The Asynchronous Remote Key Generation (ARKG) algorithm

1.1. Requirements Language

The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT", "SHOULD", "SHOULD NOT", "RECOMMENDED", "NOT RECOMMENDED", "MAY", and "OPTIONAL" in this document are to be interpreted as described in BCP 14 [RFC2119] [RFC8174] when, and only when, they appear in all capitals, as shown here.¶

1.2. Notation

The following notation is used throughout this document:¶

• The symbol || represents octet string concatenation.¶

• Literal text strings and octet strings are denoted using the CDDL syntax defined in Section 3.1 of [RFC8610].¶

• Elliptic curve operations are written in additive notation: + denotes point addition, i.e., the curve group operation; * denotes point multiplication, i.e., repeated point addition; and + also denotes scalar addition modulo the curve order. * has higher precedence than +, i.e., a + b * C is equivalent to a + (b * C).¶

• LEN(x) is the length, in octets, of the octet string x.¶

• The function I2OSP converts a nonnegative integer into an octet string as defined in Section 4.1 of [RFC8017].¶

2.1. Instance parameters

ARKG is composed of a suite of other algorithms. The parameters of an ARKG instance are:¶

• BL: An asymmetric key blinding scheme [Wilson], consisting of:¶

  • Function BL-Derive-Key-Pair(ikm) -> (pk, sk): Derive a blinding key pair.¶

    Input consists of input keying material entropy ikm.¶

    Output consists of a blinding public key pk and a blinding private key sk.¶

  • Function BL-Blind-Public-Key(pk, tau, ctx) -> pk_tau: Deterministically compute a blinded public key.¶

    Input consists of a blinding public key pk, a blinding factor tau and a domain separation parameter ctx.¶

    Output consists of the blinded public key pk_tau.¶

  • Function BL-Blind-Private-Key(sk, tau, ctx) -> sk_tau: Deterministically compute a blinded private key.¶

    Input consists of a blinding private key sk, a blinding factor tau and a domain separation parameter ctx.¶

    Output consists of the blinded private key sk_tau.¶

  ikm is an opaque octet string of a suitable length as defined by the ARKG instance. tau and ctx are opaque octet strings of arbitrary length. The representations of pk and pk_tau are defined by the protocol that invokes ARKG. The representations of sk and sk_tau are an undefined implementation detail.¶

  See [Wilson] for definitions of security properties required of the key blinding scheme BL.¶

• KEM: A key encapsulation mechanism [Shoup], consisting of the functions:¶

  • KEM-Derive-Key-Pair(ikm) -> (pk, sk): Derive a key encapsulation key pair.¶

    Input consists of input keying material entropy ikm.¶

    Output consists of public key pk and private key sk.¶

  • KEM-Encaps(pk, ikm, ctx) -> (k, c): Derive a key encapsulation.¶

    Input consists of an encapsulation public key pk, input entropy ikm and a domain separation parameter ctx.¶

    Output consists of a shared secret k and an encapsulation ciphertext c.¶

  • KEM-Decaps(sk, c, ctx) -> k: Decapsulate a shared secret.¶

    Input consists of encapsulation private key sk, encapsulation ciphertext c and a domain separation parameter ctx.¶

    Output consists of the shared secret k on success, or an error otherwise.¶

  ikm is an opaque octet string of a suitable length as defined by the ARKG instance. k, c and ctx are opaque octet strings of arbitrary length. The representation of pk is defined by the protocol that invokes ARKG. The representation of sk is an undefined implementation detail.¶

  The KEM MUST guarantee integrity of the ciphertext, meaning that knowledge of the public key pk and the domain separation parameter ctx is required in order to create any ciphertext c that can be successfully decapsulated by the corresponding private key sk. Section 3.2 describes a general formula for how any KEM can be adapted to include this guarantee. Section 9.1 discusses the reasons for this requirement.¶

  See [Wilson] for definitions of additional security properties required of the key encapsulation mechanism KEM.¶

A concrete ARKG instantiation MUST specify the instantiation of each of the above functions.¶

The output keys of the BL scheme are also the output keys of the ARKG instance as a whole. For example, if BL-Blind-Public-Key and BL-Blind-Private-Key output ECDSA keys, then the ARKG instance will also output ECDSA keys.¶

We denote a concrete ARKG instance by the pattern ARKG-NAME, substituting for NAME some description of the chosen instantiation for BL and KEM. Note that this pattern cannot in general be unambiguously parsed; implementations MUST NOT attempt to construct an ARKG instance by parsing such a pattern string. Concrete ARKG instances MUST always be identified by lookup in a registry of fully specified ARKG instances. This is to prevent usage of algorithm combinations that may be incompatible or insecure.¶

2.2. The function ARKG-Derive-Seed

This function is performed by the delegating party. The delegating party derives the ARKG seed pair (pk, sk) and keeps the private seed sk secret, while the public seed pk is provided to the subordinate party. The subordinate party will then be able to derive public keys on behalf of the delegating party.¶

ARKG-Derive-Seed(ikm_bl, ikm_kem) -> (pk, sk)
    ARKG instance parameters:
        BL        A key blinding scheme.
        KEM       A key encapsulation mechanism.

    Inputs:
        ikm_bl    Input keying material entropy for BL.
        ikm_kem   Input keying material entropy for KEM.

    Output:
        (pk, sk)  An ARKG seed pair with public seed pk
                    and private seed sk.

    The output (pk, sk) is calculated as follows:

    (pk_kem, sk_kem) = KEM-Derive-Key-Pair(ikm_bl)
    (pk_bl,  sk_bl)  = BL-Derive-Key-Pair(ikm_kem)
    pk = (pk_bl, pk_kem)
    sk = (sk_bl, sk_kem)

2.3. The function ARKG-Derive-Public-Key

This function is performed by the subordinate party, which holds the ARKG public seed pk = (pk_bl, pk_kem). The resulting public key pk' can be provided to external parties to use in asymmetric cryptography protocols, and the resulting key handle kh can be used by the delegating party to derive the private key corresponding to pk'.¶

This function may be invoked any number of times with the same public seed, using different ikm or ctx arguments, in order to generate any number of public keys.¶

ARKG-Derive-Public-Key((pk_bl, pk_kem), ikm, ctx) -> (pk', kh)
    ARKG instance parameters:
        BL        A key blinding scheme.
        KEM       A key encapsulation mechanism.

