Pairing-Friendly Curves
draft-irtf-cfrg-pairing-friendly-curves-01

Abstract

This memo introduces pairing-friendly curves used for constructing pairing-based cryptography. It describes recommended parameters for each security level and recent implementations of pairing-friendly curves.

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Table of Contents

• 1. Introduction
• 1.1. Pairing-Based Cryptography
• 1.2. Applications of Pairing-Based Cryptography
• 1.3. Goal
• 1.4. Requirements Terminology
• 2. Preliminaries
• 2.1. Elliptic Curve
• 2.2. Pairing
• 2.3. Barreto-Naehrig Curve
• 2.4. Barreto-Lynn-Scott Curve
• 2.5. Representation Convention for an Extension Field
• 3. Security of Pairing-Friendly Curves
• 3.1. Evaluating the Security of Pairing-Friendly Curves
• 3.2. Impact of the Recent Attack
• 4. Security Evaluation of Pairing-Friendly Curves
• 4.1. For 100 Bits of Security
• 4.2. For 128 Bits of Security
• 4.2.1. BN Curves
• 4.2.2. BLS Curves
• 4.3. For 192 Bits of Security
• 4.4. For 256 Bits of Security
• 5. Implementations of Pairing-Friendly Curves
• 6. Security Considerations
• 7. IANA Considerations
• 8. Acknowledgements
• 9. References
• 9.1. Normative References
• 9.2. Informative References
• Appendix A. Computing Optimal Ate Pairing
• A.1. Optimal Ate Pairings over Barreto-Naehrig Curves
• A.2. Optimal Ate Pairings over Barreto-Lynn-Scott Curves
• Appendix B. Test Vectors of Optimal Ate Pairing
• Authors' Addresses

1. Introduction

1.1. Pairing-Based Cryptography

Elliptic curve cryptography is one of the important areas in recent cryptography. The cryptographic algorithms based on elliptic curve cryptography, such as ECDSA (Elliptic Curve Digital Signature Algorithm), are widely used in many applications.

Pairing-based cryptography, a variant of elliptic curve cryptography, has attracted the attention for its flexible and applicable functionality. Pairing is a special map defined over elliptic curves. Thanks to the characteristics of pairing, it can be applied to construct several cryptographic algorithms and protocols such as identity-based encryption (IBE), attribute-based encryption (ABE), authenticated key exchange (AKE), short signatures and so on. Several applications of pairing-based cryptography are now in practical use.

As the importance of pairing grows, elliptic curves where pairing is efficiently computable are studied and the special curves called pairing-friendly curves are proposed.

1.2. Applications of Pairing-Based Cryptography

Several applications using pairing-based cryptography are standardized and implemented. We show example applications available in the real world.

IETF publishes RFCs for pairing-based cryptography such as Identity-Based Cryptography [RFC5091], Sakai-Kasahara Key Encryption (SAKKE) [RFC6508], and Identity-Based Authenticated Key Exchange (IBAKE) [RFC6539]. SAKKE is applied to Multimedia Internet KEYing (MIKEY) [RFC6509] and used in 3GPP [SAKKE].

Pairing-based key agreement protocols are standardized in ISO/IEC [ISOIEC11770-3]. In [ISOIEC11770-3], a key agreement scheme by Joux [Joux00], identity-based key agreement schemes by Smart-Chen-Cheng [CCS07] and by Fujioka-Suzuki-Ustaoglu [FSU10] are specified.

MIRACL implements M-Pin, a multi-factor authentication protocol [M-Pin]. M-Pin protocol includes a kind of zero-knowledge proof, where pairing is used for its construction.

Trusted Computing Group (TCG) specifies ECDAA (Elliptic Curve Direct Anonymous Attestation) in the specification of Trusted Platform Module (TPM) [TPM]. ECDAA is a protocol for proving the attestation held by a TPM to a verifier without revealing the attestation held by that TPM. Pairing is used for constructing ECDAA. FIDO Alliance [FIDO] and W3C [W3C] also published ECDAA algorithm similar to TCG.

Intel introduces Intel Enhanced Privacy ID (EPID) which enables remote attestation of a hardware device while preserving the privacy of the device as a functionality of Intel Software Guard Extensions (SGX) [EPID]. They extend TPM ECDAA to realize such functionality. A pairing-based EPID has been proposed [BL10] and distributed along with Intel SGX applications.

Zcash implements their own zero-knowledge proof algorithm named zk-SNARKs (Zero-Knowledge Succinct Non-Interactive Argument of Knowledge) [Zcash]. zk-SNARKs is used for protecting privacy of transactions of Zcash. They use pairing for constructing zk-SNARKS.

Cloudflare introduces Geo Key Manager [Cloudflare] to restrict distribution of customers’ private keys to the subset of their data centers. To achieve this functionality, attribute-based encryption is used and pairing takes a role as a building block.

Recently, Boneh-Lynn-Shacham (BLS) signature schemes are being standardized [I-D.boneh-bls-signature] and utilized in several blockchain projects such as Ethereum [Ethereum], Algorand [Algorand], Chia Network [Chia] and DFINITY [DFINITY]. The aggregation functionality of BLS signatures is effective for their applications of decentralization and scalability.

1.3. Goal

The goal of this memo is to consider the security of pairing-friendly curves used in pairing-based cryptography and introduce secure parameters of pairing-friendly curves. Specifically, we explain the recent attack against pairing-friendly curves and how much the security of the curves is reduced. We show how to evaluate the security of pairing-friendly curves and give the parameters for 100 bits of security, which is no longer secure, 128, 192 and 256 bits of security.

1.4. Requirements Terminology

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. Preliminaries

2.1. Elliptic Curve

Let p > 3 be a prime and q = p^n for a natural number n. Let F_q be a finite field. The curve defined by the following equation E is called an elliptic curve.

   E : y^2 = x^3 + A * x + B,

where x and y are in F_q, and A and B in F_q satisfy the discriminant inequality 4 * A^3 + 27 * B^2 != 0 mod q. This is called Weierstrass normal form of an elliptic curve.

Solutions (x, y) for an elliptic curve E, as well as the point at infinity, O_E, are called F_q-rational points. If P and Q are two points on the curve E, we can define R = P + Q as the opposite point of the intersection between the curve E and the line that passes through P and Q.
We can define P + O_E = P = O_E + P as well. Similarly, we can define 2P = P + P and a scalar multiplication S = [a]P for a positive integer a can be defined as an (a-1)-time addition of P.

The additive group, denoted by E(F_q), is constructed by the set of F_q-rational points and the addition law described above. We can define the cyclic additive group with a prime order r by taking a base point BP in E(F_q) as a generator. This group is used for the elliptic curve cryptography.

We define terminology used in this memo as follows.

O_E:

the point at infinity over an elliptic curve E.

E(F_q):

a group constructed by F_q-rational points of E.

#E(F_q):

the number of F_q-rational points of E.

h:

a cofactor such that h = #E(F_q) / r.

2.2. Pairing

Pairing is a kind of the bilinear map defined over two elliptic curves E and E’. Examples include Weil pairing, Tate pairing, optimal Ate pairing [Ver09] and so on. Especially, optimal Ate pairing is considered to be efficient to compute and mainly used for practical implementation.

Let E be an elliptic curve defined over a prime field F_p and E’ be an elliptic curve defined over an extension field of F_p. Let k be a minimum integer such that r is a divisor of p^k - 1, which is called an embedding degree. Let G_1 be a cyclic subgroup on the elliptic curve E with order r, and G_2 be a cyclic subgroup on the elliptic curve E’ with order r. Let G_T be an order r subgroup of a multiplicative group (F_p^k)^*.

