efc8bc77bc0e5c650de 04a000b2c334a9e80b85282a00f3148dfdface0865 Faz-Hernandez, et al. Expires 14 May 2022 [Page 139] Internet-Draft hash-to-curve November 2021 Q1.x = 06493fb68f0d513af08be0372f849436a787e7b701ae31cb964d96 8021d6ba6bd7d26a38aaa5a68e8c21a6b17dc8b579 Q1.y = 02e98f2ccf5802b05ffaac7c20018bc0c0b2fd580216c4aa2275d2 909dc0c92d0d0bdc979226adeb57a29933536b6bb4 msg = a512_aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa P.x = 082aabae8b7dedb0e78aeb619ad3bfd9277a2f77ba7fad20ef6aab dc6c31d19ba5a6d12283553294c1825c4b3ca2dcfe P.y = 05b84ae5a942248eea39e1d91030458c40153f3b654ab7872d779a d1e942856a20c438e8d99bc8abfbf74729ce1f7ac8 u[0] = 0a8ffa7447f6be1c5a2ea4b959c9454b431e29ccc0802bc052413a 9c5b4f9aac67a93431bd480d15be1e057c8a08e8c6 u[1] = 05d487032f602c90fa7625dbafe0f4a49ef4a6b0b33d7bb349ff4c f5410d297fd6241876e3e77b651cfc8191e40a68b7 Q0.x = 0cf97e6dbd0947857f3e578231d07b309c622ade08f2c08b32ff37 2bd90db19467b2563cc997d4407968d4ac80e154f8 Q0.y = 127f0cddf2613058101a5701f4cb9d0861fd6c2a1b8e0afe194fcc f586a3201a53874a2761a9ab6d7220c68661a35ab3 Q1.x = 092f1acfa62b05f95884c6791fba989bbe58044ee6355d100973bf 9553ade52b47929264e6ae770fb264582d8dce512a Q1.y = 028e6d0169a72cfedb737be45db6c401d3adfb12c58c619c82b93a 5dfcccef12290de530b0480575ddc8397cda0bbebf J.9.2. BLS12381G1_XMD:SHA-256_SSWU_NU_ suite = BLS12381G1_XMD:SHA-256_SSWU_NU_ dst = QUUX-V01-CS02-with-BLS12381G1_XMD:SHA-256_SSWU_NU_ msg = P.x = 184bb665c37ff561a89ec2122dd343f20e0f4cbcaec84e3c3052ea 81d1834e192c426074b02ed3dca4e7676ce4ce48ba P.y = 04407b8d35af4dacc809927071fc0405218f1401a6d15af775810e 4e460064bcc9468beeba82fdc751be70476c888bf3 u[0] = 156c8a6a2c184569d69a76be144b5cdc5141d2d2ca4fe341f011e2 5e3969c55ad9e9b9ce2eb833c81a908e5fa4ac5f03 Q.x = 11398d3b324810a1b093f8e35aa8571cced95858207e7f49c4fd74 656096d61d8a2f9a23cdb18a4dd11cd1d66f41f709 Q.y = 19316b6fb2ba7717355d5d66a361899057e1e84a6823039efc7bec cefe09d023fb2713b1c415fcf278eb0c39a89b4f72 Faz-Hernandez, et al. Expires 14 May 2022 [Page 140] Internet-Draft hash-to-curve November 2021 msg = abc P.x = 009769f3ab59bfd551d53a5f846b9984c59b97d6842b20a2c565ba a167945e3d026a3755b6345df8ec7e6acb6868ae6d P.y = 1532c00cf61aa3d0ce3e5aa20c3b531a2abd2c770a790a26138183 03c6b830ffc0ecf6c357af3317b9575c567f11cd2c u[0] = 147e1ed29f06e4c5079b9d14fc89d2820d32419b990c1c7bb7dbea 2a36a045124b31ffbde7c99329c05c559af1c6cc82 Q.x = 1998321bc27ff6d71df3051b5aec12ff47363d81a5e9d2dff55f44 4f6ca7e7d6af45c56fd029c58237c266ef5cda5254 Q.y = 034d274476c6307ae584f951c82e7ea85b84f72d28f4d647173235 6121af8d62a49bc263e8eb913a6cf6f125995514ee msg = abcdef0123456789 P.x = 1974dbb8e6b5d20b84df7e625e2fbfecb2cdb5f77d5eae5fb2955e 5ce7313cae8364bc2fff520a6c25619739c6bdcb6a P.y = 15f9897e11c6441eaa676de141c8d83c37aab8667173cbe1dfd6de 74d11861b961dccebcd9d289ac633455dfcc7013a3 u[0] = 04090815ad598a06897dd89bcda860f25837d54e897298ce31e694 7378134d3761dc59a572154963e8c954919ecfa82d Q.x = 17d502fa43bd6a4cad2859049a0c3ecefd60240d129be65da271a4 c03a9c38fa78163b9d2a919d2beb57df7d609b4919 Q.y = 109019902ae93a8732abecf2ff7fecd2e4e305eb91f41c9c3267f1 6b6c19de138c7272947f25512745da6c466cdfd1ac msg = q128_qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqq P.x = 0a7a047c4a8397b3446450642c2ac64d7239b61872c9ae7a59707a 8f4f950f101e766afe58223b3bff3a19a7f754027c P.y = 1383aebba1e4327ccff7cf9912bda0dbc77de048b71ef8c8a81111 d71dc33c5e3aa6edee9cf6f5fe525d50cc50b77cc9 u[0] = 08dccd088ca55b8bfbc96fb50bb25c592faa867a8bb78d4e94a8cc 2c92306190244532e91feba2b7fed977e3c3bb5a1f Q.x = 112eb92dd2b3aa9cd38b08de4bef603f2f9fb0ca226030626a9a2e 47ad1e9847fe0a5ed13766c339e38f514bba143b21 Q.y = 17542ce2f8d0a54f2c5ba8c4b14e10b22d5bcd7bae2af3c965c8c8 72b571058c720eac448276c99967ded2bf124490e1 msg = a512_aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa Faz-Hernandez, et al. Expires 14 May 2022 [Page 141] Internet-Draft hash-to-curve November 2021 P.x = 0e7a16a975904f131682edbb03d9560d3e48214c9986bd50417a77 108d13dc957500edf96462a3d01e62dc6cd468ef11 P.y = 0ae89e677711d05c30a48d6d75e76ca9fb70fe06c6dd6ff988683d 89ccde29ac7d46c53bb97a59b1901abf1db66052db u[0] = 0dd824886d2123a96447f6c56e3a3fa992fbfefdba17b6673f9f63 0ff19e4d326529db37e1c1be43f905bf9202e0278d Q.x = 1775d400a1bacc1c39c355da7e96d2d1c97baa9430c4a3476881f8 521c09a01f921f592607961efc99c4cd46bd78ca19 Q.y = 1109b5d59f65964315de65a7a143e86eabc053104ed289cf480949 317a5685fad7254ff8e7fe6d24d3104e5d55ad6370 J.10. BLS12-381 G2 J.10.1. BLS12381G2_XMD:SHA-256_SSWU_RO_ suite = BLS12381G2_XMD:SHA-256_SSWU_RO_ dst = QUUX-V01-CS02-with-BLS12381G2_XMD:SHA-256_SSWU_RO_ msg = P.x = 0141ebfbdca40eb85b87142e130ab689c673cf60f1a3e98d693352 66f30d9b8d4ac44c1038e9dcdd5393faf5c41fb78a + I * 05cb8437535e20ecffaef7752baddf98034139c38452458baeefab 379ba13dff5bf5dd71b72418717047f5b0f37da03d P.y = 0503921d7f6a12805e72940b963c0cf3471c7b2a524950ca195d11 062ee75ec076daf2d4bc358c4b190c0c98064fdd92 + I * 12424ac32561493f3fe3c260708a12b7c620e7be00099a974e259d dc7d1f6395c3c811cdd19f1e8dbf3e9ecfdcbab8d6 u[0] = 03dbc2cce174e91ba93cbb08f26b917f98194a2ea08d1cce75b2b9 cc9f21689d80bd79b594a613d0a68eb807dfdc1cf8 + I * 05a2acec64114845711a54199ea339abd125ba38253b70a92c876d f10598bd1986b739cad67961eb94f7076511b3b39a u[1] = 02f99798e8a5acdeed60d7e18e9120521ba1f47ec090984662846b c825de191b5b7641148c0dbc237726a334473eee94 + I * 145a81e418d4010cc027a68f14391b30074e89e60ee7a22f87217b 2f6eb0c4b94c9115b436e6fa4607e95a98de30a435 Q0.x = 019ad3fc9c72425a998d7ab1ea0e646a1f6093444fc6965f1cad5a 3195a7b1e099c050d57f45e3fa191cc6d75ed7458c + I * 171c88b0b0efb5eb2b88913a9e74fe111a4f68867b59db252ce586 8af4d1254bfab77ebde5d61cd1a86fb2fe4a5a1c1d Q0.y = 0ba10604e62bdd9eeeb4156652066167b72c8d743b050fb4c1016c 31b505129374f76e03fa127d6a156213576910fef3 + I * 0eb22c7a543d3d376e9716a49b72e79a89c9bfe9feee8533ed931c bb5373dde1fbcd7411d8052e02693654f71e15410a Q1.x = 113d2b9cd4bd98aee53470b27abc658d91b47a78a51584f3d4b950 677cfb8a3e99c24222c406128c91296ef6b45608be + I * 13855912321c5cb793e9d1e88f6f8d342d49c0b0dbac613ee9e17e 3c0b3c97dfbb5a49cc3fb45102fdbaf65e0efe2632 Q1.y = 0fd3def0b7574a1d801be44fde617162aa2e89da47f464317d9bb5 Faz-Hernandez, et al. Expires 14 May 2022 [Page 142] Internet-Draft hash-to-curve November 2021 abc3a7071763ce74180883ad7ad9a723a9afafcdca + I * 056f617902b3c0d0f78a9a8cbda43a26b65f602f8786540b9469b0 60db7b38417915b413ca65f875c130bebfaa59790c msg = abc P.x = 02c2d18e033b960562aae3cab37a27ce00d80ccd5ba4b7fe0e7a21 0245129dbec7780ccc7954725f4168aff2787776e6 + I * 139cddbccdc5e91b9623efd38c49f81a6f83f175e80b06fc374de9 eb4b41dfe4ca3a230ed250fbe3a2acf73a41177fd8 P.y = 1787327b68159716a37440985269cf584bcb1e621d3a7202be6ea0 5c4cfe244aeb197642555a0645fb87bf7466b2ba48 + I * 00aa65dae3c8d732d10ecd2c50f8a1baf3001578f71c694e03866e 9f3d49ac1e1ce70dd94a733534f106d4cec0eddd16 u[0] = 15f7c0aa8f6b296ab5ff9c2c7581ade64f4ee6f1bf18f55179ff44 a2cf355fa53dd2a2158c5ecb17d7c52f63e7195771 + I * 01c8067bf4c0ba709aa8b9abc3d1cef589a4758e09ef53732d670f d8739a7274e111ba2fcaa71b3d33df2a3a0c8529dd u[1] = 187111d5e088b6b9acfdfad078c4dacf72dcd17ca17c82be35e79f 8c372a693f60a033b461d81b025864a0ad051a06e4 + I * 