InternetDraft  Threshold Modes in Elliptic Curves  June 2023 
HallamBaker  Expires 30 December 2023  [Page] 
 Workgroup:
 Network Working Group
 InternetDraft:
 drafthallambakerthreshold
 Published:
 Intended Status:
 Informational
 Expires:
Threshold Modes in Elliptic Curves
Abstract
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 .¶
Status of This Memo
This InternetDraft is submitted in full conformance with the provisions of BCP 78 and BCP 79.¶
InternetDrafts are working documents of the Internet Engineering Task Force (IETF). Note that other groups may also distribute working documents as InternetDrafts. The list of current InternetDrafts is at https://datatracker.ietf.org/drafts/current/.¶
InternetDrafts are draft documents valid for a maximum of six months and may be updated, replaced, or obsoleted by other documents at any time. It is inappropriate to use InternetDrafts as reference material or to cite them other than as "work in progress."¶
This InternetDraft will expire on 30 December 2023.¶
Copyright Notice
Copyright (c) 2023 IETF Trust and the persons identified as the document authors. All rights reserved.¶
This document is subject to BCP 78 and the IETF Trust's Legal Provisions Relating to IETF Documents (https://trustee.ietf.org/licenseinfo) in effect on the date of publication of this document. Please review these documents carefully, as they describe your rights and restrictions with respect to this document.¶
1. Introduction
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 [RFC7748] and [RFC8032]. 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).¶
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 [Shamir79] is then described to support the case (n > t). For convenience, we refer to the non Shamir secret sharing case as 'direct key sharing'.¶
1.1. Applications
Development of the threshold modes described in this document were motivated by the following applications.¶
1.1.1. Cloud control of decryption
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.¶
1.1.2. Device Onboarding
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.¶
1.1.3. Cryptographic coprocessor
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.¶
1.1.4. Side Channel Resistance
The same techniques that make threshold cryptography possible are the basis for Kocher's sidechannel resistance technique [Kocher96]. 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.¶
2. Definitions
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.¶
2.1. 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 [RFC2119].¶
2.2. Defined Terms
The following terms are used as terms of art in this document and the accompanying specification [drafthallambakerthresholdsigs].¶
 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.¶
2.4. Implementation Status
The implementation status of the reference code base is described in the companion document [drafthallambakermeshdeveloper].¶
3. Threshold Cryptography in DiffieHellman
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.¶
3.1. Application to Diffie Hellman (not normative)
Diffie Hellman in a modular field provides a concise demonstration of the key combination and result combination laws [RFC2631]. 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:¶
Proof:¶
Let z = x + y¶
By definition, X = e^{x} mod p, Y = e^{y} mod p, and Z = e^{z }mod p.¶
Therefore,¶
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}.¶
3.2. Threshold decryption
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.¶
3.2.1. Direct Key Splitting