Pairing is defined as a bilinear map e: (G_1, G_2) -> G_T satisfying the following properties:

1. Bilinearity: for any S in G_1, T in G_2, and integers a and b, e([a]S, [b]T) = e(S, T)^{a * b}.
2. Non-degeneracy: for any T in G_2, e(S, T) = 1 if and only if S = O_E. Similarly, for any S in G_1, e(S, T) = 1 if and only if T = O_E.
3. Computability: for any S in G_1 and T in G_2, the bilinear map is efficiently computable.

2.3. Barreto-Naehrig Curve

A BN curve [BN05] is one of the instantiations of pairing-friendly curves proposed in 2005. A pairing over BN curves constructs optimal Ate pairings.

A BN curve is defined by elliptic curves E and E’ parameterized by a well chosen integer t. E is defined over F_p, where p is a prime more than or equal to 5, and E(F_p) has a subgroup of prime order r. The characteristic p and the order r are parameterized by

    p = 36 * t^4 + 36 * t^3 + 24 * t^2 + 6 * t + 1
    r = 36 * t^4 + 36 * t^3 + 18 * t^2 + 6 * t + 1

for an integer t.

The elliptic curve E has an equation of the form E: y^2 = x^3 + b, where b is an element of multiplicative group of order p.

BN curves always have order 6 twists. If m is an element which is neither a square nor a cube in an extension field F_p^2, the twisted curve E’ of E is defined over an extension field F_p^2 by the equation E’: y^2 = x^3 + b’ with b’ = b / m or b’ = b * m. BN curves are called D-type if b’ = b / m, and M-type if b’ = b * m. The embedded degree k is 12.

A pairing e is defined by taking G_1 as a subgroup of E(F_p) of order r, G_2 as a subgroup of E’(F_p^2), and G_T as a subgroup of a multiplicative group (F_p^12)^* of order r.

2.4. Barreto-Lynn-Scott Curve

A BLS curve [BLS02] is another instantiations of pairings proposed in 2002. Similar to BN curves, a pairing over BLS curves constructs optimal Ate pairings.

A BLS curve is elliptic curves E and E’ parameterized by a well chosen integer t. E is defined over a finite field F_p by an equation of the form E: y^2 = x^3 + b, and its twisted curve, E’: y^2 = x^3 + b’, is defined in the same way as BN curves. In contrast to BN curves, E(F_p) does not have a prime order. Instead, its order is divisible by a large parameterized prime r and denoted by h * r with cofactor h. The pairing will be defined on the r-torsions points. In the same way as BN curves, BLS curves can be categorized into D-type and M-type.

BLS curves vary according to different embedding degrees. In this memo, we deal with BLS12 and BLS48 families with embedding degrees 12 and 48 with respect to r, respectively.

In BLS curves, parameterized p and r are given by the following equations:

   BLS12:
       p = (t - 1)^2 * (t^4 - t^2 + 1) / 3 + t
       r = t^4 - t^2 + 1
   BLS48:
       p = (t - 1)^2 * (t^16 - t^8 + 1) / 3 + t
       r = t^16 - t^8 + 1

for a well chosen integer t.

A pairing e is defined by taking G_1 as a subgroup of E(F_p) of order r, G_2 as an order r subgroup of E’(F_p^2) for BLS12 and of E’(F_p^8) for BLS48, and G_T as an order r subgroup of a multiplicative group (F_p^12)^* for BLS12 and of a multiplicative group (F_p^48)^* for BLS48.

2.5. Representation Convention for an Extension Field

Pairing-friendly curves use a tower of some extension fields. In order to encode an element of an extension field, we adopt the representation convention shown in Appendix J.4 of [I-D.draft-lwig-curve-representations] .

Let F_p be a finite field of characteristic p and F_p^d be an extension field of F_p of degree d and an indeterminate i.

For an element s in F_p^d such that s = s_0 + s_1 * i + ... + s_{d - 1} *  i^{d - 1} for s_0, s_1, ... , s_{d - 1} in a basefield F_p, s is represented as octet string by oct(s) = s_0 || s_1 || ... || s_{d - 1}.

Let F_p^d’ be an extension field of F_p^d of degree d’ / d and an indeterminate j.

For an element s’ in F_p^d’ such that s’ = s’_0 + s’_1 * j + ... + s’_{d’ / d - 1} * j^{d’ / d - 1} for s’_0, s’_1, ..., s’_{d’ / d - 1} in a basefield F_p^d, s’ is represented as integer by oct(s’) = oct(s’_0) || oct(s’_1) || ... || oct(s’_{d’ / d - 1}), where oct(s’_0), ... , oct(s’_{d’ / d - 1}) are octet strings encoded by above convention.

In general, one can define encoding between integer and an element of any finite field tower by inductively applying the above convention.

The parameters and test vectors of extension fields described in this memo are encoded by this convention and represented in octet stream.

3. Security of Pairing-Friendly Curves

3.1. Evaluating the Security of Pairing-Friendly Curves

The security of pairing-friendly curves is evaluated by the hardness of the following discrete logarithm problems.

• The elliptic curve discrete logarithm problem (ECDLP) in G_1 and G_2
• The finite field discrete logarithm problem (FFDLP) in G_T

There are other hard problems over pairing-friendly curves used for proving the security of pairing-based cryptography. Such problems include computational bilinear Diffie-Hellman (CBDH) problem and bilinear Diffie-Hellman (BDH) Problem, decision bilinear Diffie-Hellman (DBDH) problem, gap DBDH problem, etc [ECRYPT]. Almost all of these variants are reduced to the hardness of discrete logarithm problems described above and believed to be easier than the discrete logarithm problems.

There would be the case where the attacker solves these reduced problems to break pairing-based cryptography. Since such attacks have not been discovered yet, we discuss the hardness of the discrete logarithm problems in this memo.

The security level of pairing-friendly curves is estimated by the computational cost of the most efficient algorithm to solve the above discrete logarithm problems. The well-known algorithms for solving the discrete logarithm problems include Pollard’s rho algorithm [Pollard78], Index Calculus [HR83] and so on. In order to make index calculus algorithms more efficient, number field sieve (NFS) algorithms are utilized.

3.2. Impact of the Recent Attack

In 2016, Kim and Barbulescu proposed a new variant of the NFS algorithms, the extended tower number field sieve (exTNFS), which drastically reduces the complexity of solving FFDLP [KB16]. Due to exTNFS, the security level of pairing-friendly curves asymptotically dropped down. For instance, Barbulescu and Duquesne estimated that the security of the BN curves which had been believed to provide 128 bits of security (BN256, for example) dropped down to approximately 100 bits [BD18].

Some papers showed the minimum bit length of the parameters of pairing-friendly curves for each security level when applying exTNFS as an attacking method for FFDLP. For 128 bits of security, Menezes, Sarkar and Singh estimated the minimum bit length of p of BN curves after exTNFS as 383 bits, and that of BLS12 curves as 384 bits [MSS17]. For 256 bits of security, Kiyomura et al. estimated the minimum bit length of p^k of BLS48 curves as 27,410 bits, which implied 572 bits of p [KIK17].

4. Security Evaluation of Pairing-Friendly Curves

We give security evaluation for pairing-friendly curves based on the evaluating method presented in Section 3. We also introduce secure parameters of pairing-friendly curves for each security level. The parameters introduced here are chosen with the consideration of security, efficiency and global acceptance.

For security, we introduce the parameters with 100 bits, 128 bits, 192 bits and 256 bits of security. We note that 100 bits of security is no longer secure and recommend 128 bits, 192 bits and 256 bits of security for secure applications. We follow TLS 1.3 [RFC8446] which specifies the cipher suites with 128 bits and 256 bits of security as mandatory-to-implement for the choice of the security level.

Implementers of the applications have to choose the parameters with appropriate security level according to the security requirements of the applications. For efficiency, we refer to the benchmark by mcl [mcl] for 128 bits of security, and by Kiyomura et al. [KIK17] for 256 bits of security, and then choose sufficiently efficient parameters. For global acceptance, we give the implementations of pairing-friendly curves in Section 5.