08b852331c96ed983e497ebc6dee9b75e373d923b729194af8e72a 051ea586f3538a6ebb1e80881a082fa2b24df9f566 Q0.x = 12b2e525281b5f4d2276954e84ac4f42cf4e13b6ac4228624e1776 0faf94ce5706d53f0ca1952f1c5ef75239aeed55ad + I * 05d8a724db78e570e34100c0bc4a5fa84ad5839359b40398151f37 cff5a51de945c563463c9efbdda569850ee5a53e77 Q0.y = 02eacdc556d0bdb5d18d22f23dcb086dd106cad713777c7e640794 3edbe0b3d1efe391eedf11e977fac55f9b94f2489c + I * 04bbe48bfd5814648d0b9e30f0717b34015d45a861425fabc1ee06 fdfce36384ae2c808185e693ae97dcde118f34de41 Q1.x = 19f18cc5ec0c2f055e47c802acc3b0e40c337256a208001dde14b2 5afced146f37ea3d3ce16834c78175b3ed61f3c537 + I * 15b0dadc256a258b4c68ea43605dffa6d312eef215c19e6474b3e1 01d33b661dfee43b51abbf96fee68fc6043ac56a58 Q1.y = 05e47c1781286e61c7ade887512bd9c2cb9f640d3be9cf87ea0bad 24bd0ebfe946497b48a581ab6c7d4ca74b5147287f + I * 19f98db2f4a1fcdf56a9ced7b320ea9deecf57c8e59236b0dc21f6 ee7229aa9705ce9ac7fe7a31c72edca0d92370c096 msg = abcdef0123456789 P.x = 121982811d2491fde9ba7ed31ef9ca474f0e1501297f68c298e9f4 c0028add35aea8bb83d53c08cfc007c1e005723cd0 + I * 190d119345b94fbd15497bcba94ecf7db2cbfd1e1fe7da034d26cb ba169fb3968288b3fafb265f9ebd380512a71c3f2c P.y = 05571a0f8d3c08d094576981f4a3b8eda0a8e771fcdcc8ecceaf13 56a6acf17574518acb506e435b639353c2e14827c8 + I * 0bb5e7572275c567462d91807de765611490205a941a5a6af3b169 1bfe596c31225d3aabdf15faff860cb4ef17c7c3be u[0] = 0313d9325081b415bfd4e5364efaef392ecf69b087496973b22930 Faz-Hernandez, et al. Expires 14 May 2022 [Page 143] Internet-Draft hash-to-curve November 2021 3e1816d2080971470f7da112c4eb43053130b785e1 + I * 062f84cb21ed89406890c051a0e8b9cf6c575cf6e8e18ecf63ba86 826b0ae02548d83b483b79e48512b82a6c0686df8f u[1] = 1739123845406baa7be5c5dc74492051b6d42504de008c635f3535 bb831d478a341420e67dcc7b46b2e8cba5379cca97 + I * 01897665d9cb5db16a27657760bbea7951f67ad68f8d55f7113f24 ba6ddd82caef240a9bfa627972279974894701d975 Q0.x = 0f48f1ea1318ddb713697708f7327781fb39718971d72a9245b973 1faaca4dbaa7cca433d6c434a820c28b18e20ea208 + I * 06051467c8f85da5ba2540974758f7a1e0239a5981de441fdd8768 0a995649c211054869c50edbac1f3a86c561ba3162 Q0.y = 168b3d6df80069dbbedb714d41b32961ad064c227355e1ce5fac8e 105de5e49d77f0c64867f3834848f152497eb76333 + I * 134e0e8331cee8cb12f9c2d0742714ed9eee78a84d634c9a95f6a7 391b37125ed48bfc6e90bf3546e99930ff67cc97bc Q1.x = 004fd03968cd1c99a0dd84551f44c206c84dcbdb78076c5bfee24e 89a92c8508b52b88b68a92258403cbe1ea2da3495f + I * 1674338ea298281b636b2eb0fe593008d03171195fd6dcd4531e8a 1ed1f02a72da238a17a635de307d7d24aa2d969a47 Q1.y = 0dc7fa13fff6b12558419e0a1e94bfc3cfaf67238009991c5f24ee 94b632c3d09e27eca329989aee348a67b50d5e236c + I * 169585e164c131103d85324f2d7747b23b91d66ae5d947c449c819 4a347969fc6bbd967729768da485ba71868df8aed2 msg = q128_qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqq P.x = 19a84dd7248a1066f737cc34502ee5555bd3c19f2ecdb3c7d9e24d c65d4e25e50d83f0f77105e955d78f4762d33c17da + I * 0934aba516a52d8ae479939a91998299c76d39cc0c035cd18813be c433f587e2d7a4fef038260eef0cef4d02aae3eb91 P.y = 14f81cd421617428bc3b9fe25afbb751d934a00493524bc4e06563 5b0555084dd54679df1536101b2c979c0152d09192 + I * 09bcccfa036b4847c9950780733633f13619994394c23ff0b32fa6 b795844f4a0673e20282d07bc69641cee04f5e5662 u[0] = 025820cefc7d06fd38de7d8e370e0da8a52498be9b53cba9927b2e f5c6de1e12e12f188bbc7bc923864883c57e49e253 + I * 034147b77ce337a52e5948f66db0bab47a8d038e712123bb381899 b6ab5ad20f02805601e6104c29df18c254b8618c7b u[1] = 0930315cae1f9a6017c3f0c8f2314baa130e1cf13f6532bff0a8a1 790cd70af918088c3db94bda214e896e1543629795 + I * 10c4df2cacf67ea3cb3108b00d4cbd0b3968031ebc8eac4b1ebcef e84d6b715fde66bef0219951ece29d1facc8a520ef Q0.x = 09eccbc53df677f0e5814e3f86e41e146422834854a224bf5a83a5 0e4cc0a77bfc56718e8166ad180f53526ea9194b57 + I * 0c3633943f91daee715277bd644fba585168a72f96ded64fc5a384 cce4ec884a4c3c30f08e09cd2129335dc8f67840ec Q0.y = 0eb6186a0457d5b12d132902d4468bfeb7315d83320b6c32f1c875 Faz-Hernandez, et al. Expires 14 May 2022 [Page 144] Internet-Draft hash-to-curve November 2021 f344efcba979952b4aa418589cb01af712f98cc555 + I * 119e3cf167e69eb16c1c7830e8df88856d48be12e3ff0a40791a5c d2f7221311d4bf13b1847f371f467357b3f3c0b4c7 Q1.x = 0eb3aabc1ddfce17ff18455fcc7167d15ce6b60ddc9eb9b59f8d40 ab49420d35558686293d046fc1e42f864b7f60e381 + I * 198bdfb19d7441ebcca61e8ff774b29d17da16547d2c10c273227a 635cacea3f16826322ae85717630f0867539b5ed8b Q1.y = 0aaf1dee3adf3ed4c80e481c09b57ea4c705e1b8d25b897f0ceeec 3990748716575f92abff22a1c8f4582aff7b872d52 + I * 0d058d9061ed27d4259848a06c96c5ca68921a5d269b078650c882 cb3c2bd424a8702b7a6ee4e0ead9982baf6843e924 msg = a512_aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa P.x = 01a6ba2f9a11fa5598b2d8ace0fbe0a0eacb65deceb476fbbcb64f d24557c2f4b18ecfc5663e54ae16a84f5ab7f62534 + I * 11fca2ff525572795a801eed17eb12785887c7b63fb77a42be46ce 4a34131d71f7a73e95fee3f812aea3de78b4d01569 P.y = 0b6798718c8aed24bc19cb27f866f1c9effcdbf92397ad6448b5c9 db90d2b9da6cbabf48adc1adf59a1a28344e79d57e + I * 03a47f8e6d1763ba0cad63d6114c0accbef65707825a511b251a66 0a9b3994249ae4e63fac38b23da0c398689ee2ab52 u[0] = 190b513da3e66fc9a3587b78c76d1d132b1152174d0b83e3c11140 66392579a45824c5fa17649ab89299ddd4bda54935 + I * 12ab625b0fe0ebd1367fe9fac57bb1168891846039b4216b9d9400 7b674de2d79126870e88aeef54b2ec717a887dcf39 u[1] = 0e6a42010cf435fb5bacc156a585e1ea3294cc81d0ceb81924d950 40298380b164f702275892cedd81b62de3aba3f6b5 + I * 117d9a0defc57a33ed208428cb84e54c85a6840e7648480ae42883 8989d25d97a0af8e3255be62b25c2a85630d2dddd8 Q0.x = 17cadf8d04a1a170f8347d42856526a24cc466cb2ddfd506cff011 91666b7f944e31244d662c904de5440516a2b09004 + I * 0d13ba91f2a8b0051cf3279ea0ee63a9f19bc9cb8bfcc7d78b3cbd 8cc4fc43ba726774b28038213acf2b0095391c523e Q0.y = 17ef19497d6d9246fa94d35575c0f8d06ee02f21a284dbeaa78768 cb1e25abd564e3381de87bda26acd04f41181610c5 + I * 12c3c913ba4ed03c24f0721a81a6be7430f2971ffca8fd1729aafe 496bb725807531b44b34b59b3ae5495e5a2dcbd5c8 Q1.x = 16ec57b7fe04c71dfe34fb5ad84dbce5a2dbbd6ee085f1d8cd17f4 5e8868976fc3c51ad9eeda682c7869024d24579bfd Faz-Hernandez, et al. Expires 14 May 2022 [Page 145] Internet-Draft hash-to-curve November 2021 + I * 13103f7aace1ae1420d208a537f7d3a9679c287208026e4e3439ab 8cd534c12856284d95e27f5e1f33eec2ce656533b0 Q1.y = 0958b2c4c2c10fcef5a6c59b9e92c4a67b0fae3e2e0f1b6b5edad9 c940b8f3524ba9ebbc3f2ceb3cfe377655b3163bd7 + I * 0ccb594ed8bd14ca64ed9cb4e0aba221be540f25dd0d6ba15a4a4b e5d67bcf35df7853b2d8dad3ba245f1ea3697f66aa J.10.2. BLS12381G2_XMD:SHA-256_SSWU_NU_ suite = BLS12381G2_XMD:SHA-256_SSWU_NU_ dst = QUUX-V01-CS02-with-BLS12381G2_XMD:SHA-256_SSWU_NU_ msg = P.x = 00e7f4568a82b4b7dc1f14c6aaa055edf51502319c723c4dc2688c 7fe5944c213f510328082396515734b6612c4e7bb7 + I * 126b855e9e69b1f691f816e48ac6977664d24d99f8724868a18418 6469ddfd4617367e94527d4b74fc86413483afb35b P.y = 0caead0fd7b6176c01436833c79d305c78be307da5f6af6c133c47 311def6ff1e0babf57a0fb5539fce7ee12407b0a42 + I * 1498aadcf7ae2b345243e281ae076df6de84455d766ab6fcdaad71 fab60abb2e8b980a440043cd305db09d283c895e3d u[0] = 07355d25caf6e7f2f0cb2812ca0e513bd026ed09dda65b177500fa 31714e09ea0ded3a078b526bed3307f804d4b93b04 + I * 