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.¶
3.2.2. Direct Decryption
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.¶
3.3. Direct threshold key generation
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.¶
3.3.1. Device Provisioning
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.]¶
3.3.2. Key Rollover
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:¶
3.3.3. Host Activation
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.¶
3.3.4. Separation of Duties
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.¶
3.4. Side Channel Resistance
Sidechannel attacks, present a major concern in the implementation of public key cryptosystems. Of particular concern are the timing attacks identified by Paul Kocher [Kocher96] 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.¶
4. Shamir Secret Sharing
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.¶
4.2. Lagrange basis recovery
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¶
4.3. Verifiable Secret Sharing
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]¶
5. Application to Elliptic Curves
For elliptic curve cryptosystems, the operators [x] and [.] are point addition.¶
Implementing a robust Key CoGeneration for the Elliptic Curve Cryptography schemes described in [RFC7748] and [RFC8032] 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 [RFC7748], the means of performing point addition is not.¶
5.1. Implementation for Ed25519 and Ed448
[RFC8032] provides all the cryptographic operations required to perform threshold operations and corresponding public keys.¶
The secret scalars used in [RFC8032] 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 [RFC8032] 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.¶
5.1.1. Ed25519
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¶
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.¶
5.1.2. Ed448
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.¶
5.2. Implementation for X25519 and X448
[RFC7748] 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 [RFC7748] 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 [RFC7748] is followed using {u, v} to represent the coordinates on the Montgomery curve and {x, y} for coordinates on the corresponding Edwards curve.¶
5.2.1. Point Encoding
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 . [RFC8032]¶
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.¶
5.2.2. X25519 Point Encoding
The extended point encoding allowing the v coordinate to be recovered is as given in [draftietflwigcurverepresentations]¶
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.¶
5.2.3. X448 Point Encoding
The extended point encoding allowing the v coordinate to be recovered is as given in [draftietflwigcurverepresentations]¶
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.¶
5.2.4. Point Addition
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.¶
5.2.5. Montgomery Ladder with Coordinate Recovery
As originally described, the Montgomery Ladder only provides the u coordinate as output. L?pez and Dahab [Lopez99] 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 [Okeya01] to prime field Montgomery curves such as X25519 and X448. The realization of this result described by Costello and Smith [Costello17] is applied here.¶
The scalar multiplication function specified in [RFC7748] 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 [RFC7748] 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¶
6. Test Vectors
6.1. Threshold Key Generation
6.1.1. X25519
The key parameters of the first key contribution are:¶
