Threshold Modes in Elliptic Curves
ThresholdShare.com
phill@hallambaker.com
Threshold Cryptography
Elliptic Curve
Threshold Encryption
Threshold Key Generation
Side Channel Resistance
Threshold cryptography operation modes are described with application to the Ed25519, Ed448, X25519 and X448 Elliptic Curves. Threshold key generation allows generation of keypairs to be divided between two or more parties with verifiable security guaranties. Threshold decryption allows elliptic curve key agreement to be divided between two or more parties such that all the parties must cooperate to complete a private key agreement operation. The same primitives may be applied to improve resistance to side channel attacks. A Threshold signature scheme is described in a separate document.
https://mailarchive.ietf.org/arch/browse/cfrg/Discussion of this draft should take place on the CFRG mailing list (cfrg@irtf.org), which is archived at .
Public key cryptography provides greater functionality than symmetric key cryptography by introducing separate keys for separate roles. Knowledge of the public encryption key does not provide the ability to decrypt. Knowledge of the public signature verification key does not provide the ability to sign. Threshold cryptography extends the scope of traditional public key cryptography with further separation of roles by splitting the private key. This allows greater control of (e.g.) decryption operations by requiring the use of two decryption key shares rather than just the decryption key.
This document describes threshold modes for decryption and key generation for the elliptic curves described in and . Both schemes are interchangeable in their own right but require minor modifications to the underlying elliptic curve systems to encode the necessary information in the public (X25519/X448) or private key (Ed25519/Ed448). The companion document describes a threshold mode for signatures.
In its most general form, threshold cryptography allows a private key to be divided between (n, t) shares such that n is the total number of shares created and t is the threshold number of shares required to perform the operation. For most applications however, the number of shares is the same as the threshold (n = t) and the most common case is (n = t = 2).
This document sets out the principles that support the definition threshold modes in elliptic curve Diffie Hellman system first using simple additive key sharing, an approach which is limited to the case (n = t). The use of Shamir secret sharing is then described to support the case (n > t). For convenience, we refer to the non Shamir secret sharing case as 'direct key sharing'.
Development of the threshold modes described in this document and the companion document were motivated by the following applications.
The security of data at rest is of increasing concern in enterprises and for individual users. Transport layer security such as TLS and SSH provide effective confidentiality controls for data in motion but not for data at rest.
Of particular concern is the case in which a large store of confidential data is held on a server. Encryption provides a simple and effective means of protecting the confidentiality of such data. But the real challenge is how to effect decryption of the data on demand for the parties authorized to access it.
Storing the decryption keys on the server that holds the data provides no real security advantage as any compromise that affects the server is likely to result in disclosure of the keys. Use of endtoend security in which each document is encrypted under the public key of each person authorized to access it provides the desired security but introduces a complex key management problem and provides no effective means of revoking access after it is granted.
Threshold encryption allows a key service to control decryption of stored data without having the ability to decrypt. The data decryption key is split into two (or more) parts with a different split being created for each user. One decryption share is held at the server allowing it to control access to the data without being able to decrypt. The other decryption share is encrypted under the public encryption key of the corresponding user allowing them to decrypt the stored data but only with the cooperation of the key service.
The term 'onboarding' is used to refer to the configuration of a device for use by a particular user or within a particular enterprise. One of the typical concerns of onboarding is to initialize the device with a set of public key pairs for various purposes and to issue credentials (e.g. PKIX certificates) to enable their use.
One of the concerns that arises in such cases is whether keys should be generated on the device on which they are to be used or on another device administering the onboarding process.
Both approaches are unsatisfactory. While generation of keys on the device on which they are to be used is generally preferred, experience has shown that many devices, particularly IoT devices use insufficiently random processes to generate such keys and this has led to numerous cases of duplicate and otherwise weak keys being discovered in running systems.
Generation of keys on an administration device and transferring them to the device on which they are to be used means that the administration device used for key generation represents a single point of failure. Compromise of this device or of the means used to install the keys will lead to compromise of the device.
Threshold key generation allows the advantages of both approaches to be realized avoiding dependence on either the target device or the administration device.
Most realworld compromises of cryptographic security systems involve the disclosure of a private key. Common means of disclosure being inadvertent inclusion in backup tapes, keys being stored on public file shares and (increasingly) keys being inadvertently uploaded to source code management systems.
A new and emerging set of key disclosure threats come from the recent generation of hardware vulnerabilities being discovered in CPUs including ROWHAMMER and SPECTRE.
The advantages of Hardware Security Modules (HSMs) for storing and using private keys are well established. An HSM allows a private key to be used in an isolated environment that is designed to be resistant to side channel attacks.
Regrettably, the 'black box' nature of HSMs means that their use introduces a new security concern  the possibility that the device itself has been compromised during manufacture or in the supply chain.
Threshold approaches allows a key exchange or signature operation to be divided between two private keys, one of which is generated by the application that is to use it and the other of which is tightly bound to a specific host such that it cannot be extracted.
The same techniques that make threshold cryptography possible are the basis for Kocher's sidechannel resistance technique . Differential side channel attacks are a powerful tool capable of revealing a private key value that is used repeatedly in practically any algorithm. The claims made with respect to 'constant time' algorithms such as the Montgomery ladder depend upon the actual implementation performing operations in constant time.
This section presents the related specifications and standard, the terms that are used as terms of art within the documents and the terms used as requirements language.
The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT", "SHOULD", "SHOULD NOT", "RECOMMENDED", "MAY", and "OPTIONAL" in this document are to be interpreted as described in .
The following terms are used as terms of art in this document and the accompanying specification .
 Dealer

A party that coordinates the actions of a group of participants performing a threshold operation.
 MultiEncryption

The use of multiple decryption fields to allow a document encrypted under a session key to be decrypted by multiple parties under different decryption keys.
MultiEncryption allows a document to be shared with multiple recipients but does not allow the decryption role to be divided between multiple parties.
 MultiSignatures

The use of multiple independently verifiable digital signatures to authenticate a document.
MultiSignatures allow separation of the signing roles and thus achieve a threshold capability. But they are not true threshold signatures as the set of signers is visible to external parties.
 Onboarding

The process by which an embedded device is provisioned for deployment in a local network.
 Threshold Key Generation

An aggregate public, private key pair is constructed from a set of contributions such that the private key is a function of the private key of all the contributions.
A Threshold Key Generation function is auditable if and only if the public component of the aggregate key can be computed from the public keys of the contributions alone.
 Threshold Decryption

Threshold decryption divides the decryption role between two or more parties.
 Threshold Key Agreement

A bilateral key agreement exchange in which one or both sides present multiple public keys and the key agreement value is a function of all of them.
This approach allows a party to present multiple credentials in a single exchange, a de
 Threshold Signatures

Threshold signatures divide the signature role between two or more parties in such a way that the parties and their roles is not visible to an external observer.
This document extends the elliptic curve cryptography systems described in and to provide threshold capabilities.