02829ce3c021339ccb5caf3e187f6370e1e2a311dec9b753631170 63ab2015603ff52c3d3b98f19c2f65575e99e8b78c Q.x = 18ed3794ad43c781816c523776188deafba67ab773189b8f18c49b c7aa841cd81525171f7a5203b2a340579192403bef + I * 0727d90785d179e7b5732c8a34b660335fed03b913710b60903cf4 954b651ed3466dc3728e21855ae822d4a0f1d06587 Q.y = 00764a5cf6c5f61c52c838523460eb2168b5a5b43705e19cb612e0 06f29b717897facfd15dd1c8874c915f6d53d0342d + I * 19290bb9797c12c1d275817aa2605ebe42275b66860f0e4d04487e bc2e47c50b36edd86c685a60c20a2bd584a82b011a msg = abc P.x = 108ed59fd9fae381abfd1d6bce2fd2fa220990f0f837fa30e0f279 14ed6e1454db0d1ee957b219f61da6ff8be0d6441f + I * 0296238ea82c6d4adb3c838ee3cb2346049c90b96d602d7bb1b469 b905c9228be25c627bffee872def773d5b2a2eb57d P.y = 033f90f6057aadacae7963b0a0b379dd46750c1c94a6357c99b65f 63b79e321ff50fe3053330911c56b6ceea08fee656 + I * 153606c417e59fb331b7ae6bce4fbf7c5190c33ce9402b5ebe2b70 e44fca614f3f1382a3625ed5493843d0b0a652fc3f u[0] = 138879a9559e24cecee8697b8b4ad32cced053138ab913b9987277 2dc753a2967ed50aabc907937aefb2439ba06cc50c + I * 0a1ae7999ea9bab1dcc9ef8887a6cb6e8f1e22566015428d220b7e ec90ffa70ad1f624018a9ad11e78d588bd3617f9f2 Q.x = 0f40e1d5025ecef0d850aa0bb7bbeceab21a3d4e85e6bee857805b Faz-Hernandez, et al. Expires 14 May 2022 [Page 146] Internet-Draft hash-to-curve November 2021 09693051f5b25428c6be343edba5f14317fcc30143 + I * 02e0d261f2b9fee88b82804ec83db330caa75fbb12719cfa71ccce 1c532dc4e1e79b0a6a281ed8d3817524286c8bc04c Q.y = 0cf4a4adc5c66da0bca4caddc6a57ecd97c8252d7526a8ff478e0d fed816c4d321b5c3039c6683ae9b1e6a3a38c9c0ae + I * 11cad1646bb3768c04be2ab2bbe1f80263b7ff6f8f9488f5bc3b68 50e5a3e97e20acc583613c69cf3d2bfe8489744ebb msg = abcdef0123456789 P.x = 038af300ef34c7759a6caaa4e69363cafeed218a1f207e93b2c70d 91a1263d375d6730bd6b6509dcac3ba5b567e85bf3 + I * 0da75be60fb6aa0e9e3143e40c42796edf15685cafe0279afd2a67 c3dff1c82341f17effd402e4f1af240ea90f4b659b P.y = 19b148cbdf163cf0894f29660d2e7bfb2b68e37d54cc83fd4e6e62 c020eaa48709302ef8e746736c0e19342cc1ce3df4 + I * 0492f4fed741b073e5a82580f7c663f9b79e036b70ab3e51162359 cec4e77c78086fe879b65ca7a47d34374c8315ac5e u[0] = 18c16fe362b7dbdfa102e42bdfd3e2f4e6191d479437a59db4eb71 6986bf08ee1f42634db66bde97d6c16bbfd342b3b8 + I * 0e37812ce1b146d998d5f92bdd5ada2a31bfd63dfe18311aa91637 b5f279dd045763166aa1615e46a50d8d8f475f184e Q.x = 13a9d4a738a85c9f917c7be36b240915434b58679980010499b9ae 8d7a1bf7fbe617a15b3cd6060093f40d18e0f19456 + I * 16fa88754e7670366a859d6f6899ad765bf5a177abedb2740aacc9 252c43f90cd0421373fbd5b2b76bb8f5c4886b5d37 Q.y = 0a7fa7d82c46797039398253e8765a4194100b330dfed6d7fbb46d 6fbf01e222088779ac336e3675c7a7a0ee05bbb6e3 + I * 0c6ee170ab766d11fa9457cef53253f2628010b2cffc102b3b2835 1eb9df6c281d3cfc78e9934769d661b72a5265338d msg = q128_qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqq P.x = 0c5ae723be00e6c3f0efe184fdc0702b64588fe77dda152ab13099 a3bacd3876767fa7bbad6d6fd90b3642e902b208f9 + I * 12c8c05c1d5fc7bfa847f4d7d81e294e66b9a78bc9953990c35894 5e1f042eedafce608b67fdd3ab0cb2e6e263b9b1ad P.y = 04e77ddb3ede41b5ec4396b7421dd916efc68a358a0d7425bddd25 3547f2fb4830522358491827265dfc5bcc1928a569 + I * 11c624c56dbe154d759d021eec60fab3d8b852395a89de497e4850 4366feedd4662d023af447d66926a28076813dd646 u[0] = 08d4a0997b9d52fecf99427abb721f0fa779479963315fe21c6445 250de7183e3f63bfdf86570da8929489e421d4ee95 + I * 16cb4ccad91ec95aab070f22043916cd6a59c4ca94097f7f510043 d48515526dc8eaaea27e586f09151ae613688d5a89 Q.x = 0a08b2f639855dfdeaaed972702b109e2241a54de198b2b4cd12ad 9f88fa419a6086a58d91fc805de812ea29bee427c2 + I * 04a7442e4cb8b42ef0f41dac9ee74e65ecad3ce0851f0746dc4756 Faz-Hernandez, et al. Expires 14 May 2022 [Page 147] Internet-Draft hash-to-curve November 2021 8b0e7a8134121ed09ba054509232c49148aef62cda Q.y = 05d60b1f04212b2c87607458f71d770f43973511c260f0540eef3a 565f42c7ce59aa1cea684bb2a7bcab84acd2f36c8c + I * 1017aa5747ba15505ece266a86b0ca9c712f41a254b76ca04094ca 442ce45ecd224bd5544cd16685d0d1b9d156dd0531 msg = a512_aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa P.x = 0ea4e7c33d43e17cc516a72f76437c4bf81d8f4eac69ac355d3bf9 b71b8138d55dc10fd458be115afa798b55dac34be1 + I * 1565c2f625032d232f13121d3cfb476f45275c303a037faa255f9d a62000c2c864ea881e2bcddd111edc4a3c0da3e88d P.y = 043b6f5fe4e52c839148dc66f2b3751e69a0f6ebb3d056d6465d50 d4108543ecd956e10fa1640dfd9bc0030cc2558d28 + I * 0f8991d2a1ad662e7b6f58ab787947f1fa607fce12dde171bc1790 3b012091b657e15333e11701edcf5b63ba2a561247 u[0] = 03f80ce4ff0ca2f576d797a3660e3f65b274285c054feccc3215c8 79e2c0589d376e83ede13f93c32f05da0f68fd6a10 + I * 006488a837c5413746d868d1efb7232724da10eca410b07d8b505b 9363bdccf0a1fc0029bad07d65b15ccfe6dd25e20d Q.x = 19592c812d5a50c5601062faba14c7d670711745311c879de1235a 0a11c75aab61327bf2d1725db07ec4d6996a682886 + I * 0eef4fa41ddc17ed47baf447a2c498548f3c72a02381313d13bef9 16e240b61ce125539090d62d9fbb14a900bf1b8e90 Q.y = 1260d6e0987eae96af9ebe551e08de22b37791d53f4db9e0d59da7 36e66699735793e853e26362531fe4adf99c1883e3 + I * 0dbace5df0a4ac4ac2f45d8fdf8aee45484576fdd6efc4f98ab9b9 f4112309e628255e183022d98ea5ed6e47ca00306c Appendix K. Expand test vectors This section gives test vectors for expand_message variants specified in Section 5.4. The test vectors in this section were generated using code that is available from [hash2curve-repo]. Faz-Hernandez, et al. Expires 14 May 2022 [Page 148] Internet-Draft hash-to-curve November 2021 Each test vector in this section lists the expand_message name, hash function, and DST, along with a series of tuples of the function inputs (msg and len_in_bytes), output (uniform_bytes), and intermediate values (dst_prime and msg_prime). DST and msg are represented as ASCII strings. Intermediate and output values are represented as byte strings in hexadecimal. K.1. expand_message_xmd(SHA-256) name = expand_message_xmd DST = QUUX-V01-CS02-with-expander-SHA256-128 hash = SHA256 k = 128 msg = len_in_bytes = 0x20 DST_prime = 515555582d5630312d435330322d776974682d657870616e6465 722d5348413235362d31323826 msg_prime = 0000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000 0000000000000000000000002000515555582d5630312d43533032 2d776974682d657870616e6465722d5348413235362d31323826 uniform_bytes = 68a985b87eb6b46952128911f2a4412bbc302a9d759667f8 7f7a21d803f07235 msg = abc len_in_bytes = 0x20 DST_prime = 515555582d5630312d435330322d776974682d657870616e6465 722d5348413235362d31323826 msg_prime = 0000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000 0000000000000000000000616263002000515555582d5630312d43 5330322d776974682d657870616e6465722d5348413235362d3132 3826 uniform_bytes = d8ccab23b5985ccea865c6c97b6e5b8350e794e603b4b979 02f53a8a0d605615 msg = abcdef0123456789 len_in_bytes = 0x20 DST_prime = 515555582d5630312d435330322d776974682d657870616e6465 722d5348413235362d31323826 msg_prime = 0000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000 000000000000000000000061626364656630313233343536373839 002000515555582d5630312d435330322d776974682d657870616e 6465722d5348413235362d31323826 uniform_bytes = eff31487c770a893cfb36f912fbfcbff40d5661771ca4b2c b4eafe524333f5c1 Faz-Hernandez, et