4.1. For 100 Bits of Security

Before exTNFS, BN curves with 256-bit size of underlying finite field (so-called BN256) were considered to achieve 128 bits of security. After exTNFS, however, the security level of BN curves with 256-bit size of underlying finite field fell into 100 bits.

Implementers who will newly develop the applications of pairing-based cryptography SHOULD NOT use pairing-friendly curves with 100 bits of security (i.e. BN256).

There exists applications which already implemented pairing-based cryptography with 100-bit secure pairing-friendly curves. In such a case, implementers MAY use 100 bits of security only if they need to keep interoperability with the existing applications.

4.2. For 128 Bits of Security

4.2.1. BN Curves

A BN curve with 128 bits of security is shown in [BD18], which we call BN462. BN462 is defined by a parameter

    t = 2^114 + 2^101 - 2^14 - 1 

for the definition in Section 2.3.

For the finite field F_p, the towers of extension field F_p^2, F_p^6 and F_p^12 are defined by indeterminates u, v, w as follows:

    F_p^2 = F_p[u] / (u^2 + 1)
    F_p^6 = F_p^2[v] / (v^3 - u - 2)
    F_p^12 = F_p^6[w] / (w^2 - v).

Defined by t, the elliptic curve E and its twisted curve E’ are represented by E: y^2 = x^3 + 5 and E’: y^2 = x^3 - u + 2, respectively. The size of p becomes 462-bit length. A pairing e is defined by taking G_1 as a cyclic group of order r generated by a base point BP = (x, y) in F_p, G_2 as a cyclic group of order r generated by a based point BP’ = (x’, y’) in F_p^2, and G_T as a subgroup of a multiplicative group (F_p^12)^* of order r. BN462 is D-type.

We give the following parameters for BN462.

• G_1 defined over E: y^2 = x^3 + b
  • p : a characteristic
  • r : an order
  • BP = (x, y) : a base point
  • h : a cofactor
  • b : a coefficient of E
• G_2 defined over E’: y^2 = x^3 + b’
  • r’ : an order
  • BP’ = (x’, y’) : a base point (encoded with [I-D.draft-lwig-curve-representations])
    • x’ = x’_0 + x’_1 * u (x’_0, x’_1 in F_p)
    • y’ = y’_0 + y’_1 * u (y’_0, y’_1 in F_p)
  • h’ : a cofactor
  • b’ : a coefficient of E’

p:

0x240480360120023ffffffffff6ff0cf6b7d9bfca0000000000d812908f41c8020ffffffffff6ff66fc6ff687f640000000002401b00840138013

r:

0x240480360120023ffffffffff6ff0cf6b7d9bfca0000000000d812908ee1c201f7fffffffff6ff66fc7bf717f7c0000000002401b007e010800d

x:

0x21a6d67ef250191fadba34a0a30160b9ac9264b6f95f63b3edbec3cf4b2e689db1bbb4e69a416a0b1e79239c0372e5cd70113c98d91f36b6980d

y:

0x0118ea0460f7f7abb82b33676a7432a490eeda842cccfa7d788c659650426e6af77df11b8ae40eb80f475432c66600622ecaa8a5734d36fb03de

h:

1

b:

5

r’:

0x240480360120023ffffffffff6ff0cf6b7d9bfca0000000000d812908ee1c201f7fffffffff6ff66fc7bf717f7c0000000002401b007e010800d

x’_0:

0x0257ccc85b58dda0dfb38e3a8cbdc5482e0337e7c1cd96ed61c913820408208f9ad2699bad92e0032ae1f0aa6a8b48807695468e3d934ae1e4df

x’_1:

0x1d2e4343e8599102af8edca849566ba3c98e2a354730cbed9176884058b18134dd86bae555b783718f50af8b59bf7e850e9b73108ba6aa8cd283

y’_0:

0x0a0650439da22c1979517427a20809eca035634706e23c3fa7a6bb42fe810f1399a1f41c9ddae32e03695a140e7b11d7c3376e5b68df0db7154e

y’_1:

0x073ef0cbd438cbe0172c8ae37306324d44d5e6b0c69ac57b393f1ab370fd725cc647692444a04ef87387aa68d53743493b9eba14cc552ca2a93a

h’:

0x240480360120023ffffffffff6ff0cf6b7d9bfca0000000000d812908fa1ce0227fffffffff6ff66fc63f5f7f4c0000000002401b008a0168019

b’:

-u + 2

4.2.2. BLS Curves

A BLS12 curve with 128 bits of security shown in [BLS12-381], BLS12-381, is defined by a parameter

    t = -2^63 - 2^62 - 2^60 - 2^57 - 2^48 - 2^16 

and the size of p becomes 381-bit length.

For the finite field F_p, the towers of extension field F_p^2, F_p^6 and F_p^12 are defined by indeterminates u, v, w as follows:

    F_p^2 = F_p[u] / (u^2 + 1)
    F_p^6 = F_p^2[v] / (v^3 - u - 1)
    F_p^12 = F_p^6[w] / (w^2 - v).

Defined by t, the elliptic curve E and its twisted curve E’ are represented by E: y^2 = x^3 + 4 and E’: y^2 = x^3 + 4(u + 1).

A pairing e is defined by taking G_1 as a cyclic group of order r generated by a base point BP = (x, y) in F_p, G_2 as a cyclic group of order r generated by a based point BP’ = (x’, y’) in F_p^2, and G_T as a subgroup of a multiplicative group (F_p^12)^* of order r. BLS12-381 is M-type.

We have to note that, according to [MSS17], the bit length of p for BLS12 to achieve 128 bits of security is calculated as 384 bits and more, which BLS12-381 does not satisfy. They state that BLS12-381 achieves 127-bit security level evaluated by the computational cost of Pollard’s rho, whereas NCC group estimated that the security level of BLS12-381 is between 117 and 120 bits at most [NCCG]. Therefore, we regard BN462 as a "conservative" parameter, and BLS12-381 as an "optimistic" parameter.

We give the following parameters for BLS12-381.

• G_1 defined over E: y^2 = x^3 + b
  • p : a characteristic
  • r : an order
  • BP = (x, y) : a base point
  • h : a cofactor
  • b : a coefficient of E
• G_2 defined over E’: y^2 = x^3 + b’
  • r’ : an order
  • BP’ = (x’, y’) : a base point (encoded with [I-D.draft-lwig-curve-representations])
    • x’ = x’_0 + x’_1 * u (x’_0, x’_1 in F_p)
    • y’ = y’_0 + y’_1 * u (y’_0, y’_1 in F_p)
  • h’ : a cofactor
  • b’ : a coefficient of E’

p:

0x1a0111ea397fe69a4b1ba7b6434bacd764774b84f38512bf6730d2a0f6b0f6241eabfffeb153ffffb9feffffffffaaab

r:

0x73eda753299d7d483339d80809a1d80553bda402fffe5bfeffffffff00000001

x:

0x17f1d3a73197d7942695638c4fa9ac0fc3688c4f9774b905a14e3a3f171bac586c55e83ff97a1aeffb3af00adb22c6bb

y:

0x08b3f481e3aaa0f1a09e30ed741d8ae4fcf5e095d5d00af600db18cb2c04b3edd03cc744a2888ae40caa232946c5e7e1

h:

0x396c8c005555e1568c00aaab0000aaab

b:

4

r’:

0x1a0111ea397fe69a4b1ba7b6434bacd764774b84f38512bf6730d2a0f6b0f6241eabfffeb153ffffb9feffffffffaaab

x’_0:

0x024aa2b2f08f0a91260805272dc51051c6e47ad4fa403b02b4510b647ae3d1770bac0326a805bbefd48056c8c121bdb8

x’_1:

0x13e02b6052719f607dacd3a088274f65596bd0d09920b61ab5da61bbdc7f5049334cf11213945d57e5ac7d055d042b7e

y’_0:

0x0ce5d527727d6e118cc9cdc6da2e351aadfd9baa8cbdd3a76d429a695160d12c923ac9cc3baca289e193548608b82801

y’_1:

0x0606c4a02ea734cc32acd2b02bc28b99cb3e287e85a763af267492ab572e99ab3f370d275cec1da1aaa9075ff05f79be

h’:

0x5d543a95414e7f1091d50792876a202cd91de4547085abaa68a205b2e5a7ddfa628f1cb4d9e82ef21537e293a6691ae1616ec6e786f0c70cf1c38e31c7238e5

b’:

4 * (u + 1)

4.3. For 192 Bits of Security

(TBD)

4.4. For 256 Bits of Security

As shown in Section 3.2, it is unrealistic to achieve 256 bits of security by BN curves since the minimum size of p becomes too large to implement. Hence, we consider BLS48 for 256 bits of security.