X25519Key1 (X25519) UDF: ZAAACTKGX255XXKEYX Scalar: 324858804843944019775357777134446830499336694251 10150162410603197194664525072 Encoded Private 10 BD E5 52 D6 AF 62 BE E4 5B F3 30 B8 FC 1C 51 B3 1B 10 9D 1E E9 D7 8D 04 23 39 08 55 5B D2 47 U: 394710109806205653786046834135930103397328759791 49327589691528787260753559967 V: 356194396126197144337653540363972416928636805167 03007342935403505222303874766 Encoded Public 9F C1 03 BF A0 E6 6F C7 F1 98 4F 11 99 6E 35 E8 E0 12 0A 0A D0 0D 79 97 4E 8A 1C 08 EF CC 43 57 00¶
The key parameters of the second key contribution are:¶
X25519Key2 (X25519) UDF: ZAAACTKGX255XXKEY2 Scalar: 411580799970764177643732652839689210097433579803 82961631181135147895746569008 Encoded Private 30 A3 31 35 93 F6 AD C9 AC 13 1C 27 15 83 C8 1B 00 EF 48 B9 52 14 8D 4D 3C F0 A3 C1 D2 A5 FE 5A U: 459625327459773177107083316486045330014029630693 84886488666639114650950362503 V: 257984254735960195350757294576288849253536777584 92280923754924184339462178503 Encoded Public 87 E5 CC DD 1D AA 42 EA 6F E8 6F 70 71 EE CF 86 45 52 48 50 9D B2 6A 76 3B 7A 21 A0 23 DF 9D 65 80¶
The composite private key is:¶
Scalar_A = (Scalar_1 + Scalar_2) mod L =127390470814819760217717736698366165110586381169403573357222896 2235868584190 Encoded Composite Private Key: FE 18 7D E6 61 C7 58 17 32 4F 63 FA 1A BD 2F 9C B2 0A 59 56 71 FD 64 DB 40 13 DD C9 27 01 D1 02¶
The composite public key is:¶
Point_A = Point_1 + Point_2 U: 150135516006055775898023592854502286813423441552 78788763949643404481150980325 V: 415138854802975146804397164494309448482487190852 96478905918078322266869330796 Encoded Public E5 10 7A CA 6D 63 5F 0B 96 8D C1 FF 03 88 6A 9F 5E 39 FB C7 7D 4E 0C 8F B9 BE 02 68 7B 5E 31 21¶
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.¶
6.1.2. X448
The key parameters of the first key contribution are:¶
X448Key1 (X448) UDF: ZAAAETKGX44XKEYX Scalar: 481994585826426647968618231378947367458943870175 360121942970840151506728554061197387112897908361553737348667 886278259299907241847731316 Encoded Private 74 B4 D2 F1 12 CC E7 DD F8 1A 30 80 1F 2C 19 EA EF E2 B3 8A 84 AF 60 11 0C 12 ED C3 B7 59 AE CC C9 B4 E4 9D 39 26 7C 61 5F 18 F1 24 FE 63 D6 4B BB 90 58 16 43 6E C3 A9 U: 394700438080601820949554957897111201031639217153 649316073420498416350673492846326387358441405527682364389676 084099279128049619119543974 V: 444540392869323915210590844990511294596507999517 821659576820658489772434608466545900017841967298236181114781 72629203404123869477697007 Encoded Public A6 96 1A 77 DC 39 41 5F D7 DA A5 07 45 AC 8E A4 3E AE 8C 77 BD 50 4A B0 24 64 CD EA 58 0A A3 C7 A7 80 BA A6 10 BD 57 9A FA 0C E3 EB 2F C8 BB 52 36 42 B2 58 C3 7B 04 8B 80¶
The key parameters of the second key contribution are:¶
X448Key2 (X448) UDF: ZAAAETKGX44XKEY2 Scalar: 511754555230466033071173155845990436633003100190 696628114856620555834938157360893794661792647517417178355617 271188892590944570884738624 Encoded Private 40 CE 77 E2 F2 EC 9B 7D 3E F4 62 C6 F9 99 81 B4 19 E5 4B 18 48 54 13 C9 79 D4 FF 3C ED 3B 9C A1 FE 10 7E DC 1F 56 BD 4D 27 7F 9C 70 4B 30 BE 0A 86 2A 01 3D 2A C3 3E B4 U: 150272226357947871808446816866811163361018047865 977226556625643162666620657310046951510401422900774703631401 927908349274354369426223715 V: 366137359317179726859107301681665650104046728426 432019544955478508004916807825041417161908471339553361130707 683387754680120674193097983 Encoded Public 63 F2 0D 66 B0 F9 43 1C 58 AD 56 2B C7 9A D5 83 B0 B5 B1 73 9A BE B9 1E 72 5D 4A F7 8D 45 00 A6 B3 7F AA 27 BE B4 72 44 EE D6 AA 24 5B BE B9 92 F8 8D 63 CC A1 6A ED 34 80¶