    Inputs:
        pk_bl     A key blinding public key.
        pk_kem    A key encapsulation public key.
        ikm       Input entropy for KEM encapsulation.
        ctx       An octet string of length at most 64,
                    containing optional context and
                    application specific information
                    (can be a zero-length string).

    Output:
        pk'       A blinded public key.
        kh        A key handle for deriving the blinded
                    private key sk' corresponding to pk'.

    The output (pk', kh) is calculated as follows:

    if LEN(ctx) > 64:
        Abort with an error.

    ctx'    = I2OSP(LEN(ctx), 1) || ctx
    ctx_bl  = 'ARKG-Derive-Key-BL.'  || ctx'
    ctx_kem = 'ARKG-Derive-Key-KEM.' || ctx'

    (tau, c) = KEM-Encaps(pk_kem, ikm, ctx_kem)
    pk' = BL-Blind-Public-Key(pk_bl, tau, ctx_bl)

    kh = c

If this procedure aborts due to an error, the procedure can safely be retried with the same (pk_bl, pk_kem) and ctx arguments but a new ikm argument.¶

See Section 2.5 for guidance on using ctx arguments longer than 64 bytes.¶

2.4. The function ARKG-Derive-Private-Key

This function is performed by the delegating party, which holds the ARKG private seed (sk_bl, sk_kem). The resulting private key sk' can be used in asymmetric cryptography protocols to prove possession of sk' to an external party that has the corresponding public key.¶

This function may be invoked any number of times with the same private seed, in order to derive the same or different private keys any number of times.¶

ARKG-Derive-Private-Key((sk_bl, sk_kem), kh, ctx) -> sk'
    ARKG instance parameters:
        BL        A key blinding scheme.
        KEM       A key encapsulation mechanism.

    Inputs:
        sk_bl     A key blinding private key.
        sk_kem    A key encapsulation private key.
        kh        A key handle output from ARKG-Derive-Public-Key.
        ctx       An octet string of length at most 64,
                    containing optional context and
                    application specific information
                    (can be a zero-length string).

    Output:
        sk'       A blinded private key.

    The output sk' is calculated as follows:

    if LEN(ctx) > 64:
        Abort with an error.

    ctx'    = I2OSP(LEN(ctx), 1) || ctx
    ctx_bl  = 'ARKG-Derive-Key-BL.'  || ctx'
    ctx_kem = 'ARKG-Derive-Key-KEM.' || ctx'

    tau = KEM-Decaps(sk_kem, kh, ctx_kem)
    If decapsulation failed:
        Abort with an error.

    sk' = BL-Blind-Private-Key(sk_bl, tau, ctx_bl)

Errors in this procedure are typically unrecoverable. For example, KEM-Decaps may fail to decapsulate the KEM ciphertext kh if it fails an integrity check. ARKG instantiations SHOULD be chosen in a way that such errors are impossible if kh was generated by an honest and correct implementation of ARKG-Derive-Public-Key. Incorrect or malicious implementations of ARKG-Derive-Public-Key do not degrade the security of an honest and correct implementation of ARKG-Derive-Private-Key. See also Section 9.1.¶

See Section 2.5 for guidance on using ctx arguments longer than 64 bytes.¶

2.5. Using ctx values longer than 64 bytes

The ctx parameter of ARKG-Derive-Public-Key and ARKG-Derive-Private-Key is limited to a length of at most 64 bytes. This is because this value needs to be communicated from the subordinate party to the delegating party to use the same argument value in both functions, therefore it is necessary in some contexts to limit the size of this parameter in order to limit the size of overall protocol messages.¶

If applications require ctx values longer than 64 bytes, implementors MAY use techniques such as that described in Section 5.3.3 of [RFC9380]. Precise procedure definitions are left as an application-specific implementation detail.¶

3.1. Using elliptic curve addition for key blinding

Instantiations of ARKG whose output keys are elliptic curve keys can use elliptic curve addition as the key blinding scheme BL [Frymann2020] [Wilson]. This section defines a general formula for such instantiations of BL.¶

This formula has the following parameters:¶

• crv: An elliptic curve.¶

• hash-to-crv-suite: A hash-to-curve suite [RFC9380] suitable for hashing to the scalar field of crv.¶

• DST_ext: A domain separation tag.¶

Then the BL parameter of ARKG may be instantiated as follows:¶

• G is the generator of the prime order subgroup of crv.¶

• N is the order of G.¶

• The function hash_to_field is defined in Section 5 of [RFC9380].¶

BL-Derive-Key-Pair(ikm) -> (pk, sk)

    sk = hash_to_field(ikm, 1) with the parameters:
        DST: 'ARKG-BL-EC-KG.' || DST_ext
        F: GF(N), the scalar field
           of the prime order subgroup of crv
        p: N
        m: 1
        L: The L defined in hash-to-crv-suite
        expand_message: The expand_message function
                        defined in hash-to-crv-suite

    pk = sk * G


BL-Blind-Public-Key(pk, tau, ctx) -> pk_tau

    tau' = hash_to_field(tau, 1) with the parameters:
        DST: 'ARKG-BL-EC.' || DST_ext || ctx
        F: GF(N), the scalar field
           of the prime order subgroup of crv
        p: N
        m: 1
        L: The L defined in hash-to-crv-suite
        expand_message: The expand_message function
                        defined in hash-to-crv-suite

    pk_tau = pk + tau' * G


BL-Blind-Private-Key(sk, tau, ctx) -> sk_tau

    tau' = hash_to_field(tau, 1) with the parameters:
        DST: 'ARKG-BL-EC.' || DST_ext || ctx
        F: GF(N), the scalar field
           of the prime order subgroup of crv.
        p: N
        m: 1
        L: The L defined in hash-to-crv-suite
        expand_message: The expand_message function
                        defined in hash-to-crv-suite

    sk_tau_tmp = sk + tau'
    If sk_tau_tmp = 0, abort with an error.
    sk_tau = sk_tau_tmp

3.2. Using HMAC to adapt a KEM without ciphertext integrity

Not all key encapsulation mechanisms guarantee ciphertext integrity, meaning that a valid KEM ciphertext can be created only with knowledge of the KEM public key. This section defines a general formula for adapting any KEM to guarantee ciphertext integrity by prepending a MAC to the KEM ciphertext.¶

For example, ECDH does not guarantee ciphertext integrity - any elliptic curve point is a valid ECDH ciphertext and can be successfully decapsulated using any elliptic curve private scalar.¶