A BLS48 curve with 256 bits of security is shown in [KIK17], which we call BLS48-581. It is defined by a parameter

    t = -1 + 2^7 - 2^10 - 2^30 - 2^32. 

For the finite field F_p, the towers of extension field F_p^2, F_p^4, F_p^8, F_p^24 and F_p^48 are defined by indeterminates u, v, w, z, s as follows:

    F_p^2 = F_p[u] / (u^2 + 1)
    F_p^4 = F_p^2[v] / (v^2 + u + 1)
    F_p^8 = F_p^4[w] / (w^2 + v)
    F_p^24 = F_p^8[z] / (z^3 + w)
    F_p^48 = F_p^24[s] / (s^2 + z).

The elliptic curve E and its twisted curve E’ are represented by E: y^2 = x^3 + 1 and E’: y^2 = x^3 - 1 / w. A pairing e is defined by taking G_1 as a cyclic group of order r generated by a base point BP = (x, y) in F_p, G_2 as a cyclic group of order r generated by a based point BP’ = (x’, y’) in F_p^8, and G_T as a subgroup of a multiplicative group (F_p^48)^* of order r. The size of p becomes 581-bit length. BLS48-581 is D-type.

We then give the parameters for BLS48-581 as follows.

• G_1 defined over E: y^2 = x^3 + b
  • p : a characteristic
  • r : a prime which divides an order of G_1
  • BP = (x, y) : a base point
  • h : a cofactor
  • b : a coefficient of E
• G_2 defined over E’: y^2 = x^3 + b’
  • r’ : an order
  • BP’ = (x’, y’) : a base point (encoded with [I-D.draft-lwig-curve-representations])
    • x’ = x’_0 + x’_1 * u + x’_2 * v + x’_3 * u * v + x’_4 * w + x’_5 * u * w + x’_6 * v * w + x’_7 * u * v * w (x’_0, …, x’_7 in F_p)
    • y’ = y’_0 + y’_1 * u + y’_2 * v + y’_3 * u * v + y’_4 * w + y’_5 * u * w + y’_6 * v * w + y’_7 * u * v * w (y’_0, …, y’_7 in F_p)
  • h’ : a cofactor
  • b’ : a coefficient of E’

p:

0x1280f73ff3476f313824e31d47012a0056e84f8d122131bb3be6c0f1f3975444a48ae43af6e082acd9cd30394f4736daf68367a5513170ee0a578fdf721a4a48ac3edc154e6565912b

r:

0x2386f8a925e2885e233a9ccc1615c0d6c635387a3f0b3cbe003fad6bc972c2e6e741969d34c4c92016a85c7cd0562303c4ccbe599467c24da118a5fe6fcd671c01

x:

0x02af59b7ac340f2baf2b73df1e93f860de3f257e0e86868cf61abdbaedffb9f7544550546a9df6f9645847665d859236ebdbc57db368b11786cb74da5d3a1e6d8c3bce8732315af640

y:

0x0cefda44f6531f91f86b3a2d1fb398a488a553c9efeb8a52e991279dd41b720ef7bb7beffb98aee53e80f678584c3ef22f487f77c2876d1b2e35f37aef7b926b576dbb5de3e2587a70

x’_0:

0x05d615d9a7871e4a38237fa45a2775debabbefc70344dbccb7de64db3a2ef156c46ff79baad1a8c42281a63ca0612f400503004d80491f510317b79766322154dec34fd0b4ace8bfab

x’_1:

0x07c4973ece2258512069b0e86abc07e8b22bb6d980e1623e9526f6da12307f4e1c3943a00abfedf16214a76affa62504f0c3c7630d979630ffd75556a01afa143f1669b36676b47c57

x’_2:

0x01fccc70198f1334e1b2ea1853ad83bc73a8a6ca9ae237ca7a6d6957ccbab5ab6860161c1dbd19242ffae766f0d2a6d55f028cbdfbb879d5fea8ef4cded6b3f0b46488156ca55a3e6a

x’_3:

0x0be2218c25ceb6185c78d8012954d4bfe8f5985ac62f3e5821b7b92a393f8be0cc218a95f63e1c776e6ec143b1b279b9468c31c5257c200ca52310b8cb4e80bc3f09a7033cbb7feafe

x’_4:

0x038b91c600b35913a3c598e4caa9dd63007c675d0b1642b5675ff0e7c5805386699981f9e48199d5ac10b2ef492ae589274fad55fc1889aa80c65b5f746c9d4cbb739c3a1c53f8cce5

x’_5:

0x0c96c7797eb0738603f1311e4ecda088f7b8f35dcef0977a3d1a58677bb037418181df63835d28997eb57b40b9c0b15dd7595a9f177612f097fc7960910fce3370f2004d914a3c093a

x’_6:

0x0b9b7951c6061ee3f0197a498908aee660dea41b39d13852b6db908ba2c0b7a449cef11f293b13ced0fd0caa5efcf3432aad1cbe4324c22d63334b5b0e205c3354e41607e60750e057

x’_7:

0x0827d5c22fb2bdec5282624c4f4aaa2b1e5d7a9defaf47b5211cf741719728a7f9f8cfca93f29cff364a7190b7e2b0d4585479bd6aebf9fc44e56af2fc9e97c3f84e19da00fbc6ae34

y’_0:

0x00eb53356c375b5dfa497216452f3024b918b4238059a577e6f3b39ebfc435faab0906235afa27748d90f7336d8ae5163c1599abf77eea6d659045012ab12c0ff323edd3fe4d2d7971

y’_1:

0x0284dc75979e0ff144da6531815fcadc2b75a422ba325e6fba01d72964732fcbf3afb096b243b1f192c5c3d1892ab24e1dd212fa097d760e2e588b423525ffc7b111471db936cd5665

y’_2:

0x0b36a201dd008523e421efb70367669ef2c2fc5030216d5b119d3a480d370514475f7d5c99d0e90411515536ca3295e5e2f0c1d35d51a652269cbc7c46fc3b8fde68332a526a2a8474

y’_3:

0x0aec25a4621edc0688223fbbd478762b1c2cded3360dcee23dd8b0e710e122d2742c89b224333fa40dced2817742770ba10d67bda503ee5e578fb3d8b8a1e5337316213da92841589d

y’_4:

0x0d209d5a223a9c46916503fa5a88325a2554dc541b43dd93b5a959805f1129857ed85c77fa238cdce8a1e2ca4e512b64f59f430135945d137b08857fdddfcf7a43f47831f982e50137

y’_5:

0x07d0d03745736b7a513d339d5ad537b90421ad66eb16722b589d82e2055ab7504fa83420e8c270841f6824f47c180d139e3aafc198caa72b679da59ed8226cf3a594eedc58cf90bee4

y’_6:

0x0896767811be65ea25c2d05dfdd17af8a006f364fc0841b064155f14e4c819a6df98f425ae3a2864f22c1fab8c74b2618b5bb40fa639f53dccc9e884017d9aa62b3d41faeafeb23986

y’_7:

0x035e2524ff89029d393a5c07e84f981b5e068f1406be8e50c87549b6ef8eca9a9533a3f8e69c31e97e1ad0333ec719205417300d8c4ab33f748e5ac66e84069c55d667ffcb732718b6

h:

0x85555841aaaec4ac

b:

1

r’:

0x2386f8a925e2885e233a9ccc1615c0d6c635387a3f0b3cbe003fad6bc972c2e6e741969d34c4c92016a85c7cd0562303c4ccbe599467c24da118a5fe6fcd671c01

h’:

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

b’:

-1 / w

5. Implementations of Pairing-Friendly Curves

We show the pairing-friendly curves selected by existing standards, cryptographic libraries and applications.