This work was originally motivated by the requirements of the Mathematical Mesh .
Threshold modes for generating signatures compatible with are described in .
The implementation status of the reference code base is described in the companion document .
The threshold modes described in this specification are made possible by the fact that Diffie Hellman key agreement and elliptic curve variants thereof support properties we call the Key Combination Law and the Result Combination Law.
Let {X, x}, {Y, y}, {A, a} be {public, private} key pairs and r [.] S represent the Diffie Hellman operation applying the private key r to the public key S.
The Key Combination law states that we can define an operator [x] such that there is a keypair {Z, z} such that:
Z = X [x] Y and z = (x + y) mod o (where o is the order of the group)
The Result Combination Law states that we can define an operator [+] such that:
(x [.] A) [+] (y [.] A) = (z [.] A) = (a [.] Z)
It will be noted that each of these laws is interchangeable. The output of the key combination law to a Diffie Hellman key pair is a Diffie Hellman key pair and the output of the result combination law is a Diffie Hellman result. This allows modular and recursive application of these principles.
Diffie Hellman in a modular field provides a concise demonstration of the key combination and result combination laws . The realization of the threshold schemes in a modular field is outside the scope of this document.
For the Diffie Hellman system in a modular field p, with exponent e:
 r [.] S = S^{r} mod p
 o = p1
 A [x] B = A [.] B = AB mod p.
Proof:
Let z = x + y
By definition, X = e^{x} mod p, Y = e^{y} mod p, and Z = e^{z }mod p.
Therefore,
 Z = e^{z} mod p

= e^{x+y} mod p
= (e^{x}e^{y}) mod p
= e^{x }mod p.e^{y} mod p
= X.Y
Moreover, A = e^{a} mod p,
Therefore,
 (A^{x} mod p).(A^{y} mod p)

= (A^{x}A^{y}) mod p)
= (A^{x+y}) mod p)
= A^{z} mod p
= e^{az} mod p
= (e^{z})^{a} mod p
= Z^{a} mod p
Since e^{o} mod p = 1, the same result holds for z = (x + y) mod o since e^{x+y+no} = e^{x+y}.e^{no} = e^{x+y}.1 = e^{x+y}.
Threshold decryption allows a decryption key to be divided into two or more parts, allowing these roles to be assigned to different parties. This capability can be employed within a machine to divide use of a private key between an implementation running in the user mode and a process running in a privileged mode that is bound to the machine. Alternatively, threshold cryptography can be employed to
The key combination law and result combination law provide a basis for threshold decryption.
We begin by creating a base key pair { X, x }. The public component X is used to perform encryption operations by means of a key agreement against an ephemeral key in the usual fashion. The private component x may be used for decryption in the normal fashion or to provide the source material for a key splitting operation.
To split the base key into n shares { S_{1}, s_{1} }, { S_{2}, s_{2} }, ... { S_{n}, s_{n} }, we begin by generating the first n1 private keys in the normal fashion. It is not necessary to generate the corresponding public keys as these are not required.
The private key of the final key share s_{n} is given by:
s_{n} = (x  s1  s2  ...  sn1) mod o
Thus, the base private key x is equal to the sum of the private key shares modulo the group order.
To encrypt a document, we first generate an ephemeral key pair { Y, y }. The key agreement value e^{xy} is calculated from the base public key X = e^{x} and the ephemeral private key y. A key derivation function is then used to obtain the session key to be used to encrypt the document under a symmetric cipher.
To decrypt a document using the threshold key shares, each key share holder first performs a Diffie Hellman operation using their private key on the ephemeral public key. The key shares are then combined using the result combination law to obtain the key exchange value from which the session key is recovered.
The key contribution c_{i} for the holder of the i^{th} key share is calculated as:
c_{i} = Y^{si}
The key agreement value is thus
A = c_{1} . c_{2} . ... . c_{n}
This value is equal to the encryption key agreement value according to the group law.
The key combination law allows an aggregate private key to be derived from private key contributions provided by two or more parties such that the corresponding aggregate public key may be derived from the public keys corresponding to the contributions. The resulting key generation mechanism is thus, auditable and interoperable.
Auditable Threshold Key Generation provides a simple and efficient means of securely provisioning keys to devices. This is encountered in the IoT space as a concern in 'onboarding' and in the provisioning of unique keys for use with cryptographic applications (e.g. SSH, S/MIME, OpenPGP, etc.).
Consider the case in which Alice purchases an IoT connected Device D which requires a unique device key pair { X , x } for its operation. The process of provisioning (aka 'onboarding') is performed using an administration device. Traditional key pair generation gives us three options:
 Generate and install a key pair during manufacture.
 Generate a new key pair during device provisioning.
 Generate a key pair on the administration device and transfer it to the device being provisioned.
The first approach has the obvious disadvantage that the manufacturer has knowledge of the private key. This represents a liability for both the user and the manufacturer. Less obvious is the fact that the second approach doesn't actually provide a solution unless the process of generating keys on the device is auditable. The third approach is auditable with respect to the device being provisioned but not with respect to the administration device being used for provisioning. If that device were to be compromised, so could every device it was used to provision.
Threshold key generation allows the administration device and the joining device being provisioned to jointly provision a key pair as follows:
 The joining device has public, private key pair { D, d }.
 A provisioning request is made for the device which includes the joining device public key D.
 The administration device generates a key pair contribution { A, a }.
 The administration device private key is transmitted to the Device by means of a secure channel.
 The joining device calculates the aggregate key pair { A [x] D, a+d }.
 The administration device authorizes the joining device to participate in the local network using the public key A [x] D.
The Device key pair MAY be installed during manufacture or generated during provisioning or be derived from a combination of both using threshold key generation recursively. The provisioning request MAY be originated by the device or be generated by a purchasing system.
Note that the provisioning protocol does not require either party to authenticate the aggregate key pair. The protocol provides security by ensuring that both parties obtain the correct results if and only if the values each communicated to the other were correct.
Out of band authentication of the joining device public key D is sufficient to prevent ManintheMiddle attack. This may be achieved by means of a QR code printed on the device itself that discloses or provides a means of obtaining D.
[Note add serious warning that a party providing a private contribution MUST make sure they see the public key first. Otherwise a rogue key attack is possible. The Mesh protocols ensure that this is the case but other implementations may overlook this detail.]
Traditional key rollover protocols in PKIX and other PKIs following the Kohnfelder model, require a multistep interaction between the key holder and the Certificate Authority.
Threshold key generation allows a Certificate Authority to implement key rollover with a single communication:
Consider the case in which the service host has a base key pair { B , b }. A CA that has knowledge of the Host public key B may generate a certificate for the time period t with a fresh key as follows:
 Generate a key pair contribution { A_{t}, a_{t} }.
 Generate and sign an end entity certificate C_{t} for the key B [x] A_{t}.