The composite private key is:¶
Scalar_A = (Scalar_1 + Scalar_2) mod L =852007356873840678531366273649321361498952695069091747059647117 316116469037241626007959140974170910991528166120088403669830 33434221045 Encoded Composite Private Key: F5 29 91 7B 28 EC 27 AA 8D 42 B7 81 DC F9 7A F7 38 B7 D0 38 5C BB E9 04 F5 32 FA 90 A7 95 4A 6E C8 C5 62 7A 59 7C 39 AF 86 97 8D 95 49 94 94 56 41 BB 59 53 6D 31 02 1E¶
The composite public key is:¶
Point_A = Point_1 + Point_2 U: 992302561314692402358105531637482692313485564268 012269312838533303200135230457785976232221579845336021490878 11249911833804897105796187 V: 456075089062293144799088747749397731190530292504 203343525695648243447672023318462080397566120195617049762303 280841894269351301464260878 Encoded Public 5B DC 74 39 94 08 79 2C D5 F0 F1 E0 5F 7F 87 4D 4D 3B 92 96 AB 62 FF EC CB 3C 74 42 48 D2 D0 30 95 45 37 89 5E 53 5D 47 72 DD D8 1A 24 2C 65 76 1F 7A FB 2E 15 2D F3 22¶
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.¶
6.1.3. Ed25519
The key parameters of the first key contribution are:¶
ED25519Key1 (ED25519) UDF: ZAAAGTKGED255XXKEYX Scalar: 566707288876750955042011684517553468890396806254 32020135051411766560411209648 Encoded Private 1D 04 C0 18 98 F0 31 CA 3B A1 F0 C3 AD 2B FC 3B CF F1 DC DC 07 FC 61 5F B1 63 75 35 CE C4 EA B4 X: 102242812372232316143929541945425934190366349837 479493246432658075153118475011602747352787206173426322413798 12049009970952489882776094157952001361119144 Y: 789031640540855004178762236065881312872127689877 983340977132004355159711145586361923666616187290905159854593 272855769333002825303166428484255810207910288 Encoded Public D8 4E 98 3E F7 21 87 CC C6 47 CB 6E 5A 57 1F 79 F5 B9 22 D9 A8 00 D3 69 91 E1 B6 E6 10 FF 59 08¶
The key parameters of the second key contribution are:¶
ED25519Key2 (ED25519) UDF: ZAAAGTKGED255XXKEY2 Scalar: 566707288876750955042011684517553468890396806254 32020135051411766560411209648 Encoded Private 1D 04 C0 18 98 F0 31 CA 3B A1 F0 C3 AD 2B FC 3B CF F1 DC DC 07 FC 61 5F B1 63 75 35 CE C4 EA B4 X: 102242812372232316143929541945425934190366349837 479493246432658075153118475011602747352787206173426322413798 12049009970952489882776094157952001361119144 Y: 789031640540855004178762236065881312872127689877 983340977132004355159711145586361923666616187290905159854593 272855769333002825303166428484255810207910288 Encoded Public D8 4E 98 3E F7 21 87 CC C6 47 CB 6E 5A 57 1F 79 F5 B9 22 D9 A8 00 D3 69 91 E1 B6 E6 10 FF 59 08¶
The composite private key is:¶
Scalar_A = (Scalar_1 + Scalar_2) mod L =478637411536625779880453845786578016522261586016542618007355945 8839008654461 Encoded Composite Private Key: 7D 34 94 4B 39 80 D9 34 DB EB E1 7C 6C 7E BB FA 77 03 B0 DC 59 BD DB 30 E9 EC 02 15 E3 FD 94 0A¶
The composite public key is:¶
Point_A = Point_1 + Point_2 X: 156803891315589086286215914260988096641777056734 949191292807298262090581551508321205906817723758419038014448 221785557483058278412455441230622087321788265 Y: 797171707400790781155409731019930644260359827339 627382830182820599849038258772654275070450097319504001761725 123426865448110684459259776487353068850001481 Encoded Public 7B 1C 48 02 66 17 79 32 B3 02 7B 21 8E D8 FD 6C A1 D5 EC 8E 28 5D E8 D3 E2 08 1A F9 EB FA AC 32¶
6.1.4. Ed448
The key parameters of the first key contribution are:¶