al. Expires 14 May 2022 [Page 149] Internet-Draft hash-to-curve November 2021 msg = q128_qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqq len_in_bytes = 0x20 DST_prime = 515555582d5630312d435330322d776974682d657870616e6465 722d5348413235362d31323826 msg_prime = 0000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000 0000000000000000000000713132385f7171717171717171717171 717171717171717171717171717171717171717171717171717171 717171717171717171717171717171717171717171717171717171 717171717171717171717171717171717171717171717171717171 717171717171717171717171717171717171717171717171717171 717171717171717171002000515555582d5630312d435330322d77 6974682d657870616e6465722d5348413235362d31323826 uniform_bytes = b23a1d2b4d97b2ef7785562a7e8bac7eed54ed6e97e29aa5 1bfe3f12ddad1ff9 msg = a512_aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa len_in_bytes = 0x20 DST_prime = 515555582d5630312d435330322d776974682d657870616e6465 722d5348413235362d31323826 msg_prime = 0000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000 0000000000000000000000613531325f6161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 Faz-Hernandez, et al. Expires 14 May 2022 [Page 150] Internet-Draft hash-to-curve November 2021 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161002000515555582d5630312d 435330322d776974682d657870616e6465722d5348413235362d31 323826 uniform_bytes = 4623227bcc01293b8c130bf771da8c298dede7383243dc09 93d2d94823958c4c msg = len_in_bytes = 0x80 DST_prime = 515555582d5630312d435330322d776974682d657870616e6465 722d5348413235362d31323826 msg_prime = 0000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000 0000000000000000000000008000515555582d5630312d43533032 2d776974682d657870616e6465722d5348413235362d31323826 uniform_bytes = af84c27ccfd45d41914fdff5df25293e221afc53d8ad2ac0 6d5e3e29485dadbee0d121587713a3e0dd4d5e69e93eb7cd4f5df4 cd103e188cf60cb02edc3edf18eda8576c412b18ffb658e3dd6ec8 49469b979d444cf7b26911a08e63cf31f9dcc541708d3491184472 c2c29bb749d4286b004ceb5ee6b9a7fa5b646c993f0ced msg = abc len_in_bytes = 0x80 DST_prime = 515555582d5630312d435330322d776974682d657870616e6465 722d5348413235362d31323826 msg_prime = 0000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000 0000000000000000000000616263008000515555582d5630312d43 5330322d776974682d657870616e6465722d5348413235362d3132 3826 uniform_bytes = abba86a6129e366fc877aab32fc4ffc70120d8996c88aee2 fe4b32d6c7b6437a647e6c3163d40b76a73cf6a5674ef1d890f95b 664ee0afa5359a5c4e07985635bbecbac65d747d3d2da7ec2b8221 b17b0ca9dc8a1ac1c07ea6a1e60583e2cb00058e77b7b72a298425 cd1b941ad4ec65e8afc50303a22c0f99b0509b4c895f40 msg = abcdef0123456789 len_in_bytes = 0x80 DST_prime = 515555582d5630312d435330322d776974682d657870616e6465 722d5348413235362d31323826 msg_prime = 0000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000 000000000000000000000061626364656630313233343536373839 008000515555582d5630312d435330322d776974682d657870616e 6465722d5348413235362d31323826 Faz-Hernandez, et al. Expires 14 May 2022 [Page 151] Internet-Draft hash-to-curve November 2021 uniform_bytes = ef904a29bffc4cf9ee82832451c946ac3c8f8058ae97d8d6 29831a74c6572bd9ebd0df635cd1f208e2038e760c4994984ce73f 0d55ea9f22af83ba4734569d4bc95e18350f740c07eef653cbb9f8 7910d833751825f0ebefa1abe5420bb52be14cf489b37fe1a72f7d e2d10be453b2c9d9eb20c7e3f6edc5a60629178d9478df msg = q128_qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqq len_in_bytes = 0x80 DST_prime = 515555582d5630312d435330322d776974682d657870616e6465 722d5348413235362d31323826 msg_prime = 0000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000 0000000000000000000000713132385f7171717171717171717171 717171717171717171717171717171717171717171717171717171 717171717171717171717171717171717171717171717171717171 717171717171717171717171717171717171717171717171717171 717171717171717171717171717171717171717171717171717171 717171717171717171008000515555582d5630312d435330322d77 6974682d657870616e6465722d5348413235362d31323826 uniform_bytes = 80be107d0884f0d881bb460322f0443d38bd222db8bd0b0a 5312a6fedb49c1bbd88fd75d8b9a09486c60123dfa1d73c1cc3169 761b17476d3c6b7cbbd727acd0e2c942f4dd96ae3da5de368d26b3 2286e32de7e5a8cb2949f866a0b80c58116b29fa7fabb3ea7d520e e603e0c25bcaf0b9a5e92ec6a1fe4e0391d1cdbce8c68a msg = a512_aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa len_in_bytes = 0x80 DST_prime = 515555582d5630312d435330322d776974682d657870616e6465 722d5348413235362d31323826 msg_prime = 0000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000 0000000000000000000000613531325f6161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 Faz-Hernandez, et al. Expires 14 May 2022 [Page 152] Internet-Draft hash-to-curve November 2021 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161008000515555582d5630312d 435330322d776974682d657870616e6465722d5348413235362d31 323826 uniform_bytes = 546aff5444b5b79aa6148bd81728704c32decb73a3ba76e9 e75885cad9def1d06d6792f8a7d12794e90efed817d96920d72889 6a4510864370c207f99bd4a608ea121700ef01ed879745ee3e4cee f777eda6d9e5e38b90c86ea6fb0b36504ba4a45d22e86f6db5dd43 d98a294bebb9125d5b794e9d2a81181066eb954966a487 K.2. expand_message_xmd(SHA-256) (long DST) name = expand_message_xmd DST = QUUX-V01-CS02-with-expander-SHA256-128-long-DST-111111 111111111111111111111111111111111111111111111111111111 111111111111111111111111111111111111111111111111111111 111111111111111111111111111111111111111111111111111111 1111111111111111111111111111111111111111 hash = SHA256 k = 128 msg = len_in_bytes = 0x20 DST_prime = 412717974da474d0f8c420f320ff81e8432adb7c927d9bd082b4 fb4d16c0a23620 msg_prime = 0000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000 0000000000000000000000002000412717974da474d0f8c420f320 ff81e8432adb7c927d9bd082b4fb4d16c0a23620 uniform_bytes = e8dc0c8b686b7ef2074086fbdd2f30e3f8bfbd3bdf177f73 f04b97ce618a3ed3 msg = abc len_in_bytes = 0x20 DST_prime = 412717974da474d0f8c420f320ff81e8432adb7c927d9bd082b4 fb4d16c0a23620 Faz-Hernandez, et al. Expires 14 May 2022 [Page 153] Internet-Draft hash-to-curve November 2021 msg_prime = 0000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000 0000000000000000000000616263002000412717974da474d0f8c4 20f320ff81e8432adb7c927d9bd082b4fb4d16c0a23620 uniform_bytes = 52dbf4f36cf560fca57dedec2ad924ee9c266341d8f3d6af e5171733b16bbb12 msg = abcdef0123456789 len_in_bytes = 0x20 DST_prime = 412717974da474d0f8c420f320ff81e8432adb7c927d9bd082b4 fb4d16c0a23620 msg_prime = 0000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000 000000000000000000000061626364656630313233343536373839 002000412717974da474d0f8c420f320ff81e8432adb7c927d9bd0 82b4fb4d16c0a23620 uniform_bytes = 35387dcf22618f3728e6c686490f8b431f76550b0b2c61cb c1ce7001536f4521 msg = q128_qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqq len_in_bytes = 0x20 DST_prime = 412717974da474d0f8c420f320ff81e8432adb7c927d9bd082b4 fb4d16c0a23620 msg_prime = 0000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000 0000000000000000000000713132385f7171717171717171717171 717171717171717171717171717171717171717171717171717171 