This formula has the following parameters:¶

• Hash: A cryptographic hash function.¶

• DST_ext: A domain separation parameter.¶

• Sub-Kem: A key encapsulation mechanism as described for the KEM parameter in Section 2.1, except Sub-Kem MAY ignore the ctx parameter and MAY not guarantee ciphertext integrity. Sub-Kem defines the functions Sub-Kem-Derive-Key-Pair, Sub-Kem-Encaps and Sub-Kem-Decaps.¶

The KEM parameter of ARKG may be instantiated using Sub-Kem, HMAC [RFC2104] and HKDF [RFC5869] as follows:¶

• L is the output length of Hash in octets.¶

• LEFT(X, n) is the first n bytes of the byte array X.¶

• DROP_LEFT(X, n) is the byte array X without the first n bytes.¶

We truncate the HMAC output to 128 bits (16 octets) because as described in Section 9.1, ARKG needs ciphertext integrity only to ensure correctness, not for security. Extendable-output functions used as the Hash parameter SHOULD still be instantiated with an output length appropriate for the desired security level, in order to not leak information about the Sub-KEM shared secret key.¶

KEM-Derive-Key-Pair(ikm) -> (pk, sk)

    (pk, sk) = Sub-Kem-Derive-Key-Pair(ikm)


KEM-Encaps(pk, ikm, ctx) -> (k, c)

    ctx_sub = 'ARKG-KEM-HMAC.' || DST_ext || ctx
    (k', c') = Sub-Kem-Encaps(pk, ikm, ctx_sub)

    prk = HKDF-Extract with the arguments:
        Hash: Hash
        salt: not set
        IKM: k'

    mk = HKDF-Expand with the arguments:
        Hash: Hash
        PRK: prk
        info: 'ARKG-KEM-HMAC-mac.' || DST_ext || ctx
        L: L
    t = HMAC-Hash-128(K=mk, text=c')

    k = HKDF-Expand with the arguments:
        Hash: Hash
        PRK: prk
        info: 'ARKG-KEM-HMAC-shared.' || DST_ext || ctx
        L: The length of k' in octets.
    c = t || c'


KEM-Decaps(sk, c, ctx) -> k

    t = LEFT(c, 16)
    c' = DROP_LEFT(c, 16)
    ctx_sub = 'ARKG-KEM-HMAC.' || DST_ext || ctx
    k' = Sub-Kem-Decaps(sk, c', ctx_sub)

    prk = HKDF-Extract with the arguments:
        Hash: Hash
        salt: not set
        IKM: k'

    mk = HKDF-Expand with the arguments:
        Hash: Hash
        PRK: prk
        info: 'ARKG-KEM-HMAC-mac.' || DST_ext || ctx
        L: L

    t' = HMAC-Hash-128(K=mk, text=c')
    If t = t':
        k = HKDF-Expand with the arguments:
            Hash: Hash
            PRK: prk
            info: 'ARKG-KEM-HMAC-shared.' || DST_ext || ctx
            L: The length of k' in octets.
    Else:
        Abort with an error.

3.3. Using ECDH as the KEM

Instantiations of ARKG can use ECDH [RFC6090] as the key encapsulation mechanism KEM [Frymann2020] [Wilson]. This section defines a general formula for such instantiations of KEM.¶

This formula has the following parameters:¶

• crv: an elliptic curve valid for use with ECDH [RFC6090].¶

• Hash: A cryptographic hash function.¶

• hash-to-crv-suite: A hash-to-curve suite [RFC9380] suitable for hashing to the scalar field of crv.¶

• DST_ext: A domain separation parameter.¶

The KEM parameter of ARKG may be instantiated as described in section Section 3.2 with the parameters:¶

• Hash: Hash.¶

• DST_ext: 'ARKG-ECDH.' || DST_ext.¶

• Sub-Kem: The functions Sub-Kem-Derive-Key-Pair, Sub-Kem-Encaps and Sub-Kem-Decaps defined as follows:¶

  • Elliptic-Curve-Point-to-Octet-String and Octet-String-to-Elliptic-Curve-Point are the conversion routines defined in sections 2.3.3 and 2.3.4 of [SEC1], without point compression.¶

  • ECDH(pk, sk) represents the compact output of ECDH [RFC6090] using public key (curve point) pk and private key (exponent) sk.¶

  • G is the generator of the prime order subgroup of crv.¶

  • N is the order of G.¶

  Sub-Kem-Derive-Key-Pair(ikm) -> (pk, sk)
  
      sk = hash_to_field(ikm, 1) with the parameters:
          DST: 'ARKG-KEM-ECDH-KG.' || DST_ext
          F: GF(N), the scalar field
            of the prime order subgroup of crv
          p: N
          m: 1
          L: The L defined in hash-to-crv-suite
          expand_message: The expand_message function
                          defined in hash-to-crv-suite
  
      pk = sk * G
  
  
  Sub-Kem-Encaps(pk, ikm, ctx) -> (k, c)
  
      (pk', sk') = Sub-Kem-Derive-Key-Pair(ikm)
  
      k = ECDH(pk, sk')
      c = Elliptic-Curve-Point-to-Octet-String(pk')
  
  
  Sub-Kem-Decaps(sk, c, ctx) -> k
  
      pk' = Octet-String-to-Elliptic-Curve-Point(c)
      k = ECDH(pk', sk)
  

  ¶

3.4. Using X25519 or X448 as the KEM

Instantiations of ARKG can use X25519 or X448 [RFC7748] as the key encapsulation mechanism KEM. This section defines a general formula for such instantiations of KEM.¶

This formula has the following parameters:¶

• DH-Function: the function X25519 or the function X448 [RFC7748].¶

• DST_ext: A domain separation parameter.¶

The KEM parameter of ARKG may be instantiated as described in section Section 3.2 with the parameters:¶

• Hash: SHA-512 [FIPS 180-4] if DH-Function is X25519, or SHAKE256 [FIPS 202] with output length 64 octets if DH-Function is X448.¶

• DST_ext: 'ARKG-ECDHX.' || DST_ext.¶

• Sub-Kem: The functions Sub-Kem-Derive-Key-Pair, Sub-Kem-Encaps and Sub-Kem-Decaps defined as follows:¶

  • G is the octet string h'0900000000000000 0000000000000000 0000000000000000 0000000000000000' if DH-Function is X25519, or the octet string h'0500000000000000 0000000000000000 0000000000000000 0000000000000000 0000000000000000 0000000000000000 0000000000000000' if DH-Function is X448.¶

    These are the little-endian encodings of the integers 9 and 5, which is the u-coordinate of the generator point of the respective curve group.¶