ED448Key1 (ED448) UDF: ZAAAITKGED44XKEYX Scalar: 627183439506120484725959826569914487299049496696 595523009635802846606657882532934711603764178690431435900965 134411043888412886366713268 Encoded Private 03 81 D6 5A 17 53 20 CB F4 02 40 66 C8 28 A5 E7 85 C6 87 7D 41 3A 33 CD 5B 09 9F E5 DF 1A 80 4D 4F 6E B1 35 F7 46 8B 1C 46 99 6E 6D A4 B5 23 EC FD 9C DF A4 9A E5 17 C6 X: 583262368398068306064924227289526726945089383036 870654148794281596473976374037255611400384372179423639836667 405050261630569976923704901 Y: 394808842674474575092248856523729981587605779158 130628847253048358283876706462306075444313878262092034880682 915931230840640589150609812 Encoded Public BB 4B F3 61 B5 1F 95 1E 88 04 A8 5E 48 77 4C A5 25 F0 D6 49 41 07 12 C1 15 05 25 78 60 FC D9 A8 CC DB F9 6D 63 6B C3 F7 63 F2 BB 00 A0 7C 13 E1 C6 4B 72 08 EA C8 87 11 00¶
The key parameters of the second key contribution are:¶
ED448Key2 (ED448) UDF: ZAAAITKGED44XKEY2 Scalar: 627183439506120484725959826569914487299049496696 595523009635802846606657882532934711603764178690431435900965 134411043888412886366713268 Encoded Private 03 81 D6 5A 17 53 20 CB F4 02 40 66 C8 28 A5 E7 85 C6 87 7D 41 3A 33 CD 5B 09 9F E5 DF 1A 80 4D 4F 6E B1 35 F7 46 8B 1C 46 99 6E 6D A4 B5 23 EC FD 9C DF A4 9A E5 17 C6 X: 583262368398068306064924227289526726945089383036 870654148794281596473976374037255611400384372179423639836667 405050261630569976923704901 Y: 394808842674474575092248856523729981587605779158 130628847253048358283876706462306075444313878262092034880682 915931230840640589150609812 Encoded Public BB 4B F3 61 B5 1F 95 1E 88 04 A8 5E 48 77 4C A5 25 F0 D6 49 41 07 12 C1 15 05 25 78 60 FC D9 A8 CC DB F9 6D 63 6B C3 F7 63 F2 BB 00 A0 7C 13 E1 C6 4B 72 08 EA C8 87 11 00¶
The composite private key is:¶
Scalar_A = (Scalar_1 + Scalar_2) mod L =164108792568830633627933941307822173067636952362213955597036306 922337291995828355126032996607226607091940168014272113948183 237575527862 Encoded Composite Private Key: B6 4D CF FC C2 D8 08 4F 08 3C 03 5E C5 68 95 D2 65 F8 1D F2 C6 DC 4D 2E 23 B5 D5 02 AC 7A C5 A8 77 E9 D0 6E A2 1D 3D 11 91 49 AC 00 BF 0A 5B 95 59 BD 0D 94 6E 00 CD 39¶
The composite public key is:¶
Point_A = Point_1 + Point_2 X: 397684546797270425386442556916724605113037846866 341266059847555740358632067326446186925590144430667066483990 646036946928867316187394724 Y: 408779958726533701154676738846429933260841277918 126357328937237395286337382114316760480663335443832148985549 99820255282659972439867001 Encoded Public 39 DE 52 BB 22 79 72 DD 40 82 BB 45 7B FA 61 AD D0 BA 02 FF 8A FB 62 62 39 C6 C0 46 6B 35 93 E4 CD 1B 5F 79 FB D7 68 87 4C E7 D8 09 B1 3C ED 5A DE 61 5A D2 DC 19 C6 D3 00¶
6.2. Threshold Decryption
6.2.1. Direct Key Splitting X25519
The encryption key pair is¶
X25519KeyA (X25519) UDF: ZAAACTHDX255XXKEYA Scalar: 326889380688366386055157824948497405399138729191 89762489343300449845480879296 Encoded Private C0 74 51 B1 0A 11 F3 AA E9 E8 5C 99 A2 29 2F 78 88 A8 FC 3D 09 69 06 60 C2 B4 95 71 85 48 45 48 U: 298393977624490693814310584567182927861271505374 9632477801696949179285366587 V: 469514996865823666093464626932614325688577702009 92824380735949162568370305003 Encoded Public 3B E7 D1 11 EA 09 02 81 C7 88 E9 59 7A 44 D1 D5 34 AE 12 E2 3C 59 32 99 41 D1 99 B6 9D D9 98 06¶
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: 312348810422743662022326371802074910867528782383 29365764492453739245942799272 Encoded Private A8 FB C2 FD 62 20 CA 30 DA 44 87 38 BA 30 BE 5E FD 71 8F 48 33 E2 D2 9E 3F 28 AA C7 F0 50 0E 45¶
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 =602777449245290709596292717071327769980982028247986740582014668 2807789670656 This is encoded as a binary integer in little endian format: 00 D1 65 C7 9A 18 2A 1B 11 47 27 BA 67 8B F5 2F 85 1A 8C 86 3C 4B D9 FE 01 DD 3F 39 76 99 53 0D¶
