Threshold Key Generation and Decryption in Ed25519 and Ed448

Internet-Draft	Threshold Key Generation and Decryption	January 2020
Hallam-Baker	Expires 8 July 2020	[Page]

Abstract

Threshold cryptography schemes 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 co-operate to complete a private key agreement operation. The same primitives may be applied to improve resistance to side channel attacks.¶

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 .¶

3. Principles

The threshold cryptography mechanisms 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:¶

r [.] S = S^r mod p¶
o = p-1¶
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^zmod p.¶

Therefore,¶

Z = e^z mod p

= e^x+y mod p¶

= (e^xe^y) mod p¶

= e^xmod p.e^y mod p¶

= X.Y¶

Moreover, A = e^a mod p,¶

Therefore,¶

(A^x mod p).(A^y mod p)

= (A^xA^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. 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 { S₁, s₁ }, { S₂, s₂ }, ... { S_n, s_n }, we begin by generating the first n-1 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 - ... - sn-1) 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. 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 = c₁ . c₂ . ... . c_n¶

This value is equal to the encryption key agreement value according to the group law.¶

3.3. 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 Man-in-the-Middle 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 multi-step 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¶

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

Side-channel 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. Application to Elliptic Curves

For elliptic curve cryptosystems, the operators [x] and [.] are point addition.¶

Implementing a robust Key Co-Generation 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 Co-Generation 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.¶

4.1. 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.¶

4.1.1. Ed25519

Let the inputs to the threshold key generation scheme be a set of 32 byte private key values P₁, P2 ... Pn. For each private key value i in turn:¶

Hash the 32-byte private key using SHA-512, storing the digest in a 64-octet 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 little-endian 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 m₁, m₂, ... 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.¶

4.1.2. Ed448

Let the inputs to the threshold key generation scheme be a set of 57 byte private key values P₁, P2 ... Pn. For each private key value i in turn:¶

Hash the 57-byte private key using SHAKE256(x, 114), storing the digest in a 114-octet 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 little-endian 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 m₁, m₂, ... 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.¶

4.2. 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.¶

4.2.1. Point Encoding

The relationship between the u and v coordinates is specified by the Montgomery Curve formula itself:¶

v² = u³ + Au2 + u¶

An algorithm for extracting a square root of a number in a finite field is specified in . [RFC8032]¶

Since v² has a positive (v) and a negative solution (-v), it follows that v² mod p will have the solutions v, p-v. Furthermore, since p is odd, if v is odd, p-v 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.¶

4.2.2. X25519 Point Encoding

The extended point encoding allowing the v coordinate to be recovered is as given in [draft-ietf-lwig-curve-representations]¶

[TBS]¶

4.2.3. X448 Point Encoding

The extended point encoding allowing the v coordinate to be recovered is as given in [draft-ietf-lwig-curve-representations]¶

[TBS]¶

4.2.4. Point Addition

The point addition formula for the Montgomery curve is defined as follows:¶

Let P₁ = {u₁, v₁}, P₂ = {u₂, v₂}, P₃ = {u₃, v₃} = P₁ + P₂¶

By definition:¶

u₃ = B(v₂ - v₁)² / (u₂ - u₁)² - A - u₁ - u₂¶

= B((u₂v₁ - u₁v₂)² ) / u₁u₂ (u₂ - u₁)²¶

v₃ = ((2u₁ + u₂ + A)(v₂ - v₁) / (u₂ - u₁)) - B (v₂ - v₁)³ / (u₂ -u₁)³ - v₁¶

For curves X25519 and X448, B = 1 and so:¶

u₃ = ((v₂ - v₁).(u₂ - u₁)^-1)² - A - u₁ - u₂¶

v₃ = ((2u₁ + u₂ + A)(v₂ - v₁).(u₂ - u₁)^-1) - ((v₂ - v₁).(u₂ -u₁)^-1)³ - v₁¶

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 v₁, v₂. At minimum whether one is odd and the other even or if both are the same.¶

4.2.5. 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 u₁ of the point P₁ = {u₁, v₁} to be multiplied. The return value in this case is u₂ where P₂ = {u₂, v₂} = k.P₁.¶