  Sub-Kem-Derive-Key-Pair(ikm) -> (pk, sk)
  
      sk = ikm
      pk = DH-Function(sk, G)
  
  
  Sub-Kem-Encaps(pk, ikm, ctx) -> (k, c)
  
      (pk', sk') = Sub-Kem-Derive-Key-Pair(ikm)
  
      k = DH-Function(sk', pk)
      c = pk'
  
  
  Sub-Kem-Decaps(sk, c, ctx) -> k
  
      k = DH-Function(sk, c)
  

  ¶

3.5. Using the same key for both key blinding and KEM

When an ARKG instance uses the same type of key for both the key blinding and the KEM - for example, if elliptic curve arithmetic is used for key blinding as described in Section 3.1 and ECDH is used as the KEM as described in Section 3.3 [Frymann2020] - then the two keys MAY be the same key. Representations of such an ARKG seed MAY allow for omitting the second copy of the constituent key, but such representations MUST clearly identify that the single constituent key is to be used both as the key blinding key and the KEM key.¶

4.1. ARKG-P256

The identifier ARKG-P256 represents the following ARKG instance:¶

• BL: Elliptic curve addition as described in Section 3.1 with the parameters:¶

  • crv: The NIST curve secp256r1 [SEC2].¶

  • hash-to-crv-suite: P256_XMD:SHA-256_SSWU_RO_ [RFC9380].¶

  • DST_ext: 'ARKG-P256'.¶

• KEM: ECDH as described in Section 3.3 with the parameters:¶

  • crv: The NIST curve secp256r1 [SEC2].¶

  • Hash: SHA-256 [FIPS 180-4].¶

  • hash-to-crv-suite: P256_XMD:SHA-256_SSWU_RO_ [RFC9380].¶

  • DST_ext: 'ARKG-P256'.¶

Each ikm_bl, ikm_kem and ikm input to the procedures in this ARKG instance SHOULD contain at least 256 bits of entropy.¶

4.2. ARKG-P384

The identifier ARKG-P384 represents the following ARKG instance:¶

• BL: Elliptic curve addition as described in Section 3.1 with the parameters:¶

  • crv: The NIST curve secp384r1 [SEC2].¶

  • hash-to-crv-suite: P384_XMD:SHA-384_SSWU_RO_ [RFC9380].¶

  • DST_ext: 'ARKG-P384'.¶

• KEM: ECDH as described in Section 3.3 with the parameters:¶

  • crv: The NIST curve secp384r1 [SEC2].¶

  • Hash: SHA-384 [FIPS 180-4].¶

  • hash-to-crv-suite: P384_XMD:SHA-384_SSWU_RO_ [RFC9380].¶

  • DST_ext: 'ARKG-P384'.¶

Each ikm_bl, ikm_kem and ikm input to the procedures in this ARKG instance SHOULD contain at least 384 bits of entropy.¶

4.3. ARKG-P521

The identifier ARKG-P521 represents the following ARKG instance:¶

• BL: Elliptic curve addition as described in Section 3.1 with the parameters:¶

  • crv: The NIST curve secp521r1 [SEC2].¶

  • hash-to-crv-suite: P521_XMD:SHA-512_SSWU_RO_ [RFC9380].¶

  • DST_ext: 'ARKG-P521'.¶

• KEM: ECDH as described in Section 3.3 with the parameters:¶

  • crv: The NIST curve secp521r1 [SEC2].¶

  • Hash: SHA-512 [FIPS 180-4].¶

  • hash-to-crv-suite: P521_XMD:SHA-512_SSWU_RO_ [RFC9380].¶

  • DST_ext: 'ARKG-P521'.¶

Each ikm_bl, ikm_kem and ikm input to the procedures in this ARKG instance SHOULD contain at least 512 bits of entropy.¶

4.4. ARKG-P256k

The identifier ARKG-P256k represents the following ARKG instance:¶

• BL: Elliptic curve addition as described in Section 3.1 with the parameters:¶

  • crv: The SECG curve secp256k1 [SEC2].¶

  • hash-to-crv-suite: secp256k1_XMD:SHA-256_SSWU_RO_ [RFC9380].¶

  • DST_ext: 'ARKG-P256k'.¶

• KEM: ECDH as described in Section 3.3 with the parameters:¶

  • crv: The SECG curve secp256k1 [SEC2].¶

  • Hash: SHA-256 [FIPS 180-4].¶

  • hash-to-crv-suite: secp256k1_XMD:SHA-256_SSWU_RO_ [RFC9380].¶

  • DST_ext: 'ARKG-P256k'.¶

Each ikm_bl, ikm_kem and ikm input to the procedures in this ARKG instance SHOULD contain at least 256 bits of entropy.¶

5.1. COSE key type: ARKG public seed

An ARKG public seed is represented as a COSE_Key structure [RFC9052] with kty value TBD (placeholder value -65537). Table 1 defines key type parameters pkbl (-1) and pkkem (-2) for the BL and KEM public key, respectively, as well as key type parameter dkalg (-3), representing the algorithm that derived public and private keys are to be used with.¶

When dkalg (-3) is present in an ARKG public seed, the alg (3) parameter of public keys derived using ARKG-Derive-Public-Key with that seed SHOULD be set to the dkalg (-3) value of the seed.¶

The alg (3) parameter, when present, identifies the ARKG instance this public seed is to be used with. Table 2 defines an initial set of COSE algorithm identifiers for this purpose.¶

The following CDDL [RFC8610] example represents an ARKG-P256 public seed restricted to generating derived keys for use with the ESP256 [I-D.jose-fully-spec-algs] signature algorithm:¶

{
  1: -65537,   ; kty: ARKG-pub (placeholder value)
               ; kid: Opaque identifier
  2: h'60b6dfddd31659598ae5de49acb220d8
       704949e84d484b68344340e2565337d2',
  3: -65700,   ; alg: ARKG-P256 (placeholder value)

  -1: {        ; BL public key
    1: 2,      ; kty: EC2
    -1: 1,     ; crv: P256
    -2: h'69380FC1C3B09652134FEEFBA61776F9
          7AF875CE46CA20252C4165102966EBC5',
    -3: h'8B515831462CCB0BD55CBA04BFD50DA6
          3FAF18BD845433622DAF97C06A10D0F1',
  },