6.2.2. Direct Decryption X25519
The means of encryption is unchanged. We begin by generating an ephemeral key pair:¶
X25519KeyE (X25519) UDF: ZAAACTHDX255XXKEYE Scalar: 493149316568141341116267515722816577238360781396 63006284230452497981236822048 Encoded Private 20 C0 8B F4 BA DB D2 9A 69 47 45 73 4F 34 8E 35 B5 78 24 AB F6 85 29 51 37 0A CB 38 1E 43 07 6D U: 262957444446660172926134873812719769018477396438 13954077731613449391499377029 V: 275785345363973047453636076008194379259667671218 204709190728344437212344040 Encoded Public 85 F9 AB 1E 1F 07 0F F9 9A 61 9F 3A C8 34 C5 A2 44 20 2A 92 7C 06 D8 54 E7 56 83 4F 2A DD 22 3A¶
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: 288787781548525369610061893837025332891029237654 14471063569609087123262768472¶
The ucoordinate is encoded in the usual fashion (i.e. without specifying the sign of v).¶
58 85 FB 70 25 DB ED FB F4 3F C2 11 65 A7 B6 FA 1B 2F 02 B7 36 34 A3 7B F3 A0 2B 90 27 CF D8 3F¶
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: 492098238905847779621612291688488017809049229602 94749204467459229896958710447 z_2: 429527757883227955375237456769123024739869368263 91347354499451449213467668091 x_3: 116605272267998449524192895707557114882629363798 86482562941790223923418840285 z_3: 195732355890349962819915384747512066302779899283 35607999800289560050872569334¶
The coordinates of the corresponding point are:¶
u: 536730894304808282930528988436660517867298128709 04753433533361766729625453382 v: 421632078600697821949748757382882309748636068616 05021486709890742236074210497¶
The encoding of this point specifies the u coordinate and the sign (oddness) of the v coordinate:¶
85 F9 AB 1E 1F 07 0F F9 9A 61 9F 3A C8 34 C5 A2 44 20 2A 92 7C 06 D8 54 E7 56 83 4F 2A DD 22 3A¶
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: 315780533103181510115520285714808517940341134033 73677773373235419322267220550 v: 182446295500312360012159151093139263903764865208 52094490828880182704089261185¶
85 F9 AB 1E 1F 07 0F F9 9A 61 9F 3A C8 34 C5 A2 44 20 2A 92 7C 06 D8 54 E7 56 83 4F 2A DD 22 3A¶
To obtain the key agreement value, we add the two decryption contributions:¶
u: 400218389629441160369977262251603578789352483113 44007312711045713754930418024 v: 365543819982499491694061844647668737870082451479 1236363503671566229693675370¶
This returns the same u coordinate value as before, allowing us to obtain the encoding of the key agreement value and decrypt the message.¶
6.2.3. Direct Key Splitting X448
The encryption key pair is¶
X448KeyA (X448) UDF: ZAAAETHDX44XKEYA Scalar: 513061070451628243040038974375191143450906600896 180328078543208371170428980692680538107843349213517810077524 748539797940481719097207576 Encoded Private 18 AB BD 69 F6 B7 16 23 72 4E B5 28 7E F8 F1 4E DB B5 6C EF 00 CD 51 4A AD F6 24 AF 73 0B CC 37 E4 66 01 C0 B4 35 18 99 CA 31 D0 7E 5D C6 86 9F 4F 33 33 95 BB 90 B4 B4 U: 193313388884413202154350037477490868937254022660 936368668113777978992235566230625856064905970782804480638718 961661755983597338734305565 V: 632512063264494610095455928921278058920338770152 331221019494393225431601640040112765612198296012844171344559 006776860014410518640582570 Encoded Public 1D 21 53 89 F7 D8 78 AD F5 4F 66 AE F6 E4 35 57 A4 2D 0F 29 D7 ED 64 13 5A 15 5D 0C 5A 9D 78 8E 30 AA D7 ED 94 D3 0A FD 5F C9 EB C4 6E 78 CB EC 67 10 DE 1A F7 41 16 44¶
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: 584733191291060171614515657474831905352900996815 538008733617256668598608739673264046230908386003505289747870 068686374821834248930905564 Encoded Private DC CD 9E 38 94 A1 EA CA 7D 41 FA B5 FB D3 01 43 4A FB F9 1D AE 73 82 3B 4C 11 A8 F2 2A 36 87 AB 3F EB EE E8 DC AB A7 30 5E 26 9C BC 4C FC D0 C8 48 F1 D3 78 A1 F0 F2 CD¶