717171717171717171717171717171717171717171717171717171 717171717171717171717171717171717171717171717171717171 717171717171717171717171717171717171717171717171717171 717171717171717171002000412717974da474d0f8c420f320ff81 e8432adb7c927d9bd082b4fb4d16c0a23620 uniform_bytes = 01b637612bb18e840028be900a833a74414140dde0c4754c 198532c3a0ba42bc msg = a512_aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa len_in_bytes = 0x20 Faz-Hernandez, et al. Expires 14 May 2022 [Page 154] Internet-Draft hash-to-curve November 2021 DST_prime = 412717974da474d0f8c420f320ff81e8432adb7c927d9bd082b4 fb4d16c0a23620 msg_prime = 0000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000 0000000000000000000000613531325f6161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161002000412717974da474d0f8 c420f320ff81e8432adb7c927d9bd082b4fb4d16c0a23620 uniform_bytes = 20cce7033cabc5460743180be6fa8aac5a103f56d481cf36 9a8accc0c374431b msg = len_in_bytes = 0x80 DST_prime = 412717974da474d0f8c420f320ff81e8432adb7c927d9bd082b4 fb4d16c0a23620 msg_prime = 0000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000 0000000000000000000000008000412717974da474d0f8c420f320 ff81e8432adb7c927d9bd082b4fb4d16c0a23620 uniform_bytes = 14604d85432c68b757e485c8894db3117992fc57e0e136f7 1ad987f789a0abc287c47876978e2388a02af86b1e8d1342e5ce4f 7aaa07a87321e691f6fba7e0072eecc1218aebb89fb14a0662322d 5edbd873f0eb35260145cd4e64f748c5dfe60567e126604bcab1a3 ee2dc0778102ae8a5cfd1429ebc0fa6bf1a53c36f55dfc msg = abc len_in_bytes = 0x80 DST_prime = 412717974da474d0f8c420f320ff81e8432adb7c927d9bd082b4 fb4d16c0a23620 msg_prime = 0000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000 Faz-Hernandez, et al. Expires 14 May 2022 [Page 155] Internet-Draft hash-to-curve November 2021 0000000000000000000000616263008000412717974da474d0f8c4 20f320ff81e8432adb7c927d9bd082b4fb4d16c0a23620 uniform_bytes = 1a30a5e36fbdb87077552b9d18b9f0aee16e80181d5b951d 0471d55b66684914aef87dbb3626eaabf5ded8cd0686567e503853 e5c84c259ba0efc37f71c839da2129fe81afdaec7fbdc0ccd4c794 727a17c0d20ff0ea55e1389d6982d1241cb8d165762dbc39fb0cee 4474d2cbbd468a835ae5b2f20e4f959f56ab24cd6fe267 msg = abcdef0123456789 len_in_bytes = 0x80 DST_prime = 412717974da474d0f8c420f320ff81e8432adb7c927d9bd082b4 fb4d16c0a23620 msg_prime = 0000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000 000000000000000000000061626364656630313233343536373839 008000412717974da474d0f8c420f320ff81e8432adb7c927d9bd0 82b4fb4d16c0a23620 uniform_bytes = d2ecef3635d2397f34a9f86438d772db19ffe9924e28a1ca f6f1c8f15603d4028f40891044e5c7e39ebb9b31339979ff33a424 9206f67d4a1e7c765410bcd249ad78d407e303675918f20f26ce6d 7027ed3774512ef5b00d816e51bfcc96c3539601fa48ef1c07e494 bdc37054ba96ecb9dbd666417e3de289d4f424f502a982 msg = q128_qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqq len_in_bytes = 0x80 DST_prime = 412717974da474d0f8c420f320ff81e8432adb7c927d9bd082b4 fb4d16c0a23620 msg_prime = 0000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000 0000000000000000000000713132385f7171717171717171717171 717171717171717171717171717171717171717171717171717171 717171717171717171717171717171717171717171717171717171 717171717171717171717171717171717171717171717171717171 717171717171717171717171717171717171717171717171717171 717171717171717171008000412717974da474d0f8c420f320ff81 e8432adb7c927d9bd082b4fb4d16c0a23620 uniform_bytes = ed6e8c036df90111410431431a232d41a32c86e296c05d42 6e5f44e75b9a50d335b2412bc6c91e0a6dc131de09c43110d9180d 0a70f0d6289cb4e43b05f7ee5e9b3f42a1fad0f31bac6a625b3b5c 50e3a83316783b649e5ecc9d3b1d9471cb5024b7ccf40d41d1751a 04ca0356548bc6e703fca02ab521b505e8e45600508d32 msg = a512_aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa Faz-Hernandez, et al. Expires 14 May 2022 [Page 156] Internet-Draft hash-to-curve November 2021 aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa len_in_bytes = 0x80 DST_prime = 412717974da474d0f8c420f320ff81e8432adb7c927d9bd082b4 fb4d16c0a23620 msg_prime = 0000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000 0000000000000000000000613531325f6161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161008000412717974da474d0f8 c420f320ff81e8432adb7c927d9bd082b4fb4d16c0a23620 uniform_bytes = 78b53f2413f3c688f07732c10e5ced29a17c6a16f717179f fbe38d92d6c9ec296502eb9889af83a1928cd162e845b0d3c5424e 83280fed3d10cffb2f8431f14e7a23f4c68819d40617589e4c4116 9d0b56e0e3535be1fd71fbb08bb70c5b5ffed953d6c14bf7618b35 fc1f4c4b30538236b4b08c9fbf90462447a8ada60be495 K.3. expand_message_xmd(SHA-512) name = expand_message_xmd DST = QUUX-V01-CS02-with-expander-SHA512-256 hash = SHA512 k = 256 msg = len_in_bytes = 0x20 DST_prime = 515555582d5630312d435330322d776974682d657870616e6465 Faz-Hernandez, et al. Expires 14 May 2022 [Page 157] Internet-Draft hash-to-curve November 2021 722d5348413531322d32353626 msg_prime = 0000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000002000515555 582d5630312d435330322d776974682d657870616e6465722d5348 413531322d32353626 uniform_bytes = 6b9a7312411d92f921c6f68ca0b6380730a1a4d982c50721 1a90964c394179ba msg = abc len_in_bytes = 0x20 DST_prime = 515555582d5630312d435330322d776974682d657870616e6465 722d5348413531322d32353626 msg_prime = 0000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000616263002000 515555582d5630312d435330322d776974682d657870616e646572 2d5348413531322d32353626 uniform_bytes = 0da749f12fbe5483eb066a5f595055679b976e93abe9be6f 0f6318bce7aca8dc msg = abcdef0123456789 len_in_bytes = 0x20 DST_prime = 515555582d5630312d435330322d776974682d657870616e6465 722d5348413531322d32353626 msg_prime = 0000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000616263646566 30313233343536373839002000515555582d5630312d435330322d 776974682d657870616e6465722d5348413531322d32353626 uniform_bytes = 087e45a86e2939ee8b91100af1583c4938e0f5fc6c9db4b1 07b83346bc967f58 msg = q128_qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqq len_in_bytes = 0x20 DST_prime = 515555582d5630312d435330322d776974682d657870616e6465 722d5348413531322d32353626 msg_prime = 0000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000 Faz-Hernandez, et al. Expires 14 May 2022 [Page 158] Internet-Draft hash-to-curve November 2021 000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000713132385f71 717171717171717171717171717171717171717171717171717171 717171717171717171717171717171717171717171717171717171 717171717171717171717171717171717171717171717171717171 717171717171717171717171717171717171717171717171717171 71717171717171717171717171717171717171002000515555582d 5630312d435330322d776974682d657870616e6465722d53484135 31322d32353626 uniform_bytes = 7336234ee9983902440f6bc35b348352013becd88938d2af ec44311caf8356b3 msg = a512_aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa len_in_bytes = 0x20 DST_prime = 515555582d5630312d435330322d776974682d657870616e6465 722d5348413531322d32353626 msg_prime = 0000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000613531325f61 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 Faz-Hernandez, et al. Expires 14 May 2022 [Page 159] Internet-Draft hash-to-curve November 2021 