  -2: {        ; KEM public key
    1: 2,      ; kty: EC2
    -1: 1,     ; crv: P256
    -2: h'5C099BEC31FAA581D14E208250D3FFDA
          9EC7F543043008BC84967A8D875B5D78',
    -3: h'539D57429FCB1C138DA29010A155DCA1
          4566A8F55AC2F1780810C49D4ED72D58',
  },

  -3: -9       ; Derived key algorithm: ESP256
}

The following is the same example encoded as CBOR:¶

h'a6013a0001000002582060b6dfddd31659598ae5de49acb220d8704949e84d48
  4b68344340e2565337d2033a000100a320a40102200121582069380fc1c3b096
  52134feefba61776f97af875ce46ca20252c4165102966ebc52258208b515831
  462ccb0bd55cba04bfd50da63faf18bd845433622daf97c06a10d0f121a40102
  20012158205c099bec31faa581d14e208250d3ffda9ec7f543043008bc84967a
  8d875b5d78225820539d57429fcb1c138da29010a155dca14566a8f55ac2f178
  0810c49d4ed72d582228'

5.2. COSE key reference type: ARKG derived private key

A reference to a private key derived using ARKG may be represented as a COSE_Key_Ref structure [I-D.lundberg-cose-2p-algs] whose kty is TBD (Ref-ARKG-derived, placeholder -65538). This key reference type defines key type parameters -1 and -2 respectively for the kh and ctx parameters of ARKG-Derive-Private-Key. The kid (2) parameter identifies the ARKG private seed sk. Thus the COSE_Key_Ref structure conveys all arguments to use in ARKG-Derive-Private-Key to derive the referenced private key.¶

Table 3 defines key type parameters for the Ref-ARKG-derived key type. A COSE_Key_Ref structure whose kty is TBD (Ref-ARKG-derived, placeholder -65538) MUST include the parameters kh (-1) and ctx (-2). The inst (-3) parameter MAY be used to indicate the ARKG instance whose ARKG-Derive-Private-Key procedure to use to derive the private key; its value is taken from the IANA "COSE Algorithms" registry [IANA.cose] and an initial set of values is defined in Table 1.¶

If dkalg (-3) is present in the ARKG public seed used in ARKG-Derive-Public-Key to generate the kh value, then the alg (3) parameter of the COSE_Key_Ref SHOULD be set to the dkalg (-3) value of the seed. If alg (3) is present in the seed, then the inst (-3) parameter of the COSE_Key_Ref SHOULD be set to the alg (3) value of the seed.¶

The following CDDL example represents a reference to a key derived using ARKG-P256 and restricted for use with the ESP256 [I-D.jose-fully-spec-algs] signature algorithm:¶

{
  1: -65538,   ; kty: Ref-ARKG-derived
               ; kid: Opaque identifier of ARKG-pub
  2: h'60b6dfddd31659598ae5de49acb220d8
       704949e84d484b68344340e2565337d2',
  3: -9,       ; alg: ESP256

               ; ARKG-P256 key handle
               ; (HMAC-SHA-256-128 followed by
                  SEC1 uncompressed ECDH public key)
  -1: h'ae079e9c52212860678a7cee25b6a6d4
        048219d973768f8e1adb8eb84b220b0ee3
          a2532828b9aa65254fe3717a29499e9b
          aee70cea75b5c8a2ec2eb737834f7467
          e37b3254776f65f4cfc81e2bc4747a84',

               ; ctx argument to ARKG-Derive-Private-Key
  -2: 'Example application info',

  -3: -65700   ; inst: ARKG-P256 (placeholder value)
}

The following is the same example encoded as CBOR:¶

h'a6013a0001000102582060b6dfddd31659598ae5de49acb220d8704949e84d48
  4b68344340e2565337d20328205851ae079e9c52212860678a7cee25b6a6d404
  8219d973768f8e1adb8eb84b220b0ee3a2532828b9aa65254fe3717a29499e9b
  aee70cea75b5c8a2ec2eb737834f7467e37b3254776f65f4cfc81e2bc4747a84
  2158184578616d706c65206170706c69636174696f6e20696e666f223a000100
  a3'

8.1. COSE Key Types Registrations

This section registers the following values in the IANA "COSE Key Types" registry [IANA.cose].¶

• Name: ARKG-pub¶

  • Value: TBD (Placeholder -65537)¶

  • Description: ARKG public seed¶

  • Capabilities: [kty(-65537), pk_bl, pk_kem]¶

  • Reference: Section 5.1 of this document¶

• Name: Ref-ARKG-derived¶

  • Value: TBD (Placeholder -65538)¶

  • Description: Reference to private key derived by ARKG¶

  • Capabilities: [kty(-65538), kh, ctx]¶

  • Reference: [I-D.lundberg-cose-2p-algs], Section 5.2 of this document¶

These registrations add the following choices to the CDDL [RFC8610] type socket $COSE_kty_ref [I-D.lundberg-cose-2p-algs]:¶

$COSE_kty_ref /= -65538   ; Placeholder value

8.2. COSE Key Type Parameters Registrations

This section registers the following values in the IANA "COSE Key Type Parameters" registry [IANA.cose].¶

• Key Type: TBD (ARKG-pub, placeholder -65537)¶

  • Name: pk_bl¶

  • Label: -1¶

  • CBOR Type: COSE_Key¶

  • Description: ARKG key blinding public key¶

  • Reference: Section 5.1 of this document¶

• Key Type: TBD (ARKG-pub, placeholder -65537)¶

  • Name: pk_kem¶

  • Label: -2¶

  • CBOR Type: COSE_Key¶

  • Description: ARKG key encapsulation public key¶

  • Reference: Section 5.1 of this document¶

• Key Type: TBD (Ref-ARKG-derived, placeholder -65538)¶

  • Name: kh¶

  • Label: -1¶

  • CBOR Type: bstr¶

  • Description: kh argument to ARKG-Derive-Private-Key¶

  • Reference: [I-D.lundberg-cose-2p-algs], Section 5.2 of this document¶

• Key Type: TBD (Ref-ARKG-derived, placeholder -65538)¶

  • Name: ctx¶

  • Label: -2¶

  • CBOR Type: bstr¶

  • Description: ctx argument to ARKG-Derive-Private-Key¶

  • Reference: [I-D.lundberg-cose-2p-algs], Section 5.2 of this document¶

8.3. COSE Algorithms Registrations

This section registers the following values in the IANA "COSE Algorithms" registry [IANA.cose].¶