To recover the coordinate v₂ 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 = bits-1 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 P₂ = {u₂, v₂} = k.P₁ and the values x_3, z_3 give the projective form of the u coordinate of the point P₃ = {u₃, v₃} = (k+1).P₁ = P₁ + k.P₁ = P₁ + P₂.¶

Given the coordinates {u₁, v₁} of the point P1 and the u coordinates of the points P₂, P₁ + P₂, the coordinate v₂ 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 {u₂, v₂} 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

5. Test Vectors

5.1. Threshold Key Generation

5.1.1. X25519

The key parameters of the first key contribution are:¶

X25519Key1 (X25519)
    UDF:        ZAAA-CTKG-X255-XXKE-YX
    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:        ZAAA-CTKG-X255-XXKE-Y2
    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 aggregate private key is:¶

Scalar_A = (Scalar_1 + Scalar_2) mod L
  = 70836469997123835938663996799427126600035261699252680723027418877
      4936630452

Encoded Aggrgate 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 aggregate 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 aggregate key is intended for unsigned CurveX key agreement. If the result is intended for use as a key contribution, the signed representation is required.¶

5.1.2. X448

The key parameters of the first key contribution are:¶

X448Key1 (X448)
    UDF:        ZAAA-ETKG-X44X-KEYX
    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:        ZAAA-ETKG-X44X-KEY2
    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 aggregate private key is:¶

Scalar_A = (Scalar_1 + Scalar_2) mod L
  = 87935198894654874397041717160555226349504546089353009501069716070
      586506403266723929544670861554164189887604126085304951388779109
      045747

Encoded Aggrgate 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 aggregate 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

5.1.3. Ed25519

The key parameters of the first key contribution are:¶

ED25519Key1 (ED25519)
    UDF:        ZAAA-GTKG-ED25-5XXK-EYX
    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:        ZAAA-GTKG-ED25-5XXK-EY2
    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 aggregate private key is:¶

Scalar_A = (Scalar_1 + Scalar_2) mod L
  = 66455490081190904847072782185220719282059319549388206770560479847
      89407801486

Encoded Aggrgate 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 aggregate 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

5.1.4. Ed448

The key parameters of the first key contribution are:¶

ED448Key1 (ED448)
    UDF:        ZAAA-ITKG-ED44-XKEY-X
    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:        ZAAA-ITKG-ED44-XKEY-2
    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 aggregate private key is:¶

Scalar_A = (Scalar_1 + Scalar_2) mod L
  = 16628213117375882432961168004377507211427270876895354579839960414
      666978326982600598665720267457234882718565087272340290578290296
      3178673

Encoded Aggrgate 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 aggregate 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

5.2. Threshold Decryption

5.2.1. Key Splitting X25519

The encryption key pair is¶

X25519KeyA (X25519)
    UDF:        ZAAA-CTHD-X255-XXKE-YA
    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 n-1 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:        ZAAA-CTHD-X255-XXKE-YX
    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

5.2.2. Decryption X25519

The means of encryption is unchanged. We begin by generating an ephemeral key pair:¶

X25519KeyE (X25519)
    UDF:        ZAAA-CTHD-X255-XXKE-YE
    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 u-coordinate of the result.¶

U: 247351751388894803426442650867524924086144759194834658830326105266
    22202018180

The u-coordinate 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.¶

5.2.3. Key Splitting X448

The encryption key pair is¶

X448KeyA (X448)
    UDF:        ZAAA-ETHD-X44X-KEYA
    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 n-1 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:        ZAAA-ETHD-X44X-KEYX
    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

5.2.4. Decryption X448

The means of encryption is unchanged. We begin by generating an ephemeral key pair:¶

X448KeyE (X448)
    UDF:        ZAAA-ETHD-X44X-KEYE
    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 u-coordinate of the result.¶

U: 414519841929159382730636919036875317796178034011999213235983537052
    30510678702415242441193107876747178183775523375063128358893667955
    0233

The u-coordinate 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.¶