 Transmit C_{t}, a_{t} to the host by means of a secure channel
Modern Internet service architectures frequently make use of short lived, ephemeral hosts running on virtualized machines. Provisioning cryptographic material in such environments is a significant challenge and especially so when the underlying hardware is a shared resource.
The key rollover approach described above provides a means of provisioning short lived credentials to ephemeral hosts that potentially avoids the need to build sensitive keys into the service image or configuration.
Threshold key generation provides a means of separating administration of cryptographic keys between individuals. This allows two or more administrators to control access to a private key without having the ability to use it themselves. This approach is of particular utility when used in combination with threshold decryption. Alice and Bob can be granted the ability to create key contributions allowing a user to decrypt information without having the ability to decrypt themselves.
Sidechannel attacks, present a major concern in the implementation of public key cryptosystems. Of particular concern are the timing attacks identified by Paul Kocher and related attacks in the power and emissions ranges. Performing repeated observations of the use of the same private key material provides an attacker with considerably greater opportunity to extract the private key material.
A simple but effective means of defeating such attacks is to split the private key value into two or more random shares for every private key operation and use the result combination law to recover the result.
The implementation of this approach is identical to that for Threshold Decryption except that instead of giving the key shares to different parties, they are kept by the party performing the private key operation.
While this approach doubles the number of private key operations required, the operations MAY be performed in parallel. Thus avoiding impact on the user experience.
The direct threshold modes described above may be extended to support the case (n > t) through application of Shamir secret sharing and the use of the Lagrange basis to recover the secret value.
Shamir Secret Sharing makes use of the fact that a polynomial of degree t1 is defined by t points on the curve. To share a secret s, we construct a polynomial of degree t1 in the modular field L where L > s.
f(x) = s + a_{1}.x + a_{2}.x^{2} + ... a_{t1}.x^{t1} mod L
The shares p_{1}, p_{2} ... p_{n} are given by the values f(x_{1}), f(x_{2}), ... , f(x_{n}) where x_{1}, x_{2} ... x_{n} are in the range 1 x_{i} L.
We can use the Lagrange basis function to construct a set of coefficients l_{1}, l_{2}, ... l_{t} from a set of t shares p_{1}, p_{2} ... p_{t} such that:
s = l_{1}p_{1} + l_{2}p_{2} + ... + l_{t}p_{t} mod L
Thus, if we choose the subgroup order of the curve as the value of L, the formula used to recover the secret s is a sum of terms as was used for the case where n = t. We can thus apply a set of Lagrange coefficients provided by the dealer to the secret shares to extend the threshold operations to the case where n > t.
To create n shares for the secret s with a threshold of t, the dealer constructs a polynomial of degree t in the modular field L, where L is the order of the curve subgroup:
f(x) = a_{0} + a_{1}.x + a_{2}.x^{2} + ... a_{t}.x^{t1} mod L
 where

a_{0} = s
a_{1} ... a_{t} are randomly chosen integers in the range 1 a_{i} L
The values of the key shares are the values f(x_{1}), f(x_{2}), ... , f(x_{n}). That is
p_{i} = f(x_{i})
 where

p_{1} ... p_{n} are the private key shares
x_{1} ... x_{n} are distinct integers in the range 1 x_{i} L
The means of constructing distinct values x_{1} ... x_{n} is left to the implementation. It is not necessary for these values to be secret.
Given n shares (x_{0}, y_{0}), (x_{1}, y_{1}), ... (x_{n1}, y_{n1}), The corresponding the Lagrange basis polynomials l_{0}, l_{1}, .. l_{n1} are given by:
l_{m} = ((x  x_{m0}) / (x_{m}  x_{m0})) . ((x  x_{m1}) / (x_{m}  x_{m1})) . ... . ((x  x_{mn2}) / (x_{m}  x_{mn}2))
Where the values m_{0}, m_{1}, ... m_{n2}, are the integers 0, 1, .. n1, excluding the value m.
These can be used to compute f(x) as follows:
f(x) = y_{0}l0 + y1l1 + ... yn1ln1
Since it is only the value of f(0) that we are interested in, we compute the Lagrange basis for the value x = 0:
lz_{m} = ((x_{m1}) / (x_{m}  x_{m1})) . ((x_{m2}) / (x_{m}  x_{m2}))
Hence,
a_{0} = f(0) = y_{0}lz0 + y1lz1 + ... yn1ln1
The secret share generation mechanism described above allows a private key to be split into n shares such that t shares are required for recovery. While this supports a wide variety of applications, there are many cases in which it is desirable for generation of the private key to be distributed.
Feldman's Verifiable Secret Sharing (VSS) Scheme provides a mechanism that allows the participants in a distributed generation scheme to determine that their share has been correctly formed by means of a commitment.
To generate a commitment the dealer generates the polynomial f(x) as before and in addition selects a generator g
[TBS]
For elliptic curve cryptosystems, the operators [x] and [.] are point addition.
Implementing a robust Key CoGeneration for the Elliptic Curve Cryptography schemes described in and requires some additional considerations to be addressed.
 The secret scalar used in the EdDSA algorithm is calculated from the private key using a digest function. It is therefore necessary to specify the Key CoGeneration mechanism by reference to operations on the secret scalar values rather than operations on the private keys.
 The Montgomery Ladder traditionally used to perform X25519 and X448 point multiplication does not require implementation of a function to add two arbitrary points. While the steps required to create such a function are fully constrained by , the means of performing point addition is not.
provides all the cryptographic operations required to perform threshold operations and corresponding public keys.
The secret scalars used in private key operations are derived from a private key value using a cryptographic digest function. This encoding allows the inputs to a private key combination to be described but not the output. Contrawise, the encoding allows the inputs to a private key splitting operation to be described but not the output
It is therefore necessary to provide an alternative representation for the Ed25519 and Ed448 private keys. Moreover, the signature algorithm requires both a secret scalar and a prefix value as inputs.
Since threshold signatures are out of scope for this document and does not specify a key agreement mechanism, it suffices to specify the data formats required to encode private key values generated by means of threshold key generation.
Let the inputs to the threshold key generation scheme be a set of 32 byte private key values P_{1}, P2 ... Pn. For each private key value i in turn:
 Hash the 32byte private key using SHA512, storing the digest in a 64octet large buffer, denoted h_{i}. Let n_{i} be the first 32 octets of h_{i} and m_{i} be the remaining 32 octets of h_{i}.
 Prune n_{i}: The lowest three bits of the first octet are cleared, the highest bit of the last octet is cleared, and the second highest bit of the last octet is set.
 Interpret the buffer as the littleendian integer, forming a secret scalar s_{i}.
The private key values are calculated as follows:
The aggregate secret scalar value s_{a} = s1 + s2 + ... sn mod L, where L is the order of the curve.
The aggregate prefix value is calculated by either
 Some function TBS on the values m_{1}, m_{2}, ... m_{n}, or
 Taking the SHA256 digest of s_{a}.