• Name: ARKG-P256¶

  • Value: TBD (placeholder -65700)¶

  • Description: ARKG using ECDH and additive blinding on secp256r1¶

  • Reference: Section 4.1 of this document¶

  • Recommended: TBD¶

• Name: ARKG-P384¶

  • Value: TBD (placeholder -65701)¶

  • Description: ARKG using ECDH and additive blinding on secp384r1¶

  • Reference: Section 4.2 of this document¶

  • Recommended: TBD¶

• Name: ARKG-P521¶

  • Value: TBD (placeholder -65702)¶

  • Description: ARKG using ECDH and additive blinding on secp521r1¶

  • Reference: Section 4.3 of this document¶

  • Recommended: TBD¶

• Name: ARKG-P256k¶

  • Value: TBD (placeholder -65703)¶

  • Description: ARKG using ECDH and additive blinding on secp256k1¶

  • Reference: Section 4.4 of this document¶

  • Recommended: TBD¶

9.1. Using a MAC

The ARKG construction by Wilson [Wilson] omits the MAC and instead encodes application context in the PRF labels, arguing that this leads to invalid keys/signatures in cases that would have a bad MAC. We choose to keep the MAC from the construction by Frymann et al. [Frymann2020], but allow it to be omitted in case the chosen KEM already guarantees ciphertext integrity.¶

The reason for this is to ensure that the delegating party can distinguish key handles that belong to its ARKG seed. For example, this is important for applications using the W3C Web Authentication API [WebAuthn], which do not know beforehand which authenticators are connected and available. Instead, authentication requests may include references to several eligible authenticators, and the one to use is chosen opportunistically by the WebAuthn client depending on which are available at the time. Consider using ARKG in such a scenario to sign some data with a derived private key: a user may have several authenticators and thus several ARKG seeds, so the signing request might include several well-formed ARKG key handles, but only one of them belongs to the ARKG seed of the authenticator that is currently connected. Without an integrity check, choosing the wrong key handle might cause the ARKG-Derive-Private-Key procedure to silently derive the wrong key instead of returning an explicit error, which would in turn lead to an invalid signature or similar final output. This would make it difficult or impossible to diagnose the root cause of the issue and present actionable user feedback. For this reason, we require the KEM to guarantee ciphertext integrity so that ARKG-Derive-Private-Key can fail early if the key handle belongs to a different ARKG seed.¶

It is straightforward to see that adding the MAC to the construction by Wilson does not weaken the security properties defined by Frymann et al. [Frymann2020]: the construction by Frymann et al. can be reduced to the ARKG construction in this document by instantiating BL as described in Section 3.1 and KEM as described in Section 3.3. The use of hash_to_field in Section 3.1 corresponds to the KDF₁ parameter in [Frymann2020], and the use of HMAC and HKDF in Section 3.2 corresponds to the MAC and KDF₂ parameters in [Frymann2020]. Hence if one can break PK-unlinkability or SK-security of the ARKG construction in this document, one can also break the same property of the construction by Frymann et al.¶

9.2. Implementation Status

TODO¶

10.1. Normative References

[I-D.jose-fully-spec-algs]

Jones, M. B. and O. Steele, "Fully-Specified Algorithms for JOSE and COSE", Work in Progress, Internet-Draft, draft-ietf-jose-fully-specified-algorithms-09, 2 April 2025, <https://datatracker.ietf.org/doc/html/draft-ietf-jose-fully-specified-algorithms-09>.

[I-D.lundberg-cose-2p-algs]

Lundberg, E. and M. B. Jones, "COSE Algorithms for Two-Party Signing", Work in Progress, Internet-Draft, draft-lundberg-cose-two-party-signing-algs-01, 3 March 2025, <https://datatracker.ietf.org/doc/html/draft-lundberg-cose-two-party-signing-algs-01>.

[IANA.cose]

IANA, "CBOR Object Signing and Encryption (COSE)", <https://www.iana.org/assignments/cose>.

[RFC2104]

Krawczyk, H., Bellare, M., and R. Canetti, "HMAC: Keyed-Hashing for Message Authentication", RFC 2104, DOI 10.17487/RFC2104, February 1997, <https://www.rfc-editor.org/rfc/rfc2104>.

[RFC2119]

Bradner, S., "Key words for use in RFCs to Indicate Requirement Levels", BCP 14, RFC 2119, DOI 10.17487/RFC2119, March 1997, <https://www.rfc-editor.org/rfc/rfc2119>.

[RFC4949]

Shirey, R., "Internet Security Glossary, Version 2", FYI 36, RFC 4949, DOI 10.17487/RFC4949, August 2007, <https://www.rfc-editor.org/rfc/rfc4949>.

[RFC5869]

Krawczyk, H. and P. Eronen, "HMAC-based Extract-and-Expand Key Derivation Function (HKDF)", RFC 5869, DOI 10.17487/RFC5869, May 2010, <https://www.rfc-editor.org/rfc/rfc5869>.

[RFC6090]

McGrew, D., Igoe, K., and M. Salter, "Fundamental Elliptic Curve Cryptography Algorithms", RFC 6090, DOI 10.17487/RFC6090, February 2011, <https://www.rfc-editor.org/rfc/rfc6090>.

[RFC7748]

Langley, A., Hamburg, M., and S. Turner, "Elliptic Curves for Security", RFC 7748, DOI 10.17487/RFC7748, January 2016, <https://www.rfc-editor.org/rfc/rfc7748>.

[RFC8017]

Moriarty, K., Ed., Kaliski, B., Jonsson, J., and A. Rusch, "PKCS #1: RSA Cryptography Specifications Version 2.2", RFC 8017, DOI 10.17487/RFC8017, November 2016, <https://www.rfc-editor.org/rfc/rfc8017>.

[RFC8032]

Josefsson, S. and I. Liusvaara, "Edwards-Curve Digital Signature Algorithm (EdDSA)", RFC 8032, DOI 10.17487/RFC8032, January 2017, <https://www.rfc-editor.org/rfc/rfc8032>.

[RFC8174]

Leiba, B., "Ambiguity of Uppercase vs Lowercase in RFC 2119 Key Words", BCP 14, RFC 8174, DOI 10.17487/RFC8174, May 2017, <https://www.rfc-editor.org/rfc/rfc8174>.

[RFC8610]

Birkholz, H., Vigano, C., and C. Bormann, "Concise Data Definition Language (CDDL): A Notational Convention to Express Concise Binary Object Representation (CBOR) and JSON Data Structures", RFC 8610, DOI 10.17487/RFC8610, June 2019, <https://www.rfc-editor.org/rfc/rfc8610>.