616161616161616161616161616161616161616161616161610020 00515555582d5630312d435330322d776974682d657870616e6465 722d5348413531322d32353626 uniform_bytes = 57b5f7e766d5be68a6bfe1768e3c2b7f1228b3e4b3134956 dd73a59b954c66f4 msg = len_in_bytes = 0x80 DST_prime = 515555582d5630312d435330322d776974682d657870616e6465 722d5348413531322d32353626 msg_prime = 0000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000008000515555 582d5630312d435330322d776974682d657870616e6465722d5348 413531322d32353626 uniform_bytes = 41b037d1734a5f8df225dd8c7de38f851efdb45c372887be 655212d07251b921b052b62eaed99b46f72f2ef4cc96bfaf254ebb bec091e1a3b9e4fb5e5b619d2e0c5414800a1d882b62bb5cd1778f 098b8eb6cb399d5d9d18f5d5842cf5d13d7eb00a7cff859b605da6 78b318bd0e65ebff70bec88c753b159a805d2c89c55961 msg = abc len_in_bytes = 0x80 DST_prime = 515555582d5630312d435330322d776974682d657870616e6465 722d5348413531322d32353626 msg_prime = 0000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000616263008000 515555582d5630312d435330322d776974682d657870616e646572 2d5348413531322d32353626 uniform_bytes = 7f1dddd13c08b543f2e2037b14cefb255b44c83cc397c178 6d975653e36a6b11bdd7732d8b38adb4a0edc26a0cef4bb4521713 5456e58fbca1703cd6032cb1347ee720b87972d63fbf232587043e d2901bce7f22610c0419751c065922b488431851041310ad659e4b 23520e1772ab29dcdeb2002222a363f0c2b1c972b3efe1 msg = abcdef0123456789 len_in_bytes = 0x80 DST_prime = 515555582d5630312d435330322d776974682d657870616e6465 722d5348413531322d32353626 msg_prime = 0000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000 Faz-Hernandez, et al. Expires 14 May 2022 [Page 160] Internet-Draft hash-to-curve November 2021 000000000000000000000000000000000000000000616263646566 30313233343536373839008000515555582d5630312d435330322d 776974682d657870616e6465722d5348413531322d32353626 uniform_bytes = 3f721f208e6199fe903545abc26c837ce59ac6fa45733f1b aaf0222f8b7acb0424814fcb5eecf6c1d38f06e9d0a6ccfbf85ae6 12ab8735dfdf9ce84c372a77c8f9e1c1e952c3a61b7567dd069301 6af51d2745822663d0c2367e3f4f0bed827feecc2aaf98c949b5ed 0d35c3f1023d64ad1407924288d366ea159f46287e61ac msg = q128_qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqq len_in_bytes = 0x80 DST_prime = 515555582d5630312d435330322d776974682d657870616e6465 722d5348413531322d32353626 msg_prime = 0000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000713132385f71 717171717171717171717171717171717171717171717171717171 717171717171717171717171717171717171717171717171717171 717171717171717171717171717171717171717171717171717171 717171717171717171717171717171717171717171717171717171 71717171717171717171717171717171717171008000515555582d 5630312d435330322d776974682d657870616e6465722d53484135 31322d32353626 uniform_bytes = b799b045a58c8d2b4334cf54b78260b45eec544f9f2fb5bd 12fb603eaee70db7317bf807c406e26373922b7b8920fa29142703 dd52bdf280084fb7ef69da78afdf80b3586395b433dc66cde048a2 58e476a561e9deba7060af40adf30c64249ca7ddea79806ee5beb9 a1422949471d267b21bc88e688e4014087a0b592b695ed msg = a512_aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa len_in_bytes = 0x80 DST_prime = 515555582d5630312d435330322d776974682d657870616e6465 722d5348413531322d32353626 msg_prime = 0000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000 Faz-Hernandez, et al. Expires 14 May 2022 [Page 161] Internet-Draft hash-to-curve November 2021 000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000613531325f61 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161610080 00515555582d5630312d435330322d776974682d657870616e6465 722d5348413531322d32353626 uniform_bytes = 05b0bfef265dcee87654372777b7c44177e2ae4c13a27f10 3340d9cd11c86cb2426ffcad5bd964080c2aee97f03be1ca18e30a 1f14e27bc11ebbd650f305269cc9fb1db08bf90bfc79b42a952b46 daf810359e7bc36452684784a64952c343c52e5124cd1f71d474d5 197fefc571a92929c9084ffe1112cf5eea5192ebff330b K.4. expand_message_xof(SHAKE128) name = expand_message_xof DST = QUUX-V01-CS02-with-expander-SHAKE128 hash = SHAKE128 k = 128 msg = len_in_bytes = 0x20 DST_prime = 515555582d5630312d435330322d776974682d657870616e6465 722d5348414b4531323824 msg_prime = 0020515555582d5630312d435330322d776974682d657870616e 6465722d5348414b4531323824 uniform_bytes = 86518c9cd86581486e9485aa74ab35ba150d1c75c88e26b7 043e44e2acd735a2 msg = abc len_in_bytes = 0x20 Faz-Hernandez, et al. Expires 14 May 2022 [Page 162] Internet-Draft hash-to-curve November 2021 DST_prime = 515555582d5630312d435330322d776974682d657870616e6465 722d5348414b4531323824 msg_prime = 6162630020515555582d5630312d435330322d776974682d6578 70616e6465722d5348414b4531323824 uniform_bytes = 8696af52a4d862417c0763556073f47bc9b9ba43c99b5053 05cb1ec04a9ab468 msg = abcdef0123456789 len_in_bytes = 0x20 DST_prime = 515555582d5630312d435330322d776974682d657870616e6465 722d5348414b4531323824 msg_prime = 616263646566303132333435363738390020515555582d563031 2d435330322d776974682d657870616e6465722d5348414b453132 3824 uniform_bytes = 912c58deac4821c3509dbefa094df54b34b8f5d01a191d1d 3108a2c89077acca msg = q128_qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqq len_in_bytes = 0x20 DST_prime = 515555582d5630312d435330322d776974682d657870616e6465 722d5348414b4531323824 msg_prime = 713132385f717171717171717171717171717171717171717171 717171717171717171717171717171717171717171717171717171 717171717171717171717171717171717171717171717171717171 717171717171717171717171717171717171717171717171717171 717171717171717171717171717171717171717171717171717100 20515555582d5630312d435330322d776974682d657870616e6465 722d5348414b4531323824 uniform_bytes = 1adbcc448aef2a0cebc71dac9f756b22e51839d348e031e6 3b33ebb50faeaf3f msg = a512_aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa len_in_bytes = 0x20 DST_prime = 515555582d5630312d435330322d776974682d657870616e6465 722d5348414b4531323824 msg_prime = 613531325f616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 Faz-Hernandez, et al. Expires 14 May 2022 [Page 163] Internet-Draft hash-to-curve November 2021 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 61616161610020515555582d5630312d435330322d776974682d65 7870616e6465722d5348414b4531323824 uniform_bytes = df3447cc5f3e9a77da10f819218ddf31342c310778e0e4ef 72bbaecee786a4fe msg = len_in_bytes = 0x80 DST_prime = 515555582d5630312d435330322d776974682d657870616e6465 722d5348414b4531323824 msg_prime = 0080515555582d5630312d435330322d776974682d657870616e 6465722d5348414b4531323824 uniform_bytes = 7314ff1a155a2fb99a0171dc71b89ab6e3b2b7d59e38e644 19b8b6294d03ffee42491f11370261f436220ef787f8f76f5b26bd cd850071920ce023f3ac46847744f4612b8714db8f5db83205b2e6 25d95afd7d7b4d3094d3bdde815f52850bb41ead9822e08f22cf41 d615a303b0d9dde73263c049a7b9898208003a739a2e57 msg = abc len_in_bytes = 0x80 DST_prime = 515555582d5630312d435330322d776974682d657870616e6465 722d5348414b4531323824 msg_prime = 6162630080515555582d5630312d435330322d776974682d6578 70616e6465722d5348414b4531323824 uniform_bytes = c952f0c8e529ca8824acc6a4cab0e782fc3648c563ddb00d a7399f2ae35654f4860ec671db2356ba7baa55a34a9d7f79197b60 ddae6e64768a37d699a78323496db3878c8d64d909d0f8a7de4927 dcab0d3dbbc26cb20a49eceb0530b431cdf47bc8c0fa3e0d88f53b 