The second approach is the simplest and the most robust. It does however mean that the prefix is a function of the secret scalar rather than both being functions of the same seed.
Let the inputs to the threshold key generation scheme be a set of 57 byte private key values P_{1}, P2 ... Pn. For each private key value i in turn:
 Hash the 57byte private key using SHAKE256(x, 114), storing the digest in a 114octet large buffer, denoted h_{i}. Let n_{i} be the first 57 octets of h_{i} and m_{i} be the remaining 57 octets of h_{i}.
 Prune n_{i}: The two least significant bits of the first octet are cleared, all eight bits the last octet are cleared, and the highest bit of the second to last octet is set.
 Interpret the buffer as the littleendian integer, forming a secret scalar s_{i}.
The private key values are calculated as follows:
The aggregate secret scalar value s_{a} = s1 + s2 + ... sn mod L, where L is the order of the curve.
The aggregate prefix value is calculated by either
 Some function TBS on the values m_{1}, m_{2}, ... m_{n}, or
 Taking the SHAKE256(x, 57) digest of s_{a}.
The second approach is the simplest and the most robust. It does however mean that the prefix is a function of the secret scalar rather than both being functions of the same seed.
defines all the cryptographic operations required to perform threshold key generation and threshold decryption but does not describe how to implement them.
The Montgomery curve described in allows for efficient scalar multiplication using arithmetic operations on a single coordinate. Point addition requires both coordinate values. It is thus necessary to provide an extended representation for point encoding and provide an algorithm for recovering both coordinates from a scalar multiplication operation and an algorithm for point addition.
The notation of is followed using {u, v} to represent the coordinates on the Montgomery curve and {x, y} for coordinates on the corresponding Edwards curve.
The relationship between the u and v coordinates is specified by the Montgomery Curve formula itself:
v^{2} = u^{3} + Au2 + u
An algorithm for extracting a square root of a number in a finite field is specified in .
Since v^{2} has a positive (v) and a negative solution (v), it follows that v^{2} mod p will have the solutions v, pv. Furthermore, since p is odd, if v is odd, pv must be even and vice versa. It is thus sufficient to record whether v is odd or even to enable recovery of the v coordinate from u.
The extended point encoding allowing the v coordinate to be recovered is as given in
A curve point (u, v), with coordinates in the range 0 = u,v p, is coded as follows. First, encode the ucoordinate as a littleendian string of 57 octets. The final octet is always zero. To form the encoding of the point, copy the least significant bit of the vcoordinate to the most significant bit of the final octet.
The extended point encoding allowing the v coordinate to be recovered is as given in
A curve point (u, v), with coordinates in the range 0 = u,v p, is coded as follows. First, encode the ucoordinate as a littleendian string of 57 octets. The final octet is always zero. To form the encoding of the point, copy the least significant bit of the vcoordinate to the most significant bit of the final octet.
The point addition formula for the Montgomery curve is defined as follows:
Let P_{1} = {u_{1}, v_{1}}, P_{2} = {u_{2}, v_{2}}, P_{3} = {u_{3}, v_{3}} = P_{1} + P_{2}
By definition:
 u_{3}

= B(v_{2}  v_{1})^{2} / (u_{2}  u_{1})^{2}  A  u_{1}  u_{2}
= B((u_{2}v_{1}  u_{1}v_{2})^{2} ) / u_{1}u_{2} (u_{2}  u_{1})^{2}
v_{3} = ((2u_{1} + u_{2} + A)(v_{2}  v_{1}) / (u_{2}  u_{1}))  B (v_{2}  v_{1})^{3} / (u_{2} u_{1})^{3}  v_{1}
For curves X25519 and X448, B = 1 and so:
u_{3} = ((v_{2}  v_{1}).(u_{2}  u_{1})^{1})^{2}  A  u_{1}  u_{2}
v_{3} = ((2u_{1} + u_{2} + A)(v_{2}  v_{1}).(u_{2}  u_{1})^{1})  ((v_{2}  v_{1}).(u_{2} u_{1})^{1})^{3}  v_{1}
This may be implemented using the following code:
B = v2  v1
C = u2  u1
CINV = C^(p  2)
D = B * CINV
DD = D * D
DDD = DD * D
u3 = DD  A  u1  u2
v3 = ((u1 + u1 + u2 + A) * B * CINV)  DDD  v1
Performing point addition thus requires that we have sufficient knowledge of the values v_{1}, v_{2}. At minimum whether one is odd and the other even or if both are the same.
As originally described, the Montgomery Ladder only provides the u coordinate as output. L?pez and Dahab provided a formula for recovery of the v coordinate of the result for curves over binary fields. This result was then extended by Okeya and Sakurai to prime field Montgomery curves such as X25519 and X448. The realization of this result described by Costello and Smith is applied here.
The scalar multiplication function specified in takes as input the scalar value k and the coordinate u_{1} of the point P_{1} = {u_{1}, v_{1}} to be multiplied. The return value in this case is u_{2} where P_{2} = {u_{2}, v_{2}} = k.P_{1}.
To recover the coordinate v_{2} we require the values x_2, z_2, x_3, z_3 calculated in the penultimate step:
x_1 = u
x_2 = 1
z_2 = 0
x_3 = u
z_3 = 1
swap = 0
For t = bits1 down to 0:
k_t = (k >> t) & 1
swap ^= k_t
// Conditional swap as specified in RFC 7748
(x_2, x_3) = cswap(swap, x_2, x_3)
(z_2, z_3) = cswap(swap, z_2, z_3)
swap = k_t
A = x_2 + z_2
AA = A^2
B = x_2  z_2
BB = B^2
E = AA  BB
C = x_3 + z_3
D = x_3  z_3
DA = D * A
CB = C * B
x_3 = (DA + CB)^2
z_3 = x_1 * (DA  CB)^2
x_2 = AA * BB
z_2 = E * (AA + a24 * E)
(x_2, x_3) = cswap(swap, x_2, x_3)
(z_2, z_3) = cswap(swap, z_2, z_3)
Return x_2, z_2, x_3, z_3
The values x_2, z_2 give the projective form of the u coordinate of the point P_{2} = {u_{2}, v_{2}} = k.P_{1} and the values x_3, z_3 give the projective form of the u coordinate of the point P_{3} = {u_{3}, v_{3}} = (k+1).P_{1} = P_{1} + k.P_{1} = P_{1} + P_{2}.