[RFC8812]

Jones, M., "CBOR Object Signing and Encryption (COSE) and JSON Object Signing and Encryption (JOSE) Registrations for Web Authentication (WebAuthn) Algorithms", RFC 8812, DOI 10.17487/RFC8812, August 2020, <https://www.rfc-editor.org/rfc/rfc8812>.

[RFC9052]

Schaad, J., "CBOR Object Signing and Encryption (COSE): Structures and Process", STD 96, RFC 9052, DOI 10.17487/RFC9052, August 2022, <https://www.rfc-editor.org/rfc/rfc9052>.

[RFC9380]

Faz-Hernandez, A., Scott, S., Sullivan, N., Wahby, R. S., and C. A. Wood, "Hashing to Elliptic Curves", RFC 9380, DOI 10.17487/RFC9380, August 2023, <https://www.rfc-editor.org/rfc/rfc9380>.

[SEC1]

Certicom Research, "SEC 1: Elliptic Curve Cryptography", 2009, <http://www.secg.org/sec1-v2.pdf>.

[SEC2]

Certicom Research, "SEC 2: Recommended Elliptic Curve Domain Parameters", 2010, <http://www.secg.org/sec2-v2.pdf>.

10.2. Informative References

[BIP32]

Wuille, P., "BIP 32 Hierarchical Deterministic Wallets", 2012, <https://github.com/bitcoin/bips/blob/master/bip-0032.mediawiki>.

[Clermont]

Clermont, S. A., "Post Quantum Asynchronous Remote Key Generation. Master's thesis", 2022, <https://www.cryptoplexity.informatik.tu-darmstadt.de/media/crypt/teaching_1/theses_1/Sebastian_Clermont_Thesis.pdf>.

[Frymann2020]

Frymann, N., Gardham, D., Kiefer, F., Lundberg, E., Manulis, M., and D. Nilsson, "Asynchronous Remote Key Generation: An Analysis of Yubico's Proposal for W3C WebAuthn. CCS '20: Proceedings of the 2020 ACM SIGSAC Conference on Computer and Communications Security", 2020, <https://eprint.iacr.org/2020/1004>.

[Frymann2023]

Frymann, N., Gardham, D., and M. Manulis, "Asynchronous Remote Key Generation for Post-Quantum Cryptosystems from Lattices. 2023 IEEE 8th European Symposium on Security and Privacy", 2023, <https://eprint.iacr.org/2023/419>.

[Shoup]

Shoup, V., "A Proposal for an ISO Standard for Public Key Encryption (version 2.0)", 2001, <https://www.shoup.net/papers/iso-2.pdf>.

[WebAuthn-Recovery]

Lundberg, E. and D. Nilsson, "WebAuthn recovery extension: Asynchronous delegated key generation without shared secrets. GitHub", 2019, <https://github.com/Yubico/webauthn-recovery-extension>.

[Wilson]

Wilson, S. M., "Post-Quantum Account Recovery for Passwordless Authentication. Master's thesis", 2023, <http://hdl.handle.net/10012/19316>.

B.1. ARKG-P256

; Inputs:
ctx      = 'ARKG-P256.test vectors'
ikm_bl   = h'000102030405060708090a0b0c0d0e0f101112131415161718191a1b1c1d1e1f'
ikm_kem  = h'202122232425262728292a2b2c2d2e2f303132333435363738393a3b3c3d3e3f'
ikm      = h'404142434445464748494a4b4c4d4e4f505152535455565758595a5b5c5d5e5f'

; Derive-Seed outputs:
pk_bl    = h'046d3bdf31d0db48988f16d47048fdd24123cd286e42d0512daa9f726b4ecf18df
               65ed42169c69675f936ff7de5f9bd93adbc8ea73036b16e8d90adbfabdaddba7'
pk_kem   = h'042eff91b46617d0628b979405bb871a7593e4b02ec533712bc1cf80d0b0a1ccf3
               0ec3b161632183ceedf94fbe35a96e60a17c2c79c6379b141eeeba521ea8030f'
sk_bl    = 0xd959500a78ccf850ce46c80a8c5043c9a2e33844232b3829df37d05b3069f455
sk_kem   = 0x4253051878eac98187f1394605a3ef5ce1981e664cea41e8094c7d12c606d906

; Derive-Public-Key outputs:
pk_prime = h'04018fcbb2f920282a321da180efe321307d03ed476883c02199cc563ccc66a077
               ec03e52a66d4de13c85187323f0a06b9d90c287ea774457b9362c1f66b6a177e'
;kh      = (implementation defined)

; Derive-Private-Key outputs:
sk_prime = 0x52cb5af8edfb25fe5e945f5e83cb7929de9459bda95ef68085b5cb9018c5cacc'

; Inputs:
ctx      = 'ARKG-P256.test vectors'
ikm_bl   = h'000102030405060708090a0b0c0d0e0f101112131415161718191a1b1c1d1e1f'
ikm_kem  = h'202122232425262728292a2b2c2d2e2f303132333435363738393a3b3c3d3e3f'
ikm      = h'00'

; Derive-Seed outputs:
pk_bl    = h'046d3bdf31d0db48988f16d47048fdd24123cd286e42d0512daa9f726b4ecf18df
               65ed42169c69675f936ff7de5f9bd93adbc8ea73036b16e8d90adbfabdaddba7'
pk_kem   = h'042eff91b46617d0628b979405bb871a7593e4b02ec533712bc1cf80d0b0a1ccf3
               0ec3b161632183ceedf94fbe35a96e60a17c2c79c6379b141eeeba521ea8030f'
sk_bl    = 0xd959500a78ccf850ce46c80a8c5043c9a2e33844232b3829df37d05b3069f455
sk_kem   = 0x4253051878eac98187f1394605a3ef5ce1981e664cea41e8094c7d12c606d906

; Derive-Public-Key outputs:
pk_prime = h'04d54c794c2aaf1883a1e52581320ec70caeba03a5b57f8ebf3ae8b80db1e016a3
               79269f8d519890fcc67439b196b0a8e8518b4794feed03b3ded413cd0d3050e3'
;kh      = (implementation defined)

; Derive-Private-Key outputs:
sk_prime = 0x35e5bae9b0c15765ca4fef9e01433cbd1dd3ba4633c3dd025967fbc1b4421d1c'