318b6739fbed7d7634974f1b5c386d6230c76260d5337a msg = abcdef0123456789 len_in_bytes = 0x80 Faz-Hernandez, et al. Expires 14 May 2022 [Page 164] Internet-Draft hash-to-curve November 2021 DST_prime = 515555582d5630312d435330322d776974682d657870616e6465 722d5348414b4531323824 msg_prime = 616263646566303132333435363738390080515555582d563031 2d435330322d776974682d657870616e6465722d5348414b453132 3824 uniform_bytes = 19b65ee7afec6ac06a144f2d6134f08eeec185f1a890fe34 e68f0e377b7d0312883c048d9b8a1d6ecc3b541cb4987c26f45e0c 82691ea299b5e6889bbfe589153016d8131717ba26f07c3c14ffbe f1f3eff9752e5b6183f43871a78219a75e7000fbac6a7072e2b83c 790a3a5aecd9d14be79f9fd4fb180960a3772e08680495 msg = q128_qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqq len_in_bytes = 0x80 DST_prime = 515555582d5630312d435330322d776974682d657870616e6465 722d5348414b4531323824 msg_prime = 713132385f717171717171717171717171717171717171717171 717171717171717171717171717171717171717171717171717171 717171717171717171717171717171717171717171717171717171 717171717171717171717171717171717171717171717171717171 717171717171717171717171717171717171717171717171717100 80515555582d5630312d435330322d776974682d657870616e6465 722d5348414b4531323824 uniform_bytes = ca1b56861482b16eae0f4a26212112362fcc2d76dcc80c93 c4182ed66c5113fe41733ed68be2942a3487394317f3379856f482 2a611735e50528a60e7ade8ec8c71670fec6661e2c59a09ed36386 513221688b35dc47e3c3111ee8c67ff49579089d661caa29db1ef1 0eb6eace575bf3dc9806e7c4016bd50f3c0e2a6481ee6d msg = a512_aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa len_in_bytes = 0x80 DST_prime = 515555582d5630312d435330322d776974682d657870616e6465 722d5348414b4531323824 msg_prime = 613531325f616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 Faz-Hernandez, et al. Expires 14 May 2022 [Page 165] Internet-Draft hash-to-curve November 2021 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 61616161610080515555582d5630312d435330322d776974682d65 7870616e6465722d5348414b4531323824 uniform_bytes = 9d763a5ce58f65c91531b4100c7266d479a5d9777ba76169 3d052acd37d149e7ac91c796a10b919cd74a591a1e38719fb91b72 03e2af31eac3bff7ead2c195af7d88b8bc0a8adf3d1e90ab9bed6d dc2b7f655dd86c730bdeaea884e73741097142c92f0e3fc1811b69 9ba593c7fbd81da288a29d423df831652e3a01a9374999 K.5. expand_message_xof(SHAKE128) (long DST) name = expand_message_xof DST = QUUX-V01-CS02-with-expander-SHAKE128-long-DST-11111111 111111111111111111111111111111111111111111111111111111 111111111111111111111111111111111111111111111111111111 111111111111111111111111111111111111111111111111111111 1111111111111111111111111111111111111111 hash = SHAKE128 k = 128 msg = len_in_bytes = 0x20 DST_prime = acb9736c0867fdfbd6385519b90fc8c034b5af04a95897321295 0132d035792f20 msg_prime = 0020acb9736c0867fdfbd6385519b90fc8c034b5af04a9589732 12950132d035792f20 uniform_bytes = 827c6216330a122352312bccc0c8d6e7a146c5257a776dbd 9ad9d75cd880fc53 msg = abc len_in_bytes = 0x20 DST_prime = acb9736c0867fdfbd6385519b90fc8c034b5af04a95897321295 0132d035792f20 msg_prime = 6162630020acb9736c0867fdfbd6385519b90fc8c034b5af04a9 58973212950132d035792f20 Faz-Hernandez, et al. Expires 14 May 2022 [Page 166] Internet-Draft hash-to-curve November 2021 uniform_bytes = 690c8d82c7213b4282c6cb41c00e31ea1d3e2005f93ad19b bf6da40f15790c5c msg = abcdef0123456789 len_in_bytes = 0x20 DST_prime = acb9736c0867fdfbd6385519b90fc8c034b5af04a95897321295 0132d035792f20 msg_prime = 616263646566303132333435363738390020acb9736c0867fdfb d6385519b90fc8c034b5af04a958973212950132d035792f20 uniform_bytes = 979e3a15064afbbcf99f62cc09fa9c85028afcf3f825eb07 11894dcfc2f57057 msg = q128_qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqq len_in_bytes = 0x20 DST_prime = acb9736c0867fdfbd6385519b90fc8c034b5af04a95897321295 0132d035792f20 msg_prime = 713132385f717171717171717171717171717171717171717171 717171717171717171717171717171717171717171717171717171 717171717171717171717171717171717171717171717171717171 717171717171717171717171717171717171717171717171717171 717171717171717171717171717171717171717171717171717100 20acb9736c0867fdfbd6385519b90fc8c034b5af04a95897321295 0132d035792f20 uniform_bytes = c5a9220962d9edc212c063f4f65b609755a1ed96e62f9db5 d1fd6adb5a8dc52b msg = a512_aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa len_in_bytes = 0x20 DST_prime = acb9736c0867fdfbd6385519b90fc8c034b5af04a95897321295 0132d035792f20 msg_prime = 613531325f616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 Faz-Hernandez, et al. Expires 14 May 2022 [Page 167] Internet-Draft hash-to-curve November 2021 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 61616161610020acb9736c0867fdfbd6385519b90fc8c034b5af04 a958973212950132d035792f20 uniform_bytes = f7b96a5901af5d78ce1d071d9c383cac66a1dfadb508300e c6aeaea0d62d5d62 msg = len_in_bytes = 0x80 DST_prime = acb9736c0867fdfbd6385519b90fc8c034b5af04a95897321295 0132d035792f20 msg_prime = 0080acb9736c0867fdfbd6385519b90fc8c034b5af04a9589732 12950132d035792f20 uniform_bytes = 3890dbab00a2830be398524b71c2713bbef5f4884ac2e6f0 70b092effdb19208c7df943dc5dcbaee3094a78c267ef276632ee2 c8ea0c05363c94b6348500fae4208345dd3475fe0c834c2beac7fa 7bc181692fb728c0a53d809fc8111495222ce0f38468b11becb15b 32060218e285c57a60162c2c8bb5b6bded13973cd41819 msg = abc len_in_bytes = 0x80 DST_prime = acb9736c0867fdfbd6385519b90fc8c034b5af04a95897321295 0132d035792f20 msg_prime = 6162630080acb9736c0867fdfbd6385519b90fc8c034b5af04a9 58973212950132d035792f20 uniform_bytes = 41b7ffa7a301b5c1441495ebb9774e2a53dbbf4e54b9a1af 6a20fd41eafd69ef7b9418599c5545b1ee422f363642b01d4a5344 9313f68da3e49dddb9cd25b97465170537d45dcbdf92391b5bdff3 44db4bd06311a05bca7dcd360b6caec849c299133e5c9194f4e15e 3e23cfaab4003fab776f6ac0bfae9144c6e2e1c62e7d57 msg = abcdef0123456789 len_in_bytes = 0x80 DST_prime = acb9736c0867fdfbd6385519b90fc8c034b5af04a95897321295 0132d035792f20 msg_prime = 616263646566303132333435363738390080acb9736c0867fdfb d6385519b90fc8c034b5af04a958973212950132d035792f20 uniform_bytes = 55317e4a21318472cd2290c3082957e1242241d9e0d04f47 Faz-Hernandez, et al. Expires 14 May 2022 [Page 168] Internet-Draft hash-to-curve November 2021 026f03401643131401071f01aa03038b2783e795bdfa8a3541c194 ad5de7cb9c225133e24af6c86e748deb52e560569bd54ef4dac034 65111a3a44b0ea490fb36777ff8ea9f1a8a3e8e0de3cf0880b4b2f 8dd37d3a85a8b82375aee4fa0e909f9763319b55778e71 msg = q128_qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqq len_in_bytes = 0x80 DST_prime = acb9736c0867fdfbd6385519b90fc8c034b5af04a95897321295 0132d035792f20 msg_prime = 713132385f717171717171717171717171717171717171717171 717171717171717171717171717171717171717171717171717171 717171717171717171717171717171717171717171717171717171 717171717171717171717171717171717171717171717171717171 717171717171717171717171717171717171717171717171717100 80acb9736c0867fdfbd6385519b90fc8c034b5af04a95897321295 0132d035792f20 uniform_bytes = 19fdd2639f082e31c77717ac9bb032a22ff0958382b2dbb3 9020cdc78f0da43305414806abf9a561cb2d0067eb2f7bc544482f 75623438ed4b4e39dd9e6e2909dd858bd8f1d57cd0fce2d3150d90 aa67b4498bdf2df98c0100dd1a173436ba5d0df6be1defb0b2ce55 