ISO/IEC 15946-5 [ISOIEC15946-5] shows examples of BN curves with the size of 160, 192, 224, 256, 384 and 512 bits of p. There is no action so far after the proposal of exTNFS.

TCG adopts an BN curve of 256 bits specified in ISO/IEC 15946-5 (TPM_ECC_BN_P256) and that of 638 bits specified by their own (TPM_ECC_BN_P638). FIDO Alliance [FIDO] and W3C [W3C] adopt the same BN curves as TCG, a 512-bit BN curve shown in ISO/IEC 15946-5 and another 256-bit BN curve.

Cryptographic libraries which implement pairings include PBC [PBC], mcl [mcl], RELIC [RELIC], TEPLA [TEPLA], AMCL [AMCL], Intel IPP [Intel-IPP] and a library by Kyushu University [BLS48].

Cloudflare published a new cryptographic library CIRCL (Cloudflare Interoperable, Reusable Cryptographic Library) in 2019 [CIRCL]. The plan for the implementation of secure pairing-friendly curves is stated in their roadmap.

MIRACL implements BN curves and BLS12 curves [MIRACL].

Zcash implements a BN curve (named BN128) in their library libsnark [libsnark]. After exTNFS, they propose a new parameter of BLS12 as BLS12-381 [BLS12-381] and publish its experimental implementation [zkcrypto].

Ethereum 2.0 adopts BLS12-381 (BLS12_381), BN curves with 254 bits of p (CurveFp254BNb) and 382 bits of p (CurveFp382_1 and CurveFp382_2) [go-bls]. Their implementation calls mcl [mcl] for pairing computation. Chia Network publishs their implementation [Chia] by integrating the RELIC toolkit [RELIC].

Table 1 shows the adoption of pairing-friendly curves in existing standards, cryptographic libraries and applications. In this table, the curves marked as (*) indicate that the security level is evaluated less than the one labeld in the table.

6. Security Considerations

This memo entirely describes the security of pairing-friendly curves, and introduces secure parameters of pairing-friendly curves. We give these parameters in terms of security, efficiency and global acceptance. The parameters for 100, 128, 192 and 256 bits of security are introduced since the security level will different in the requirements of the pairing-based applications. Implementers can select these parameters according to their security requirements.

7. IANA Considerations

This document has no actions for IANA.

8. Acknowledgements

The authors would like to thank Akihiro Kato and Shoko Yonezawa for their significant contribution to the early version of this memo. The authors would also like to acknowledge Sakae Chikara, Hoeteck Wee, Sergey Gorbunov and Michael Scott for their valuable comments.

9. References

9.1. Normative References

9.2. Informative References

Appendix A. Computing Optimal Ate Pairing

Before presenting the computation of optimal Ate pairing e(P, Q) satisfying the properties shown in Section 2.2, we give subfunctions used for pairing computation.

The following algorithm Line_Function shows the computation of the line function. It takes A = (A[1], A[2]), B = (B[1], B[2]) in G_2 and P = ((P[1], P[2])) in G_1 as input and outputs an element of G_T.

    if (A = B) then
        l := (3 * A[1]^2) / (2 * A[2]);
    else if (A = -B) then
        return P[1] - A[1];
    else
        l := (B[2] - A[2]) / (B[1] - A[1]);
    end if;
    return (l * (P[1] -A[1]) + A[2] -P[2]);

When implementing the line function, implementers should consider the isomorphism of E and its twisted curve E’ so that one can reduce the computational cost of operations in G_2. We note that the function Line_function does not consider such isomorphism.

Computation of optimal Ate pairing for BN curves uses Frobenius map. Let a Frobenius map pi for a point Q = (x, y) over E’ be pi(p, Q) = (x^p, y^p).

A.1. Optimal Ate Pairings over Barreto-Naehrig Curves

Let c = 6 * t + 2 for a parameter t and c_0, c_1, … , c_L in {-1,0,1} such that the sum of c_i * 2^i (i = 0, 1, …, L) equals to c.

The following algorithm shows the computation of optimal Ate pairing over Barreto-Naehrig curves. It takes P in G_1, Q in G_2, an integer c, c_0, …,c_L in {-1,0,1} such that the sum of c_i * 2^i (i = 0, 1, …, L) equals to c, and an order r as input, and outputs e(P, Q).

    f := 1; T := Q;
    if (c_L = -1)
        T := -T;
    end if
    for i = L-1 to 0
        f := f^2 * Line_function(T, T, P); T := 2 * T;
        if (c_i = 1 | c_i = -1)
            f := f * Line_function(T, c_i * Q); T := T + c_i * Q;
        end if
    end for
    Q_1 := pi(p, Q); Q_2 := pi(p, Q_1);
    f := f * Line_function(T, Q_1, P); T := T + Q_1;
    f := f * Line_function(T, -Q_2, P);
    f := f^{(p^k - 1) / r}
    return f;

A.2. Optimal Ate Pairings over Barreto-Lynn-Scott Curves

Let c = t for a parameter t and c_0, c_1, … , c_L in {-1,0,1} such that the sum of c_i * 2^i (i = 0, 1, …, L) equals to c. The following algorithm shows the computation of optimal Ate pairing over Barreto-Lynn-Scott curves. It takes P in G_1, Q in G_2, a parameter c, c_0, c_1, …, c_L in {-1,0,1} such that the sum of c_i * 2^i (i = 0, 1, …, L), and an order r as input, and outputs e(P, Q).

    f := 1; T := Q;
    if (c_L = -1)
        T := -T;
    end if
    for i = L-1 to 0
        f := f^2 * Line_function(T, T, P); T := 2 * T;
        if (c_i = 1 | c_i = -1)
            f := f * Line_function(T, c_i * Q, P); T := T + c_i * Q;
        end if
    end for
    f := f^{(p^k - 1) / r};
    return f;

Appendix B. Test Vectors of Optimal Ate Pairing

We provide test vectors for Optimal Ate Pairing e(P, Q) given in Appendix A for the curves BN462, BLS12-381 and BLS48-581 given in Section 4. Here, the inputs P = (x, y) and Q = (x’, y’) are the corresponding base points BP and BP’ given in Section 4.

For BN462 and BLS12-381, Q = (x’, y’) is given by

    x' = x'_0 + x'_1 * u and
    y' = y'_0 + y'_1 * u,

where u is a indeterminate and x’_0, x’_1, y’_0, y’_1 are elements of F_p.

For BLS48-581, Q = (x’, y’) is given by

    x' = x'_0 + x'_1 * u + x'_2 * v + x'_3 * u * v 
        + x'_4 * w + x'_5 * u * w + x'_6 * v * w + x'_7 * u * v * w and 
    y' = y'_0 + y'_1 * u + y'_2 * v + y'_3 * u * v 
        + y'_4 * w + y'_5 * u * w + y'_6 * v * w + y'_7 * u * v * w,

where u, v and w are indeterminates and x’_0, …, x’_7 and y’_0, …, y’_7 are elements of F_p. The representation of Q = (x’, y’) given below is followed by [I-D.draft-lwig-curve-representations].