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 =753617529927807883056892001801624727334555668074124638992516626 889301395112843028716421998506276731996363256267619893367343 2870214466 This is encoded as a binary integer in little endian format: 42 DB 4A 9E 1A CA 2C 19 F1 33 0E 8C CA 3D 67 C9 C4 69 61 F4 F4 1C FB EB 7E 30 10 B5 A1 41 53 E3 23 52 F0 A8 91 E1 BF C9 28 58 6C 3B AA C2 57 68 98 24 07 0E 5D 81 A7 02¶
6.2.4. Direct Decryption X448
The means of encryption is unchanged. We begin by generating an ephemeral key pair:¶
X448KeyE (X448) UDF: ZAAAETHDX44XKEYE Scalar: 375092101281685084138772040470324089521485504818 151786170217259465369930000693812877111168523934570244151290 043203531788355356164832452 Encoded Private C4 3C 47 59 CD E7 17 95 B4 7B 93 AA 69 B8 B6 B7 ED FE 18 D7 F4 7F 60 65 F1 89 C7 35 8D B5 43 37 1B 7F 29 3E C2 DE EF 30 B6 C6 B5 53 17 C5 53 34 E1 86 A9 88 60 7B 1C 84 U: 563581233819606639218664767140161390782939076635 909204332105344559772692562144684584035704830171693935746990 510749935485351255733775569 V: 361864251157310605646771236064439317468860292821 265311098908535377847605233578670208219924720551821972711559 36724947885353521079579125 Encoded Public D1 2C A9 6B 5E 97 F8 F0 18 2A BF 33 E8 14 65 23 A9 F1 06 9B D5 F0 DB 06 01 E5 1F 87 07 7D 69 63 0A FD 05 FB 7A 65 4C D5 81 FC 63 11 5B D6 40 A1 40 2F A5 FE B3 C1 7F C6¶
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: 467011616546325364170545331693527825162957894564 752019329166269756954567903847269575078419988391203956276277 502444715801151733990260662¶
The ucoordinate is encoded in the usual fashion (i.e. without specifying the sign of v).¶
B6 7F 79 43 2A 13 43 58 EB A5 F5 7E 0E 58 9B AA BB D7 B1 7E 07 3E 42 F1 ED F4 C0 09 0C 5C 4E 88 C9 81 21 E5 31 53 40 2F DE 7B 91 FE E4 47 A2 A7 9B F8 E8 B0 AC 7A 7C A4 00¶
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: 506558348563097978312390313048113345168022212711 849349435150224631183990403335052324912391443123499702764248 849278356671151441957790278 z_2: 213125180906412917064601709491242258781796680209 496987386274744929820708257576147081840825053136082223498470 326539519004870996749115274 x_3: 203497752532008592529275846876889307392855130050 266584525256476121907823114215626360048014875021016264668899 873822741730779444877210235 z_3: 571082612271586102575474047974979930626798015601 680732025993048573542014881343408880669710315066232049715159 504508304710290054762104158¶
The coordinates of the corresponding point are:¶
u: 612112921160648939091711280397522831608649375644 855484627685573574796097573324948060246221854728690332168796 528561330174015687677222868 v: 201043408411452768324992701951900381185869686931 750056109757993947816262634088142979592355613308839469219652 899581533430916970635983432¶
The encoding of this point specifies the u coordinate and the sign (oddness) of the v coordinate:¶
D1 2C A9 6B 5E 97 F8 F0 18 2A BF 33 E8 14 65 23 A9 F1 06 9B D5 F0 DB 06 01 E5 1F 87 07 7D 69 63 0A FD 05 FB 7A 65 4C D5 81 FC 63 11 5B D6 40 A1 40 2F A5 FE B3 C1 7F C6¶
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: 290543592387303668402000444540573819834556168336 146133899156492196788195733788135568819938481333767701019721 625502303783518942583538850 v: 545084635544186338165791882421333945771369831404 654834318355031722653832417877619017774120725462178582001609 731098147709911715676858744¶