; Inputs:
ctx      = 'ARKG-P256.test vectors.0'
ikm_bl   = h'000102030405060708090a0b0c0d0e0f101112131415161718191a1b1c1d1e1f'
ikm_kem  = h'202122232425262728292a2b2c2d2e2f303132333435363738393a3b3c3d3e3f'
ikm      = h'404142434445464748494a4b4c4d4e4f505152535455565758595a5b5c5d5e5f'

; Derive-Seed outputs:
pk_bl    = h'046d3bdf31d0db48988f16d47048fdd24123cd286e42d0512daa9f726b4ecf18df
               65ed42169c69675f936ff7de5f9bd93adbc8ea73036b16e8d90adbfabdaddba7'
pk_kem   = h'042eff91b46617d0628b979405bb871a7593e4b02ec533712bc1cf80d0b0a1ccf3
               0ec3b161632183ceedf94fbe35a96e60a17c2c79c6379b141eeeba521ea8030f'
sk_bl    = 0xd959500a78ccf850ce46c80a8c5043c9a2e33844232b3829df37d05b3069f455
sk_kem   = 0x4253051878eac98187f1394605a3ef5ce1981e664cea41e8094c7d12c606d906

; Derive-Public-Key outputs:
pk_prime = h'047dab2c6ed6cd827750f20487c99d5ac113b6539d0d326bc0ad104a94c4ba3ff3
               6f5d3f6e82bdbcf8c404f3c64e2e0a07b1b423f85ee05683f592d63235968c51'
;kh      = (implementation defined)

; Derive-Private-Key outputs:
sk_prime = 0x02d98cb8ca1ddbe689b75c2e31ba8c1e502977d11f6e28f7493fbba00585d2f0'

; Inputs:
ctx      = 'ARKG-P256.test vectors.0'
ikm_bl   = h'000102030405060708090a0b0c0d0e0f101112131415161718191a1b1c1d1e1f'
ikm_kem  = h'202122232425262728292a2b2c2d2e2f303132333435363738393a3b3c3d3e3f'
ikm      = h'00'

; Derive-Seed outputs:
pk_bl    = h'046d3bdf31d0db48988f16d47048fdd24123cd286e42d0512daa9f726b4ecf18df
               65ed42169c69675f936ff7de5f9bd93adbc8ea73036b16e8d90adbfabdaddba7'
pk_kem   = h'042eff91b46617d0628b979405bb871a7593e4b02ec533712bc1cf80d0b0a1ccf3
               0ec3b161632183ceedf94fbe35a96e60a17c2c79c6379b141eeeba521ea8030f'
sk_bl    = 0xd959500a78ccf850ce46c80a8c5043c9a2e33844232b3829df37d05b3069f455
sk_kem   = 0x4253051878eac98187f1394605a3ef5ce1981e664cea41e8094c7d12c606d906

; Derive-Public-Key outputs:
pk_prime = h'04fe2d29db1ffd0e1ef0819500bdbb812b6d8952c0a57a3ba0548910cafeddfce3
               da32d503cd074cbed924f5bd78b7bddf0f7593a2256dd6876ecbb0e1a44bbbb6'
;kh      = (implementation defined)

; Derive-Private-Key outputs:
sk_prime = 0x154bb2b991864eef4b254cd148407f19fc32276056f8281b6c6c455cf00a38c1'

; Inputs:
ctx      = 'ARKG-P256.test vectors.1'
ikm_bl   = h'000102030405060708090a0b0c0d0e0f101112131415161718191a1b1c1d1e1f'
ikm_kem  = h'202122232425262728292a2b2c2d2e2f303132333435363738393a3b3c3d3e3f'
ikm      = h'404142434445464748494a4b4c4d4e4f505152535455565758595a5b5c5d5e5f'

; Derive-Seed outputs:
pk_bl    = h'046d3bdf31d0db48988f16d47048fdd24123cd286e42d0512daa9f726b4ecf18df
               65ed42169c69675f936ff7de5f9bd93adbc8ea73036b16e8d90adbfabdaddba7'
pk_kem   = h'042eff91b46617d0628b979405bb871a7593e4b02ec533712bc1cf80d0b0a1ccf3
               0ec3b161632183ceedf94fbe35a96e60a17c2c79c6379b141eeeba521ea8030f'
sk_bl    = 0xd959500a78ccf850ce46c80a8c5043c9a2e33844232b3829df37d05b3069f455
sk_kem   = 0x4253051878eac98187f1394605a3ef5ce1981e664cea41e8094c7d12c606d906

; Derive-Public-Key outputs:
pk_prime = h'0421df5ebc51bc67135990608349b66e799f5d7a406a404142c13910a7d488e0ca
               58bc6bcab558299b7bda9e8b1718e781dc66ca0c9b28f5da2e7a00cf2ada9765'
;kh      = (implementation defined)

; Derive-Private-Key outputs:
sk_prime = 0xb3437c08215fb083ff360b1300743fddf7aed2493dfabba718aefa60984ed09b'

; Inputs:
ctx      = 'ARKG-P256.test vectors.1'
ikm_bl   = h'000102030405060708090a0b0c0d0e0f101112131415161718191a1b1c1d1e1f'
ikm_kem  = h'202122232425262728292a2b2c2d2e2f303132333435363738393a3b3c3d3e3f'
ikm      = h'00'

; Derive-Seed outputs:
pk_bl    = h'046d3bdf31d0db48988f16d47048fdd24123cd286e42d0512daa9f726b4ecf18df
               65ed42169c69675f936ff7de5f9bd93adbc8ea73036b16e8d90adbfabdaddba7'
pk_kem   = h'042eff91b46617d0628b979405bb871a7593e4b02ec533712bc1cf80d0b0a1ccf3
               0ec3b161632183ceedf94fbe35a96e60a17c2c79c6379b141eeeba521ea8030f'
sk_bl    = 0xd959500a78ccf850ce46c80a8c5043c9a2e33844232b3829df37d05b3069f455
sk_kem   = 0x4253051878eac98187f1394605a3ef5ce1981e664cea41e8094c7d12c606d906

; Derive-Public-Key outputs:
pk_prime = h'0487c9f65395d4ddba545d6184fbb739440e1e7af008ada987eac44d58e4f9f073
               22f7099029ca1523e9ea7c07e5b4fff6ac6f7006d156bca8a6496796043a68dd'
;kh      = (implementation defined)

; Derive-Private-Key outputs:
sk_prime = 0xc9006d3cac426ecf57ced462f11fba41d2db516a5d3111e058bc0c3c2e469ed2'

B.2. Other instances

TODO¶