ccd2f4fc05eb7cb2c019c35d5398b85adc676da4238bc7 msg = a512_aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa len_in_bytes = 0x80 DST_prime = acb9736c0867fdfbd6385519b90fc8c034b5af04a95897321295 0132d035792f20 msg_prime = 613531325f616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 Faz-Hernandez, et al. Expires 14 May 2022 [Page 169] Internet-Draft hash-to-curve November 2021 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 61616161610080acb9736c0867fdfbd6385519b90fc8c034b5af04 a958973212950132d035792f20 uniform_bytes = 945373f0b3431a103333ba6a0a34f1efab2702efde41754c 4cb1d5216d5b0a92a67458d968562bde7fa6310a83f53dda138368 0a276a283438d58ceebfa7ab7ba72499d4a3eddc860595f63c93b1 c5e823ea41fc490d938398a26db28f61857698553e93f0574eb8c5 017bfed6249491f9976aaa8d23d9485339cc85ca329308 K.6. expand_message_xof(SHAKE256) name = expand_message_xof DST = QUUX-V01-CS02-with-expander-SHAKE256 hash = SHAKE256 k = 256 msg = len_in_bytes = 0x20 DST_prime = 515555582d5630312d435330322d776974682d657870616e6465 722d5348414b4532353624 msg_prime = 0020515555582d5630312d435330322d776974682d657870616e 6465722d5348414b4532353624 uniform_bytes = 2ffc05c48ed32b95d72e807f6eab9f7530dd1c2f013914c8 fed38c5ccc15ad76 msg = abc len_in_bytes = 0x20 DST_prime = 515555582d5630312d435330322d776974682d657870616e6465 722d5348414b4532353624 msg_prime = 6162630020515555582d5630312d435330322d776974682d6578 70616e6465722d5348414b4532353624 uniform_bytes = b39e493867e2767216792abce1f2676c197c0692aed06156 0ead251821808e07 msg = abcdef0123456789 len_in_bytes = 0x20 DST_prime = 515555582d5630312d435330322d776974682d657870616e6465 722d5348414b4532353624 msg_prime = 616263646566303132333435363738390020515555582d563031 2d435330322d776974682d657870616e6465722d5348414b453235 3624 Faz-Hernandez, et al. Expires 14 May 2022 [Page 170] Internet-Draft hash-to-curve November 2021 uniform_bytes = 245389cf44a13f0e70af8665fe5337ec2dcd138890bb7901 c4ad9cfceb054b65 msg = q128_qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqq len_in_bytes = 0x20 DST_prime = 515555582d5630312d435330322d776974682d657870616e6465 722d5348414b4532353624 msg_prime = 713132385f717171717171717171717171717171717171717171 717171717171717171717171717171717171717171717171717171 717171717171717171717171717171717171717171717171717171 717171717171717171717171717171717171717171717171717171 717171717171717171717171717171717171717171717171717100 20515555582d5630312d435330322d776974682d657870616e6465 722d5348414b4532353624 uniform_bytes = 719b3911821e6428a5ed9b8e600f2866bcf23c8f0515e52d 6c6c019a03f16f0e msg = a512_aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa len_in_bytes = 0x20 DST_prime = 515555582d5630312d435330322d776974682d657870616e6465 722d5348414b4532353624 msg_prime = 613531325f616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 Faz-Hernandez, et al. Expires 14 May 2022 [Page 171] Internet-Draft hash-to-curve November 2021 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 61616161610020515555582d5630312d435330322d776974682d65 7870616e6465722d5348414b4532353624 uniform_bytes = 9181ead5220b1963f1b5951f35547a5ea86a820562287d6c a4723633d17ccbbc msg = len_in_bytes = 0x80 DST_prime = 515555582d5630312d435330322d776974682d657870616e6465 722d5348414b4532353624 msg_prime = 0080515555582d5630312d435330322d776974682d657870616e 6465722d5348414b4532353624 uniform_bytes = 7a1361d2d7d82d79e035b8880c5a3c86c5afa719478c007d 96e6c88737a3f631dd74a2c88df79a4cb5e5d9f7504957c70d669e c6bfedc31e01e2bacc4ff3fdf9b6a00b17cc18d9d72ace7d6b81c2 e481b4f73f34f9a7505dccbe8f5485f3d20c5409b0310093d5d649 2dea4e18aa6979c23c8ea5de01582e9689612afbb353df msg = abc len_in_bytes = 0x80 DST_prime = 515555582d5630312d435330322d776974682d657870616e6465 722d5348414b4532353624 msg_prime = 6162630080515555582d5630312d435330322d776974682d6578 70616e6465722d5348414b4532353624 uniform_bytes = a54303e6b172909783353ab05ef08dd435a558c3197db0c1 32134649708e0b9b4e34fb99b92a9e9e28fc1f1d8860d85897a8e0 21e6382f3eea10577f968ff6df6c45fe624ce65ca25932f679a42a 404bc3681efe03fcd45ef73bb3a8f79ba784f80f55ea8a3c367408 f30381299617f50c8cf8fbb21d0f1e1d70b0131a7b6fbe msg = abcdef0123456789 len_in_bytes = 0x80 DST_prime = 515555582d5630312d435330322d776974682d657870616e6465 722d5348414b4532353624 msg_prime = 616263646566303132333435363738390080515555582d563031 2d435330322d776974682d657870616e6465722d5348414b453235 3624 uniform_bytes = e42e4d9538a189316e3154b821c1bafb390f78b2f010ea40 4e6ac063deb8c0852fcd412e098e231e43427bd2be1330bb47b403 9ad57b30ae1fc94e34993b162ff4d695e42d59d9777ea18d3848d9 d336c25d2acb93adcad009bcfb9cde12286df267ada283063de0bb 1505565b2eb6c90e31c48798ecdc71a71756a9110ff373 msg = q128_qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqq Faz-Hernandez, et al. Expires 14 May 2022 [Page 172] Internet-Draft hash-to-curve November 2021 len_in_bytes = 0x80 DST_prime = 515555582d5630312d435330322d776974682d657870616e6465 722d5348414b4532353624 msg_prime = 713132385f717171717171717171717171717171717171717171 717171717171717171717171717171717171717171717171717171 717171717171717171717171717171717171717171717171717171 717171717171717171717171717171717171717171717171717171 717171717171717171717171717171717171717171717171717100 80515555582d5630312d435330322d776974682d657870616e6465 722d5348414b4532353624 uniform_bytes = 4ac054dda0a38a65d0ecf7afd3c2812300027c8789655e47 aecf1ecc1a2426b17444c7482c99e5907afd9c25b991990490bb9c 686f43e79b4471a23a703d4b02f23c669737a886a7ec28bddb92c3 a98de63ebf878aa363a501a60055c048bea11840c4717beae7eee2 8c3cfa42857b3d130188571943a7bd747de831bd6444e0 msg = a512_aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa len_in_bytes = 0x80 DST_prime = 515555582d5630312d435330322d776974682d657870616e6465 722d5348414b4532353624 msg_prime = 613531325f616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 616161616161616161616161616161616161616161616161616161 Faz-Hernandez, et al. Expires 14 May 2022 [Page 173] Internet-Draft hash-to-curve November 2021 61616161610080515555582d5630312d435330322d776974682d65 7870616e6465722d5348414b4532353624 uniform_bytes = 09afc76d51c2cccbc129c2315df66c2be7295a231203b8ab 2dd7f95c2772c68e500bc72e20c602abc9964663b7a03a389be128 c56971ce81001a0b875e7fd17822db9d69792ddf6a23a151bf4700 79c518279aef3e75611f8f828994a9988f4a8a256ddb8bae161e65 8d5a2a09bcfe839c6396dc06ee5c8ff3c22d3b1f9deb7e Authors' Addresses Armando Faz-Hernandez Cloudflare, Inc. 101 Townsend St San Francisco, United States of America Email: armfazh@cloudflare.com Sam Scott Cornell Tech 2 West Loop Rd New York, New York 10044, United States of America Email: sam.scott@cornell.edu Nick Sullivan Cloudflare, Inc. 101 Townsend St San Francisco, United States of America Email: nick@cloudflare.com Riad S. Wahby Stanford University Email: rsw@cs.stanford.edu Faz-Hernandez, et al. Expires 14 May 2022 [Page 174] Internet-Draft hash-to-curve November 2021 Christopher A. Wood Cloudflare, Inc. 101 Townsend St San Francisco, United States of America Email: caw@heapingbits.net Faz-Hernandez, et al. Expires 14 May 2022 [Page 175]