BN462:

Input x value:

0x21a6d67ef250191fadba34a0a30160b9ac9264b6f95f63b3edbec3cf4b2e689db1bbb4e69a416a0b1e79239c0372e5cd70113c98d91f36b6980d

Input y value:

0x0118ea0460f7f7abb82b33676a7432a490eeda842cccfa7d788c659650426e6af77df11b8ae40eb80f475432c66600622ecaa8a5734d36fb03de

Input x’_0 value:

0x0257ccc85b58dda0dfb38e3a8cbdc5482e0337e7c1cd96ed61c913820408208f9ad2699bad92e0032ae1f0aa6a8b48807695468e3d934ae1e4df

Input x’_1 value:

0x1d2e4343e8599102af8edca849566ba3c98e2a354730cbed9176884058b18134dd86bae555b783718f50af8b59bf7e850e9b73108ba6aa8cd283

Input y’_0 value:

0x0a0650439da22c1979517427a20809eca035634706e23c3fa7a6bb42fe810f1399a1f41c9ddae32e03695a140e7b11d7c3376e5b68df0db7154e

Input y’_1 value:

0x073ef0cbd438cbe0172c8ae37306324d44d5e6b0c69ac57b393f1ab370fd725cc647692444a04ef87387aa68d53743493b9eba14cc552ca2a93a

e_0:

0x0cf7f0f2e01610804272f4a7a24014ac085543d787c8f8bf07059f93f87ba7e2a4ac77835d4ff10e78669be39cd23cc3a659c093dbe3b9647e8c

e_1:

0x00ef2c737515694ee5b85051e39970f24e27ca278847c7cfa709b0df408b830b3763b1b001f1194445b62d6c093fb6f77e43e369edefb1200389

e_2:

0x04d685b29fd2b8faedacd36873f24a06158742bb2328740f93827934592d6f1723e0772bb9ccd3025f88dc457fc4f77dfef76104ff43cd430bf7

e_3:

0x090067ef2892de0c48ee49cbe4ff1f835286c700c8d191574cb424019de11142b3c722cc5083a71912411c4a1f61c00d1e8f14f545348eb7462c

e_4:

0x1437603b60dce235a090c43f5147d9c03bd63081c8bb1ffa7d8a2c31d673230860bb3dfe4ca85581f7459204ef755f63cba1fbd6a4436f10ba0e

e_5:

0x13191b1110d13650bf8e76b356fe776eb9d7a03fe33f82e3fe5732071f305d201843238cc96fd0e892bc61701e1844faa8e33446f87c6e29e75f

e_6:

0x07b1ce375c0191c786bb184cc9c08a6ae5a569dd7586f75d6d2de2b2f075787ee5082d44ca4b8009b3285ecae5fa521e23be76e6a08f17fa5cc8

e_7:

0x05b64add5e49574b124a02d85f508c8d2d37993ae4c370a9cda89a100cdb5e1d441b57768dbc68429ffae243c0c57fe5ab0a3ee4c6f2d9d34714

e_8:

0x0fd9a3271854a2b4542b42c55916e1faf7a8b87a7d10907179ac7073f6a1de044906ffaf4760d11c8f92df3e50251e39ce92c700a12e77d0adf3

e_9:

0x17fa0c7fa60c9a6d4d8bb9897991efd087899edc776f33743db921a689720c82257ee3c788e8160c112f18e841a3dd9a79a6f8782f771d542ee5

e_10:

0x0c901397a62bb185a8f9cf336e28cfb0f354e2313f99c538cdceedf8b8aa22c23b896201170fc915690f79f6ba75581f1b76055cd89b7182041c

e_11:

0x20f27fde93cee94ca4bf9ded1b1378c1b0d80439eeb1d0c8daef30db0037104a5e32a2ccc94fa1860a95e39a93ba51187b45f4c2c50c16482322

BLS12-381:

Input x value:

0x17f1d3a73197d7942695638c4fa9ac0fc3688c4f9774b905a14e3a3f171bac586c55e83ff97a1aeffb3af00adb22c6bb

Input y value:

0x08b3f481e3aaa0f1a09e30ed741d8ae4fcf5e095d5d00af600db18cb2c04b3edd03cc744a2888ae40caa232946c5e7e1

Input x’_0 value:

0x024aa2b2f08f0a91260805272dc51051c6e47ad4fa403b02b4510b647ae3d1770bac0326a805bbefd48056c8c121bdb8

Input x’_1 value:

0x13e02b6052719f607dacd3a088274f65596bd0d09920b61ab5da61bbdc7f5049334cf11213945d57e5ac7d055d042b7e

Input y’_0 value:

0x0ce5d527727d6e118cc9cdc6da2e351aadfd9baa8cbdd3a76d429a695160d12c923ac9cc3baca289e193548608b82801

Input y’_1 value:

0x0606c4a02ea734cc32acd2b02bc28b99cb3e287e85a763af267492ab572e99ab3f370d275cec1da1aaa9075ff05f79be

e_0:

0x11619b45f61edfe3b47a15fac19442526ff489dcda25e59121d9931438907dfd448299a87dde3a649bdba96e84d54558

e_1:

0x153ce14a76a53e205ba8f275ef1137c56a566f638b52d34ba3bf3bf22f277d70f76316218c0dfd583a394b8448d2be7f

e_2:

0x095668fb4a02fe930ed44767834c915b283b1c6ca98c047bd4c272e9ac3f3ba6ff0b05a93e59c71fba77bce995f04692

e_3:

0x16deedaa683124fe7260085184d88f7d036b86f53bb5b7f1fc5e248814782065413e7d958d17960109ea006b2afdeb5f

e_4:

0x09c92cf02f3cd3d2f9d34bc44eee0dd50314ed44ca5d30ce6a9ec0539be7a86b121edc61839ccc908c4bdde256cd6048

e_5:

0x111061f398efc2a97ff825b04d21089e24fd8b93a47e41e60eae7e9b2a38d54fa4dedced0811c34ce528781ab9e929c7

e_6:

0x01ecfcf31c86257ab00b4709c33f1c9c4e007659dd5ffc4a735192167ce197058cfb4c94225e7f1b6c26ad9ba68f63bc

e_7:

0x08890726743a1f94a8193a166800b7787744a8ad8e2f9365db76863e894b7a11d83f90d873567e9d645ccf725b32d26f

e_8:

0x0e61c752414ca5dfd258e9606bac08daec29b3e2c57062669556954fb227d3f1260eedf25446a086b0844bcd43646c10

e_9:

0x0fe63f185f56dd29150fc498bbeea78969e7e783043620db33f75a05a0a2ce5c442beaff9da195ff15164c00ab66bdde

e_10:

0x10900338a92ed0b47af211636f7cfdec717b7ee43900eee9b5fc24f0000c5874d4801372db478987691c566a8c474978

e_11:

0x1454814f3085f0e6602247671bc408bbce2007201536818c901dbd4d2095dd86c1ec8b888e59611f60a301af7776be3d

BLS48-581:

Input x value:

0x02af59b7ac340f2baf2b73df1e93f860de3f257e0e86868cf61abdbaedffb9f7544550546a9df6f9645847665d859236ebdbc57db368b11786cb74da5d3a1e6d8c3bce8732315af640

Input y value:

0x0cefda44f6531f91f86b3a2d1fb398a488a553c9efeb8a52e991279dd41b720ef7bb7beffb98aee53e80f678584c3ef22f487f77c2876d1b2e35f37aef7b926b576dbb5de3e2587a70

x’_0:

0x05d615d9a7871e4a38237fa45a2775debabbefc70344dbccb7de64db3a2ef156c46ff79baad1a8c42281a63ca0612f400503004d80491f510317b79766322154dec34fd0b4ace8bfab

x’_1:

0x07c4973ece2258512069b0e86abc07e8b22bb6d980e1623e9526f6da12307f4e1c3943a00abfedf16214a76affa62504f0c3c7630d979630ffd75556a01afa143f1669b36676b47c57

x’_2:

0x01fccc70198f1334e1b2ea1853ad83bc73a8a6ca9ae237ca7a6d6957ccbab5ab6860161c1dbd19242ffae766f0d2a6d55f028cbdfbb879d5fea8ef4cded6b3f0b46488156ca55a3e6a

x’_3:

0x0be2218c25ceb6185c78d8012954d4bfe8f5985ac62f3e5821b7b92a393f8be0cc218a95f63e1c776e6ec143b1b279b9468c31c5257c200ca52310b8cb4e80bc3f09a7033cbb7feafe

x’_4:

0x038b91c600b35913a3c598e4caa9dd63007c675d0b1642b5675ff0e7c5805386699981f9e48199d5ac10b2ef492ae589274fad55fc1889aa80c65b5f746c9d4cbb739c3a1c53f8cce5

x’_5:

0x0c96c7797eb0738603f1311e4ecda088f7b8f35dcef0977a3d1a58677bb037418181df63835d28997eb57b40b9c0b15dd7595a9f177612f097fc7960910fce3370f2004d914a3c093a

x’_6:

0x0b9b7951c6061ee3f0197a498908aee660dea41b39d13852b6db908ba2c0b7a449cef11f293b13ced0fd0caa5efcf3432aad1cbe4324c22d63334b5b0e205c3354e41607e60750e057

x’_7:

0x0827d5c22fb2bdec5282624c4f4aaa2b1e5d7a9defaf47b5211cf741719728a7f9f8cfca93f29cff364a7190b7e2b0d4585479bd6aebf9fc44e56af2fc9e97c3f84e19da00fbc6ae34

y’_0:

0x00eb53356c375b5dfa497216452f3024b918b4238059a577e6f3b39ebfc435faab0906235afa27748d90f7336d8ae5163c1599abf77eea6d659045012ab12c0ff323edd3fe4d2d7971

y’_1:

0x0284dc75979e0ff144da6531815fcadc2b75a422ba325e6fba01d72964732fcbf3afb096b243b1f192c5c3d1892ab24e1dd212fa097d760e2e588b423525ffc7b111471db936cd5665

y’_2:

0x0b36a201dd008523e421efb70367669ef2c2fc5030216d5b119d3a480d370514475f7d5c99d0e90411515536ca3295e5e2f0c1d35d51a652269cbc7c46fc3b8fde68332a526a2a8474

y’_3:

0x0aec25a4621edc0688223fbbd478762b1c2cded3360dcee23dd8b0e710e122d2742c89b224333fa40dced2817742770ba10d67bda503ee5e578fb3d8b8a1e5337316213da92841589d

y’_4:

0x0d209d5a223a9c46916503fa5a88325a2554dc541b43dd93b5a959805f1129857ed85c77fa238cdce8a1e2ca4e512b64f59f430135945d137b08857fdddfcf7a43f47831f982e50137

y’_5:

0x07d0d03745736b7a513d339d5ad537b90421ad66eb16722b589d82e2055ab7504fa83420e8c270841f6824f47c180d139e3aafc198caa72b679da59ed8226cf3a594eedc58cf90bee4

y’_6:

0x0896767811be65ea25c2d05dfdd17af8a006f364fc0841b064155f14e4c819a6df98f425ae3a2864f22c1fab8c74b2618b5bb40fa639f53dccc9e884017d9aa62b3d41faeafeb23986

y’_7:

0x035e2524ff89029d393a5c07e84f981b5e068f1406be8e50c87549b6ef8eca9a9533a3f8e69c31e97e1ad0333ec719205417300d8c4ab33f748e5ac66e84069c55d667ffcb732718b6

e_0:

0x0e26c3fcb8ef67417814098de5111ffcccc1d003d15b367bad07cef2291a93d31db03e3f03376f3beae2bd877bcfc22a25dc51016eda1ab56ee3033bc4b4fec5962f02dffb3af5e38e

e_1:

0x069061b8047279aa5c2d25cdf676ddf34eddbc8ec2ec0f03614886fa828e1fc066b26d35744c0c38271843aa4fb617b57fa9eb4bd256d17367914159fc18b10a1085cb626e5bedb145

e_2:

0x02b9bece645fbf9d8f97025a1545359f6fe3ffab3cd57094f862f7fb9ca01c88705c26675bcc723878e943da6b56ce25d063381fcd2a292e0e7501fe572744184fb4ab4ca071a04281

e_3:

0x0080d267bf036c1e61d7fc73905e8c630b97aa05ef3266c82e7a111072c0d2056baa8137fba111c9650dfb18cb1f43363041e202e3192fced29d2b0501c882543fb370a56bfdc2435b

e_4:

0x03c6b4c12f338f9401e6a493a405b33e64389338db8c5e592a8dd79eac7720dd83dd6b0c189eeda20809160cd57cdf3e2edc82db15f553c1f6c953ea27114cb6bd8a38e273f407dae0

e_5:

0x016e46224f28bfd8833f76ac29ee6e406a9da1bde55f5e82b3bd977897a9104f18b9ee41ea9af7d4183d895102950a12ce9975669db07924e1b432d9680f5ce7e5c67ed68f381eba45

e_6:

0x008ddce7a4a1b94be5df3ceea56bef0077dcdde86d579938a50933a47296d337b7629934128e2457e24142b0eeaa978fd8e70986d7dd51fccbbeb8a1933434fec4f5bc538de2646e90

e_7:

0x060ef6eae55728e40bd4628265218b24b38cdd434968c14bfefb87f0dcbfc76cc473ae2dc0cac6e69dfdf90951175178dc75b9cc08320fcde187aa58ea047a2ee00b1968650eec2791

e_8:

0x0c3943636876fd4f9393414099a746f84b2633dfb7c36ba6512a0b48e66dcb2e409f1b9e150e36b0b4311165810a3c721525f0d43a021f090e6a27577b42c7a57bed3327edb98ba8f8

e_9:

0x02d31eb8be0d923cac2a8eb6a07556c8951d849ec53c2848ee78c5eed40262eb21822527a8555b071f1cd080e049e5e7ebfe2541d5b42c1e414341694d6f16d287e4a8d28359c2d2f9

e_10:

0x07f19673c5580d6a10d09a032397c5d425c3a99ff1dd0abe5bec40a0d47a6b8daabb22edb6b06dd8691950b8f23faefcdd80c45aa3817a840018965941f4247f9f97233a84f58b262e

e_11:

0x0d3fe01f0c114915c3bdf8089377780076c1685302279fd9ab12d07477aac03b69291652e9f179baa0a99c38aa8851c1d25ffdb4ded2c8fe8b30338c14428607d6d822610d41f51372

e_12:

0x0662eefd5fab9509aed968866b68cff3bc5d48ecc8ac6867c212a2d82cee5a689a3c9c67f1d611adac7268dc8b06471c0598f7016ca3d1c01649dda4b43531cffc4eb41e691e27f2eb

e_13:

0x0aad8f4a8cfdca8de0985070304fe4f4d32f99b01d4ea50d9f7cd2abdc0aeea99311a36ec6ed18208642cef9e09b96795b27c42a5a744a7b01a617a91d9fb7623d636640d61a6596ec

e_14:

0x0ffcf21d641fd9c6a641a749d80cab1bcad4b34ee97567d905ed9d5cfb74e9aef19674e2eb6ce3dfb706aa814d4a228db4fcd707e571259435393a27cac68b59a1b690ae8cde7a94c3

e_15:

0x0cbe92a53151790cece4a86f91e9b31644a86fc4c954e5fa04e707beb69fc60a858fed8ebd53e4cfd51546d5c0732331071c358d721ee601bfd3847e0e904101c62822dd2e4c7f8e5c

e_16:

0x0202db83b1ff33016679b6cfc8931deea6df1485c894dcd113bacf564411519a42026b5fda4e16262674dcb3f089cd7d552f8089a1fec93e3db6bca43788cdb06fc41baaa5c5098667

e_17:

0x070a617ed131b857f5b74b625c4ef70cc567f619defb5f2ab67534a1a8aa72975fc4248ac8551ce02b68801703971a2cf1cb934c9c354cadd5cfc4575cde8dbde6122bd54826a9b3e9

e_18:

0x070e1ebce457c141417f88423127b7a7321424f64119d5089d883cb953283ee4e1f2e01ffa7b903fe7a94af4bb1acb02ca6a36678e41506879069cee11c9dcf6a080b6a4a7c7f21dc9

e_19:

0x058a06be5a36c6148d8a1287ee7f0e725453fa1bb05cf77239f235b417127e370cfa4f88e61a23ea16df3c45d29c203d04d09782b39e9b4037c0c4ac8e8653e7c533ad752a640b233e

e_20:

0x0dfdfaaeb9349cf18d21b92ad68f8a7ecc509c35fcd4b8abeb93be7a204ac871f2195180206a2c340fccb69dbc30b9410ed0b122308a8fc75141f673ae5ec82b6a45fc2d664409c6b6

e_21:

0x0d06c8adfdd81275da2a0ce375b8df9199f3d359e8cf50064a3dc10a592417124a3b705b05a7ffe78e20f935a08868ecf3fc5aba0ace7ce4497bb59085ca277c16b3d53dd7dae5c857

e_22:

0x0708effd28c4ae21b6969cb9bdd0c27f8a3e341798b6f6d4baf27be259b4a47688b50cb68a69a917a4a1faf56cec93f69ac416512c32e9d5e69bd8836b6c2ba9c6889d507ad571dbc4

e_23:

0x09da7c7aa48ce571f8ece74b98431b14ae6fb4a53ae979cd6b2e82320e8d25a0ece1ca1563aa5aa6926e7d608358af8399534f6b00788e95e37ef1b549f43a58ad250a71f0b2fdb2bf

e_24:

0x0a7150a14471994833d89f41daeaa999dfc24a9968d4e33d88ed9e9f07aa2432c53e486ba6e3b6e4f4b8d9c989010a375935c06e4b8d6c31239fad6a61e2647b84a0e3f76e57005ff7

e_25:

0x084696f31ff27889d4dccdc4967964a5387a5ae071ad391c5723c9034f16c2557915ada07ec68f18672b5b2107f785c15ddf9697046dc633b5a23cc0e442d28ef6eea9915d0638d4d8

e_26:

0x0398e76e3d2202f999ac0f73e0099fe4e0fe2de9d223e78fc65c56e209cdf48f0d1ad8f6093e924ce5f0c93437c11212b7841de26f9067065b1898f48006bcc6f2ab8fa8e0b93f4ba4

e_27:

0x06d683f556022368e7a633dc6fe319fd1d4fc0e07acff7c4d4177e83a911e73313e0ed980cd9197bd17ac45942a65d90e6cb9209ede7f36c10e009c9d337ee97c4068db40e34d0e361

e_28:

0x0d764075344b70818f91b13ee445fd8c1587d1c0664002180bbac9a396ad4a8dc1e695b0c4267df4a09081c1e5c256c53fd49a73ffc817e65217a44fc0b20ef5ee92b28d4bc3e38576

e_29:

0x0aa6a32fdc4423b1c6d43e5104159bcd8e03a676d055d4496f7b1bc8761164a2908a3ff0e4c4d1f4362015c14824927011e2909531b8d87ee0acd676e7221a1ca1c21a33e2cf87dc51

e_30:

0x1147719959ac8eeab3fc913539784f1f947df47066b6c0c1beafecdb5fa784c3be9de5ab282a678a2a0cbef8714141a6c8aaa76500819a896b46af20509953495e2a85eff58348b38d

e_31:

0x11a377bcebd3c12702bb34044f06f8870ca712fb5caa6d30c48ace96898fcbcddbcf31f331c9e524684c02c90db7f30b9fc470d6e651a7e8b1f684383f3705d7a47a1b4fe463d623c8

e_32:

0x0b8b4511f451ba2cc58dc28e56d5e1d0a8f557ecb242f4d994a627e07cf3fa44e6d83cb907deacf303d2f761810b5d943b46c4383e1435ec23fec196a70e33946173c78be3c75dfc83

e_33:

0x090962d632ee2a57ce4208052ce47a9f76ea0fdad724b7256bb07f3944e9639a981d3431087241e30ae9bf5e2ea32af323ce7ed195d383b749cb25bc09f678d385a49a0c09f6d9efca

e_34:

0x0931c7befc80acd185491c68af886fa8ee39c21ed3ebd743b9168ae3b298df485bfdc75b94f0b21aecd8dca941dfc6d1566cc70dc648e6ccc73e4cbf2a1ac83c8294d447c66e74784d

e_35:

0x020ac007bf6c76ec827d53647058aca48896916269c6a2016b8c06f0130901c8975779f1672e581e2dfdbcf504e96ecf6801d0d39aad35cf79fbe7fe193c6c882c15bce593223f0c7c

e_36:

0x0c0aed0d890c3b0b673bf4981398dcbf0d15d36af6347a39599f3a22584184828f78f91bbbbd08124a97672963ec313ff142c456ec1a2fc3909fd4429fd699d827d48777d3b0e0e699

e_37:

0x0ef7799241a1ba6baaa8740d5667a1ace50fb8e63accc3bc30dc07b11d78dc545b68910c027489a0d842d1ba3ac406197881361a18b9fe337ff22d730fa44afabb9f801f759086c8e4

e_38:

0x016663c940d062f4057257c8f4fb9b35e82541717a34582dd7d55b41ebadf40d486ed74570043b2a3c4de29859fdeae9b6b456cb33bb401ecf38f9685646692300517e9b035d6665fc

e_39:

0x1184a79510edf25e3bd2dc793a5082fa0fed0d559fa14a5ce9ffca4c61f17196e1ffbb84326272e0d079368e9a735be1d05ec80c20dc6198b50a22a765defdc151d437335f1309aced

e_40:

0x120e47a747d942a593d202707c936dafa6fed489967dd94e48f317fd3c881b1041e3b6bbf9e8031d44e39c1ab5ae41e487eac9acd90e869129c38a8e6c97cf55d6666d22299951f91a

e_41:

0x026b6e374108ecb2fe8d557087f40ab7bac8c5af0644a655271765d57ad71742aa331326d871610a8c4c30ccf5d8adbeec23cdff20d9502a5005fce2593caf0682c82e4873b89d6d71

e_42:

0x041be63a2fa643e5a66faeb099a3440105c18dca58d51f74b3bf281da4e689b13f365273a2ed397e7b1c26bdd4daade710c30350318b0ae9a9b16882c29fe31ca3b884c92916d6d07a

e_43:

0x124018a12f0f0af881e6765e9e81071acc56ebcddadcd107750bd8697440cc16f190a3595633bb8900e6829823866c5769f03a306f979a3e039e620d6d2f576793d36d840b168eeedd

e_44:

0x0d422de4a83449c535b4b9ece586754c941548f15d50ada6740865be9c0b066788b6078727c7dee299acc15cbdcc7d51cdc5b17757c07d9a9146b01d2fdc7b8c562002da0f9084bde5

e_45:

0x1119f6c5468bce2ec2b450858dc073fea4fb05b6e83dd20c55c9cf694cbcc57fc0effb1d33b9b5587852d0961c40ff114b7493361e4cfdff16e85fbce667869b6f7e9eb804bcec46db

e_46:

0x061eaa8e9b0085364a61ea4f69c3516b6bf9f79f8c79d053e646ea637215cf6590203b275290872e3d7b258102dd0c0a4a310af3958165f2078ff9dc3ac9e995ce5413268d80974784

e_47:

0x0add8d58e9ec0c9393eb8c4bc0b08174a6b421e15040ef558da58d241e5f906ad6ca2aa5de361421708a6b8ff6736efbac6b4688bf752259b4650595aa395c40d00f4417f180779985

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