Given the coordinates {u_{1}, v_{1}} of the point P1 and the u coordinates of the points P_{2}, P_{1} + P_{2}, the coordinate v_{2} MAY be recovered by trial and error as follows:
v_test = SQRT (u3 + Au2 + u)
u_test = ADD_X (u, v, u_2, v_test)
if (u_test == u_3)
return u_test
else
return u_test +p
Alternatively, the following MAY be used to recover {u_{2}, v_{2}} without the need to extract the square root and using a single modular exponentiation operation to convert from the projective coordinates used in the calculation. As with the Montgomery ladder algorithm above, the expression has been modified to be consistent with the approach used in but any correct formula may be used.
x_p = u
y_p = v
B = x_p * z_2 //v1
C = x_2 + B //v2
D = X_2  B //v3
DD = D^2 //v3
E = DD. X_3 //v3
F = 2 * A * z_2 //v1
G = C + F //v2
H = x_p * x_2 //v4
I = H + z_2 //v4
J = G * I //v2
K = F * z_2 //v1
L = J  K //v2
M = L * z_3 //v2
yy_2 = M  E //Y'
N = 2 * y_p //v1
O = N * z_2 //v1
P = O * z_3 //v1
xx_2 = P * x_q //X'
zz_2 = P * z_ q //Z'
ZINV = (zz_2^(p  2))
u2 = xx_2 * ZINV
v2 = yy_2 * ZINV
return u2, v2
The key parameters of the first key contribution are:
X25519Key1 (X25519)
UDF: ZAAACTKGX255XXKEYX
Scalar: 56751742936444772792970879017152360515706108153669948
486190735258502824077920
Encoded Private
60 2A E2 12 AC 8E C8 86 A1 79 51 7E 79 90 5E C2
9B AD 10 01 B9 2D 51 33 65 DB F4 9E 23 59 78 7D
U: 25222393324990721517739552691612440154338285166262054281502859
684220669343438
V: 15622452724514925334849257786951944861130311422605147559630230
860481236780294
Encoded Public
CE 36 B9 F1 56 BD 92 5C F4 B6 F5 E1 E0 BA CA 6A
9B 7C 37 7D F8 DC 39 CC 12 2E A6 8F 64 5E C3 37
00
The key parameters of the second key contribution are:
X25519Key2 (X25519)
UDF: ZAAACTKGX255XXKEY2
Scalar: 30800688691513612134093999707357841640579640775881469
593062950189697563564400
Encoded Private
70 19 5B 38 A4 46 21 79 31 AC 48 83 60 C9 BD F8
E1 EE 04 53 67 F2 B5 D8 9E 42 53 66 6F 92 18 44
U: 35108630063567318397224393939085269372284744000330218923799041
589332061533992
V: 13827314478911339710714490558315610168380330915483870499348836
357802235649136
Encoded Public
28 37 F5 39 16 C6 10 C6 8A AC 75 E9 20 EF 67 6D
C2 6C AF 2C E4 F6 4F C9 E9 30 6C BD C9 C7 9E 4D
00
The composite private key is:
Scalar_A = (Scalar_1 + Scalar_2) mod L
= 70836469997123835938663996799427126600035261699252680723027418877
4936630452
Encoded Composite Private Key:
B4 54 B7 EF 13 30 0D DF C6 CB FE 5D 6A A3 A8 C0
7C 9C 15 54 20 20 07 0C 04 1E 48 05 93 EB 90 01
The composite public key is:
Point_A = Point_1 + Point_2
U: 416454936139914218771704724014904891682742086807613599097165979348
46285027335
V: 473400233126764321363639652645349333601103100790621508110841442520
99552212729
Encoded Public
07 98 75 38 67 9C 66 21 A3 0A D1 06 CF F5 81 04
94 C0 52 C9 9C FD AE 4E 13 3B 43 9D 9A 83 12 5C
Note that in this case, the unsigned representation of the key is used as the composite key is intended for unsigned CurveX key agreement. If the result is intended for use as a key contribution, the signed representation is required.
The key parameters of the first key contribution are:
X448Key1 (X448)
UDF: ZAAAETKGX44XKEYX
Scalar: 68165415229434843487640754974827937311214322558126978
8055715553507401814865302008262214951100710804646043741434925
630887320553400661768
Encoded Private
08 77 91 25 66 19 C6 1A 03 C7 60 9A 8C C8 10 9D
DE F5 20 E1 A7 7F 3E 83 56 57 FE A6 C9 97 79 FB
DC 85 55 6F CE 17 79 70 CA 3E B5 D1 6A B0 50 6A
60 F6 BF 3A 88 E5 15 F0
U: 60697849609835675975297341597995979787516605306209816088918249
8293453953345846660594020472424536065173283947670623780408505
23120715561
V: 60813500494147049417364978264586877278456315581444885337361828
5076034450004202627339591608123302429557097118744860203117206
220854848663
Encoded Public
29 C7 E7 1A ED 85 B5 66 F4 CA 8F 4D 07 72 EC 4B
15 42 FA 95 4D A3 25 F6 D2 BF C0 5E 11 C4 27 D3
A1 43 D8 74 B6 4C C8 22 7D 64 56 58 A4 8C C6 5D
DA F2 AA 75 DE DE 60 15 80
The key parameters of the second key contribution are:
X448Key2 (X448)
UDF: ZAAAETKGX44XKEY2
Scalar: 67824881411761849798195083121628378835623370171088982
6937962011129206719268741815680700006802689991287015918654801
310197484516725932432
Encoded Private
90 C1 CE 67 A2 88 20 95 B9 A8 8A E7 5A 12 73 C6
4C E3 B0 0E 3A A4 1A 72 03 39 FC 9B 47 D9 6A E0
A2 81 63 57 77 EB 97 E5 CE 05 2C CB EE D7 64 F6
51 C1 42 E7 FE D9 E2 EE
U: 32395912186842981800922536415382601434069282464793284039370624
1565981551756628007189667971676177150776689230729229736979561
639842244556
V: 18056267944998342850921302138832444826477211000541459460301605
4105989977716699286334066341414636204710801397092415122728296
636211077711
Encoded Public
CC 67 05 A8 AE D3 8C 6E 17 F8 7F 66 77 14 7F 32
D3 F6 12 1C E2 80 A9 BF A9 AA 41 FC 88 EF E3 F9
38 C7 1C AA 1A 14 54 EC F0 4D 6D 20 ED 4F 63 24
F2 A0 68 F5 1C 09 1A 72 80
The composite private key is:
Scalar_A = (Scalar_1 + Scalar_2) mod L
= 87935198894654874397041717160555226349504546089353009501069716070
586506403266723929544670861554164189887604126085304951388779109
045747
Encoded Composite Private Key:
F3 55 F6 DD 05 50 99 B7 68 84 84 A1 C5 89 8A 79
3A 5B F6 27 DE 24 31 97 F5 94 73 D9 14 71 E4 DB
7F 07 B9 C6 45 03 11 56 99 44 E1 9C 59 88 B5 60
B2 B7 02 22 87 BF F8 1E
The composite public key is:
Point_A = Point_1 + Point_2
U: 611634634479536677984900819195998637894371405676964036939736481366
88134799342739585406562158256601376457049422599663606975867088547
575
V: 547531628982729065710146631050685048629332114514125362339393102647
61134803271330580187933395652539791547319114595107754138802418952
4364
Encoded Public
F7 2E 68 4B 64 DC 2E 24 61 B9 28 14 2E 1D D9 41
6A 29 4F A2 5F F1 AF 07 24 6C 9B 8A 9E C0 E5 58
Note that in this case, the unsigned representation of the key is used as the composite key is intended for unsigned CurveX key agreement. If the result is intended for use as a key contribution, the signed representation is required.