D1 2C A9 6B 5E 97 F8 F0 18 2A BF 33 E8 14 65 23 A9 F1 06 9B D5 F0 DB 06 01 E5 1F 87 07 7D 69 63 0A FD 05 FB 7A 65 4C D5 81 FC 63 11 5B D6 40 A1 40 2F A5 FE B3 C1 7F C6¶
To obtain the key agreement value, we add the two decryption contributions:¶
u: 654406645663157043297342343331709836245150043735 702788665901049252553535973169012625989449576860527067131717 971914114816832035824532119 v: 104146947548847304025795777575846818531726293762 244429364569547834686063656820529540685030846449787127578128 365084621097748733067553687¶
This returns the same u coordinate value as before, allowing us to obtain the encoding of the key agreement value and decrypt the message.¶
7. Security Considerations
All the security considerations of [RFC7748] and [RFC8032] apply and are hereby incorporated by reference.¶
7.1. Complacency Risk
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.¶
7.2. Rogue Key Attack
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.¶
8. IANA Considerations
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.¶
9. Acknowledgements
Rene Struik, Tony Arcieri, Scott Fluhrer, Scott Fluhrer, Dan Brown, Mike Hamburg¶
10. Appendix A: Calculating Lagrange coefficients
The following C# code calculates the Lagrange coefficients used to recover the secret from a set of shares.¶
[TBS]¶
11. Normative References
 [draftietflwigcurverepresentations]
 Struik, R., "Alternative Elliptic Curve Representations", Work in Progress, InternetDraft, draftietflwigcurverepresentations23, , <https://datatracker.ietf.org/doc/html/draftietflwigcurverepresentations23>.
 [RFC2119]
 Bradner, S., "Key words for use in RFCs to Indicate Requirement Levels", BCP 14, RFC 2119, DOI 10.17487/RFC2119, , <https://www.rfceditor.org/rfc/rfc2119>.
 [RFC7748]
 Langley, A., Hamburg, M., and S. Turner, "Elliptic Curves for Security", RFC 7748, DOI 10.17487/RFC7748, , <https://www.rfceditor.org/rfc/rfc7748>.
 [RFC8032]
 Josefsson, S. and I. Liusvaara, "EdwardsCurve Digital Signature Algorithm (EdDSA)", RFC 8032, DOI 10.17487/RFC8032, , <https://www.rfceditor.org/rfc/rfc8032>.
12. Informative References
 [Costello17]
 Costello, C. and B. Smith, "Montgomery curves and their arithmetic", .
 [drafthallambakermesharchitecture]
 HallamBaker, P., "Mathematical Mesh 3.0 Part I: Architecture Guide", Work in Progress, InternetDraft, drafthallambakermesharchitecture21, , <https://datatracker.ietf.org/doc/html/drafthallambakermesharchitecture21>.
 [drafthallambakermeshdeveloper]
 HallamBaker, P., "Mathematical Mesh: Reference Implementation", Work in Progress, InternetDraft, drafthallambakermeshdeveloper10, , <https://datatracker.ietf.org/doc/html/drafthallambakermeshdeveloper10>.
 [drafthallambakerthresholdsigs]
 HallamBaker, P., "Threshold Signatures in Elliptic Curves", Work in Progress, InternetDraft, drafthallambakerthresholdsigs06, , <https://datatracker.ietf.org/doc/html/drafthallambakerthresholdsigs06>.
 [Kocher96]
 Kocher, P. C., "Timing attacks on implementations of DiffieHellman, RSA, DSS, and other systems.", .
 [Lopez99]
 L?opez, J. and R. Dahab, "Fast multiplication on elliptic curves over GF(2m) without precomputation.", .
 [Okeya01]
 Okeya, K. and K. Sakurai, "Efficient elliptic curve cryptosystems from a scalar multiplication algorithm with recovery of the ycoordinate on a Montgomeryform elliptic curve.", .
 [RFC2631]
 Rescorla, E., "DiffieHellman Key Agreement Method", RFC 2631, DOI 10.17487/RFC2631, , <https://www.rfceditor.org/rfc/rfc2631>.
 [Shamir79]
 Shamir, A., "How to share a secret.", .