The key parameters of the first key contribution are:
ED25519Key1 (ED25519)
UDF: ZAAAGTKGED255XXKEYX
Scalar: 39507802390720856312219571924476007168388547774368948
368537778683821975155688
Encoded Private
1C C7 DE DF 19 7B 39 5F 82 98 26 62 AA DE 6C 66
04 C3 E3 A2 C8 3D 18 58 06 2C 3E EC 7C D4 B4 F2
X: 42353721841561159243771574200946096579404715276724838688117248
3158919506245796568293273125470706294881250827210288815928170
449575540044968280671060652600
Y: 14453248808291445687399372639220007070442564445118267751942208
0837579501036794314829741330844154033441251810135221340268136
7161993702790547954006637246092
Encoded Public
6D F1 94 33 33 CC 66 4D 93 89 E2 FB 38 61 21 D5
C5 6B 29 0F 5C 12 A8 4D 99 06 31 2D 35 32 22 A5
The key parameters of the second key contribution are:
ED25519Key2 (ED25519)
UDF: ZAAAGTKGED255XXKEY2
Scalar: 39507802390720856312219571924476007168388547774368948
368537778683821975155688
Encoded Private
1C C7 DE DF 19 7B 39 5F 82 98 26 62 AA DE 6C 66
04 C3 E3 A2 C8 3D 18 58 06 2C 3E EC 7C D4 B4 F2
X: 42353721841561159243771574200946096579404715276724838688117248
3158919506245796568293273125470706294881250827210288815928170
449575540044968280671060652600
Y: 14453248808291445687399372639220007070442564445118267751942208
0837579501036794314829741330844154033441251810135221340268136
7161993702790547954006637246092
Encoded Public
6D F1 94 33 33 CC 66 4D 93 89 E2 FB 38 61 21 D5
C5 6B 29 0F 5C 12 A8 4D 99 06 31 2D 35 32 22 A5
The composite private key is:
Scalar_A = (Scalar_1 + Scalar_2) mod L
= 66455490081190904847072782185220719282059319549388206770560479847
89407801486
Encoded Composite Private Key:
8E 20 46 06 EE 61 70 82 FA 37 43 E2 5A 68 E7 3C
73 4A 36 B7 AC A4 DF 68 A7 95 5C 8E 58 3F B1 0E
The composite public key is:
Point_A = Point_1 + Point_2
X: 78285292761951767745666894197721180606214882184104422609189932681
69896558088044165355551471001743951239695738047106458517545601916
4578961762059345997440
Y: 260549350612676062448188625658154114443427558625932490186172625180
16179787773477706605100876536271693949174654809503102876480720244
0555166017893772852160
Encoded Public
8E 89 98 D0 2D 7F 76 C3 A7 FF B3 1D 2B 41 7E E9
51 6B 51 B5 F2 84 8D 17 6F 59 9B 5B 6F 01 CF 73
The key parameters of the first key contribution are:
ED448Key1 (ED448)
UDF: ZAAAITKGED44XKEYX
Scalar: 53741526827163371875813321995189037002816220481405056
0575131176561199901538761967283817989566517114321868559709090
857214825841130713784
Encoded Private
E5 0F 73 50 27 0A 2F 7D FD D0 96 E5 03 D3 35 2C
99 CB 71 7C 0B D9 49 E0 40 5E C7 FB D1 F5 05 18
18 6B 04 81 8B 4D 81 DC 33 CE DF 81 D5 EA 90 43
D9 E5 D0 A7 F1 EF 9C F3
X: 46070586985722000292864744471871508558334313374829024264732526
0256877451971853613261831326120959789917482766653217856979552
299464523257
Y: 60551781313893332009337150573641968782237423331690407655569563
6098829567124403075067563581124781122484845054468415282443786
254887063867
Encoded Public
B5 B9 3F B5 B2 5B 82 E1 08 F5 6C 79 80 A1 68 5A
5C BB 2A FD 27 B2 9A F8 DF 91 CD CA 60 B8 75 3F
62 96 40 38 78 96 77 0C 21 40 E6 D4 0B 05 8F 24
D6 FD 65 61 A2 6C C1 86 80
The key parameters of the second key contribution are:
ED448Key2 (ED448)
UDF: ZAAAITKGED44XKEY2
Scalar: 53741526827163371875813321995189037002816220481405056
0575131176561199901538761967283817989566517114321868559709090
857214825841130713784
Encoded Private
E5 0F 73 50 27 0A 2F 7D FD D0 96 E5 03 D3 35 2C
99 CB 71 7C 0B D9 49 E0 40 5E C7 FB D1 F5 05 18
18 6B 04 81 8B 4D 81 DC 33 CE DF 81 D5 EA 90 43
D9 E5 D0 A7 F1 EF 9C F3
X: 46070586985722000292864744471871508558334313374829024264732526
0256877451971853613261831326120959789917482766653217856979552
299464523257
Y: 60551781313893332009337150573641968782237423331690407655569563
6098829567124403075067563581124781122484845054468415282443786
254887063867
Encoded Public
B5 B9 3F B5 B2 5B 82 E1 08 F5 6C 79 80 A1 68 5A
5C BB 2A FD 27 B2 9A F8 DF 91 CD CA 60 B8 75 3F
62 96 40 38 78 96 77 0C 21 40 E6 D4 0B 05 8F 24
D6 FD 65 61 A2 6C C1 86 80
The composite private key is:
Scalar_A = (Scalar_1 + Scalar_2) mod L
= 16628213117375882432961168004377507211427270876895354579839960414
666978326982600598665720267457234882718565087272340290578290296
3178673
Encoded Composite Private Key:
B1 0C A6 9F 18 5C CE 75 68 CF A3 94 FD 1C 20 DC
08 4A 1A 8D EA 8F ED 45 17 68 B6 9F 55 03 DA 18
5F A8 2E F3 98 92 24 C7 C2 05 8E 86 9E BD 4E A2
6F 38 45 74 67 F6 90 3A
The composite public key is:
Point_A = Point_1 + Point_2
X: 577530061094566245645474620282168197911998758313406086914794815409
17246422939462333536659252558650386613682198540830437043888246526
5810
Y: 505836808606768081321267696834164575314792405646755177389264860926
82650383803094471233632502605998926567195957732072433352975006488
1824
Encoded Public
99 67 9B 7C 5E 1C 7E 51 CD 39 A2 41 83 62 73 3E
60 93 17 A9 20 0E E3 BA 25 B3 B5 23 A3 A7 84 2E
9D 67 6F D7 0B 33 02 1D EB 76 83 F8 77 D8 48 F8
8B A3 72 E8 A9 6F 20 18 80
The encryption key pair is
X25519KeyA (X25519)
UDF: ZAAACTHDX255XXKEYA
Scalar: 36212799908425711450656372795692477094724455418915704
216848228525958587810064
Encoded Private
10 01 D5 D1 E2 D3 DB 42 9E 40 5F D9 DB AE E8 09
DE 43 C3 E6 D1 4F 3A 31 92 BF 19 8A E9 B7 0F 50
U: 14523539712308371644546850739155588238080554014514563739095172
886049239361031
V: 56685060472089790044070522288405984326906734250304251487683593
932889808473139
Encoded Public
07 66 84 48 25 85 F6 4A 3A EE DF B7 69 1B 57 51
EC 18 BE AF 08 BA 0D FE BE F8 74 4E 3C 08 1C 20
To create n key shares we first create n1 key pairs in the normal fashion. Since these key pairs are only used for decryption operations, it is not necessary to calculate the public components:
X25519Key1 (X25519)
UDF: ZAAACTHDX255XXKEYX
Scalar: 32951726132685026729149224926255648061071804906258082
061427666995947179849152
Encoded Private
C0 B5 33 D4 F3 D0 16 4F 96 DF C3 AD 97 93 02 EF
B4 25 E2 46 A3 69 1D 22 9B 5B A2 78 1C 04 DA 48
The secret scalar of the final key share is the secret scalar of the base key minus the sum of the secret scalars of the other shares modulo the group order:
Scalar_2 = (Scalar_A  Scalar_1) mod L
= 403147584512037825404691865456117698808221309075461782425833707
7336679400315
This is encoded as a binary integer in little endian format:
7B 43 64 61 E9 28 4D 79 AB 9C 6E CC 9F 79 14 3D
92 69 A5 2D 75 B9 57 53 2D 1B BC 02 06 BC E9 08
The means of encryption is unchanged. We begin by generating an ephemeral key pair:
X25519KeyE (X25519)
UDF: ZAAACTHDX255XXKEYE
Scalar: 41955577142906312619127105554814681129195921605852142
704362465226652441661496
Encoded Private
38 50 3C 88 22 4F 61 D7 9A 2E 1D 71 F0 31 74 44
A2 3B 2B 35 21 21 CA 19 4B 11 EB F0 DF 03 C2 5C
U: 10080018124246254127076649374753145019412450363156572968151721
892767560820008
V: 43683938787921854603630290352714276342923724280578266457509078
671566344321831
Encoded Public
28 E5 5E 1D DD 1D 93 71 24 53 0A 83 B3 68 0D 28
8F 37 AC 53 B6 65 97 7E C1 54 44 41 8C 16 49 16
The key agreement result is given by multiplying the public key of the encryption pair by the secret scalar of the ephemeral key to obtain the ucoordinate of the result.
U: 247351751388894803426442650867524924086144759194834658830326105266
22202018180
The ucoordinate is encoded in the usual fashion (i.e. without specifying the sign of v).
84 39 A5 21 13 F9 13 F0 7F F4 44 C0 DF 5D 44 DD
DD F4 9B 87 4C DD E1 AB 64 00 8F A2 ED 9C AF 36
The first decryption contribution is generated from the secret scalar of the first key share and the public key of the ephemeral.
The outputs from the Montgomery Ladder are:
x_2 57800249527850149046770413207257250301842844049677844025524059085
132359257003
z_2 37229326806761131733056994095424883574786241198535734197174081138
402379671391
x_3 30722194817314627970562030033494699359853137448471883846088158083
361967945513
z_3 29143359268139878301695995826542801325089258636824690596939399658
126254126746
The coordinates of the corresponding point are:
u 2625200443692459084967263034650122583671912028244890150161521677645
8728744244
v 2340339249609928967870268630489687123941624857494487121340604194885
7707717709
The encoding of this point specifies the u coordinate and the sign (oddness) of the v coordinate:
28 E5 5E 1D DD 1D 93 71 24 53 0A 83 B3 68 0D 28
8F 37 AC 53 B6 65 97 7E C1 54 44 41 8C 16 49 16
The second decryption contribution is generated from the secret scalar of the second key share and the public key of the ephemeral in the same way:
u 2568180775076864300893967221119748767931055928591855851227298301978
9028635830
v 5237624535641756510077423429806596028526835148653096601777403098805
4910628425
28 E5 5E 1D DD 1D 93 71 24 53 0A 83 B3 68 0D 28
8F 37 AC 53 B6 65 97 7E C1 54 44 41 8C 16 49 16
To obtain the key agreement value, we add the two decryption contributions:
u 5363809193902384353244842537457427150937755976201184630020143767288
0976727749
v 2576238777948215852102595446870010694371604288066653261024661407554
3602367190
This returns the same u coordinate value as before, allowing us to obtain the encoding of the key agreement value and decrypt the message.
The encryption key pair is
X448KeyA (X448)
UDF: ZAAAETHDX44XKEYA
Scalar: 70596789123829480733485730386174565339013185647363028
6777277621057939099785091228353248522408450794128800398810019
569879502484967206280
Encoded Private
88 2D AF 58 10 66 9E 1E F9 F2 C5 76 A2 00 86 F5
B0 B9 C6 B9 E6 34 12 57 64 E3 63 B7 99 48 01 77
9B A3 49 2D 7C B8 80 D7 63 44 6B C9 CB 83 F0 01
B6 55 E0 92 1C 2A A6 F8
U: 54256629638851994806054576189463839532492460394052748417730874
4299533502601001906894660938607827805200569088593927035891085
28218439174
V: 12640494198304757803713993624351573936804262085795518571061045
0631383333635306581037675004961525698205648857075020359124084
524068583614
Encoded Public
06 FE 38 7A 1B 1E 99 D4 89 00 07 B9 88 6F 97 01
BD 88 BB 9D A9 31 30 CC 47 E6 2F 9C 44 35 AF A4
To create n key shares we first create n1 key pairs in the normal fashion. Since these key pairs are only used for decryption operations, it is not necessary to calculate the public components:
X448Key1 (X448)
UDF: ZAAAETHDX44XKEYX
Scalar: 63066265672668423343291438147840057172035337936373473
3594300758463732521976388313294665447253881782852832499090049
354258188417511652528
Encoded Private
B0 FC CE 55 87 AA A5 36 D2 5B E5 F2 5C 1B F7 9A
5A 3D 97 D8 BB C0 81 84 98 3B 7C 29 C3 02 FC AE
91 1B EA 67 68 C5 5E 87 7A ED 16 1F CB D0 20 9D
C0 D6 62 BD 0F 35 20 DE
The secret scalar of the final key share is the secret scalar of the base key minus the sum of the secret scalars of the other shares modulo the group order:
Scalar_2 = (Scalar_A  Scalar_1) mod L
= 646627804476669823064550215361382899916128546345584148789705309
5564959403070244163454368262048594523846084193642728800427461
1461310355
This is encoded as a binary integer in little endian format:
93 47 14 FF 94 BE F6 5C 77 63 44 89 DD CA 83 A6
1A 79 82 CA 9E F6 6B 7D 98 23 59 77 60 4B FD 25
2D BF 33 95 E4 7D DF 5E DE 31 82 E8 96 54 11 9F
76 2C 43 50 2C 5F C6 16
The means of encryption is unchanged. We begin by generating an ephemeral key pair:
X448KeyE (X448)
UDF: ZAAAETHDX44XKEYE
Scalar: 40831502887772840901106715270468009328116701340228919
4447950742749557088789408677311466089336893170031425082958041
776608657845012501716
Encoded Private
D4 94 79 EE 56 3A 43 D5 FC EB 88 3E F0 63 EF 2F
B0 92 B2 9D FD E1 43 8F 67 70 2A FC 2A AB A3 8B
40 5A C6 D8 DE 8E B8 81 BF AD 17 BA 14 7F A4 B0
D4 B1 9F CE D3 0D D0 8F
U: 41902542857582644710501442087876846551351583947506685319975417
2921931242764163121977613761818608927787788470856012050834001
655292441835
V: 40254626888669687592165362896510117701831215997629302922912638
4107109948063151002535904960734463095918076287222011626898597
213849340021
Encoded Public
EB 34 D3 9E 92 3E 82 CC E6 EC 77 9F 3D 11 83 3C
B6 5B 5C 04 E8 1F D6 E1 07 C0 62 FE F8 F6 34 BB
The key agreement result is given by multiplying the public key of the encryption pair by the secret scalar of the ephemeral key to obtain the ucoordinate of the result.
U: 414519841929159382730636919036875317796178034011999213235983537052
30510678702415242441193107876747178183775523375063128358893667955
0233
The ucoordinate is encoded in the usual fashion (i.e. without specifying the sign of v).
19 ED 3F 7A 63 6D AA 9A 3E 05 29 DE CC BA C7 F1
E0 A7 FA C0 C4 70 E0 E1 A5 FC DA 0A B0 52 EC 8A
The first decryption contribution is generated from the secret scalar of the first key share and the public key of the ephemeral.
The outputs from the Montgomery Ladder are:
x_2 60087238657789265874539701675840521614326868580285788286550399232
20277081132468800878653228200859196207538481852097024468118288584
83595
z_2 12573140552649037921890899942919440571502561130496393232716617155
24490660805576517881339517005970098784771472127066810694797758409
67645
x_3 40216911160845555507626938485596306260547743077930604703891475599
53207238052152872831803734397389643529057507149471429452955111471
57394
z_3 48671808823760924633626118221453591105199403417491562559814414359
34079336265430685974504655698846599309734305554546571306740770389
79747
The coordinates of the corresponding point are:
u 5310627084956226133549480379012439149584638171147187426442235629260
04186474991811830611467926523417739685244466041108245014409383321
437
v 6480384951984610654865132490606294355962228129695701220316681167173
89634554460437237795204236689985354028508124817314496063921770833
81
The encoding of this point specifies the u coordinate and the sign (oddness) of the v coordinate:
EB 34 D3 9E 92 3E 82 CC E6 EC 77 9F 3D 11 83 3C
B6 5B 5C 04 E8 1F D6 E1 07 C0 62 FE F8 F6 34 BB
The second decryption contribution is generated from the secret scalar of the second key share and the public key of the ephemeral in the same way:
u 2432029307606011854651950642127064534858415441240032460817128643691
02601495764607628214871657677482439232238254450281582409532827123
057
v 3304237495909049135999182737314299324454426462198935317955136317282
40167193898154033571033048069157608137872491595181800632292085611
537
EB 34 D3 9E 92 3E 82 CC E6 EC 77 9F 3D 11 83 3C
B6 5B 5C 04 E8 1F D6 E1 07 C0 62 FE F8 F6 34 BB
To obtain the key agreement value, we add the two decryption contributions:
u 2259776336573351712090323684006287979378142231675451855263136351631
84627019682834233607456823018602790573389381890060345691088913511
436
v 3356804465434915738990773252496451168197759413718456519751139752920
22788339143143140400339843420207702195408750068698590142923542805
226
This returns the same u coordinate value as before, allowing us to obtain the encoding of the key agreement value and decrypt the message.
All the security considerations of and apply and are hereby incorporated by reference.
Separation of duties can lead to a reduction in overall security through complacency and lack of oversight.
Consider the case in which a role that was performed by A alone is split into two roles B and C. If B and C each do their job with the same diligence as A did alone, risk should be reduced but if B and C each decide they can be careless because security is the responsibility of the other, the risk of a breach may be substantially increased.
It is therefore important that each of the participants in a threshold scheme perform their responsibilities with the same degree of diligence as if they were the sole control and for those responsible for oversight to treat single point failures that do not lead to an actual breach with the same degree of concern as if a breach had occurred.
Use of threshold operation modes mitigates but does not eliminate security considerations relating to private key operations of the underlying algorithm. For example, use of threshold key generation to generate a composite keypair {b+c, B+C} from key contributions {b, B} and {c, C} produces a strong composite private key if either of the key contributions a, b are strong. But the composite key will be weak if neither contribution is strong.
In general, threshold modes of operation provide a work factor that is at least as high as that of the work factor of the strongest private key share. The karmic tradeoff for this benefit is that the trustworthiness of a composite public key is that of the least trustworthy input.
For example, consider the case in which a client with keypair {c, C} generates an ephemeral keypair {e, E} for use in an authentication algorithm. We might decide to create an 'efficient' proof of knowledge of c and e by using the composite public key A = C+E to test for knowledge of both at the same time.
This approach fails because an attacker with knowledge of C can generate a keypair {a, A} and calculate the corresponding public key E = AC. The attacker can then use the value a in the authentication protocol.
This document requires no IANA actions (yet).
It will be necessary to define additional code points for the signed version of the X25519 and X448 public key and the threshold decryption final private keys.
Rene Struik, Tony Arcieri, Scott Fluhrer, Scott Fluhrer, Dan Brown, Mike Hamburg
The following C# code calculates the Lagrange coefficients used to recover the secret from a set of shares.
[TBS]
Elliptic Curves for Security
EdwardsCurve Digital Signature Algorithm (EdDSA)
Key words for use in RFCs to Indicate Requirement Levels
Alternative Elliptic Curve Representations
How to share a secret.
Timing attacks on implementations of DiffieHellman, RSA, DSS, and other systems.
Fast multiplication on elliptic curves over GF(2m) without precomputation.
Efficient elliptic curve cryptosystems from a scalar multiplication algorithm with recovery of the ycoordinate on a Montgomeryform elliptic curve.
Montgomery curves and their arithmetic
DiffieHellman Key Agreement Method
Threshold Signatures Using Ed25519 and Ed448
Mathematical Mesh 3.0 Part I: Architecture Guide
Mathematical Mesh: Reference Implementation