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.¶

• When literal text strings are to be interpreted as octet strings, they are encoded using UTF-8.¶

• Elliptic curve operations are written in multiplicative notation: * denotes point multiplication, i.e., the curve group operation; ^ denotes point exponentiation, i.e., repeated point multiplication of the base with itself; and + denotes scalar addition modulo the curve order.¶

• Random(min_inc, max_exc) represents a cryptographically secure random integer greater than or equal to min_inc and strictly less than max_exc.¶

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-Generate-Keypair() -> (pk, sk): Generate a blinding key pair.¶

    No input.¶

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

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

    Input consists of a blinding public key pk and a blinding factor tau.¶

    Output consists of the blinded public key pk_tau.¶

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

    Input consists of a blinding secret key sk and a blinding factor tau.¶

    Output consists of the blinded secret key sk_tau.¶

  • Integer L_bl: The length of the blinding factor tau in octets.¶

  pk and pk_tau are opaque octet strings of arbitrary length. tau is an opaque octet string of length L_bl. The representations of sk, sk_tau and L_bl are an undefined implementation detail.¶

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

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

  • KEM-Generate-Keypair() -> (pk, sk): Generate a key encapsulation key pair.¶

    No input.¶

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

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

    Input consists of an encapsulation public key pk.¶

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

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

    Input consists of encapsulation secret key sk and encapsulation ciphertext c.¶

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

  pk, k and c are opaque octet strings. The representation of sk is an undefined implementation detail.¶

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

• MAC: A message authentication code (MAC) scheme, consisting of:¶

  • Function MAC-Tag(k, m) -> t: Generate a message authentication tag for a given message using a given key.¶

    Input consists of the shared MAC key k and the message m.¶

    Output consists of the MAC tag t.¶

  • Function MAC-Verify(k, m, t) -> { 0, 1 }: Verify a message authentication tag.¶

    Input consists of the shared MAC key k, the message m and the MAC tag t.¶

    Output is 1 if and only if MAC-Tag(k, m) = t.¶

  • Integer L_mac: The length of the MAC key k in octets.¶

  k is an opaque octet string of length L_mac. m and t are opaque octet strings of arbitrary length. The representation of L_mac is an undefined implementation detail.¶

  See [Frymann2020] for definitions of security properties required of the message authentication code scheme MAC.¶

• KDF: A variable-length key derivation function with the signature: KDF(info, ikm, L) -> okm¶

  Input consists of a domain separation parameter info, input key material ikm and output length L.¶

  Output consists of output key material okm of length L in octets.¶

  info and ikm are opaque octet strings of arbitrary length. okm is an opaque octet string of length L. L is an integer with undefined representation.¶

  See [Frymann2020] for definitions of security properties required of the key derivation function KDF.¶

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

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-Secret-Key output ECDSA keys, then the ARKG instance will also output ECDSA keys.¶

Instantiations MUST satisfy the following compatibility criteria:¶

• The output shared secret k of KEM-Encaps and KEM-Decaps is a valid input key material ikm of KDF.¶

• Output key material okm of length L_bl of KDF is a valid input blinding factor tau of BL-Blind-Public-Key and BL-Blind-Secret-Key.¶

  It is permissible for some KDF outputs to not be valid blinding factors, as long as this happens with negligible probability - see Section 9.1.¶

• Output key material okm of length L_mac of KDF is a valid input MAC key k of MAC-Tag(k, m) and MAC-Verify(k, m, t).¶

  It is permissible for some KDF outputs to not be valid MAC keys, as long as this happens with negligible probability - see Section 9.1.¶

We denote a concrete ARKG instance by the pattern ARKG-BL-KEM-MAC-KDF, substituting the chosen instantiation for the BL, KEM, MAC and KDF parts. 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-Generate-Seed

This function is performed by the delegating party. The delegating party generates 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 generate public keys on behalf of the delegating party.¶

ARKG-Generate-Seed() -> (pk, sk)
    Options:
        BL        A key blinding scheme.
        KEM       A key encapsulation mechanism.

    Inputs: None

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

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

    (pk_kem, sk_kem) = KEM-Generate-Keypair()
    (pk_bl, sk_bl) = BL-Generate-Keypair()
    pk = (pk_kem, pk_bl)
    sk = (sk_kem, sk_bl)

2.3. The function ARKG-Derive-Public-Key

This function is performed by the subordinate party, which holds the ARKG public seed pk = (pk_kem, pk_bl). 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, in order to generate any number of public keys.¶

ARKG-Derive-Public-Key((pk_kem, pk_bl), info) -> (pk', kh)
    Options:
        BL        A key blinding scheme.
        KEM       A key encapsulation mechanism.
        MAC       A MAC scheme.
        KDF       A key derivation function.
        L_bl      The length in octets of the blinding factor tau
                    of the key blinding scheme BL.
        L_mac     The length in octets of the MAC key
                    of the MAC scheme MAC.

    Inputs:
        pk_kem    A key encapsulation public key.
        pk_bl     A key blinding public key.
        info      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
                    secret key sk' corresponding to pk'.

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

    (k, c) = KEM-Encaps(pk_kem)
    tau = KDF("arkg-blind" || 0x00 || info, k, L_bl)
    mk  = KDF("arkg-mac"   || 0x00 || info, k, L_mac)
    tag = MAC-Tag(mk, c || info)

    pk' = BL-Blind-Public-Key(pk_bl, tau)
    kh = (c, tag)

If this procedure aborts due to an error, for example because KDF returns an invalid tau or mk, the procedure can safely be retried with the same arguments.¶

2.4. The function ARKG-Derive-Secret-Key

This function is performed by the delegating party, which holds the ARKG private seed (sk_kem, sk_bl). The resulting secret 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 secret keys any number of times.¶

ARKG-Derive-Secret-Key((sk_kem, sk_bl), kh, info) -> sk'
    Options:
        BL        A key blinding scheme.
        KEM       A key encapsulation mechanism.
        MAC       A MAC scheme.
        KDF       A key derivation function.
        L_bl      The length in octets of the blinding factor tau
                    of the key blinding scheme BL.
        L_mac     The length in octets of the MAC key
                    of the MAC scheme MAC.

    Inputs:
        sk_kem    A key encapsulation secret key.
        sk_bl     A key blinding secret key.
        kh        A key handle output from ARKG-Derive-Public-Key.
        info      Optional context and application specific
                    information (can be a zero-length string).

    Output:
        sk'       A blinded secret key.

    The output sk' is calculated as follows:

    (c, tag) = kh
    k = KEM-Decaps(sk_kem, c)
    mk = KDF("arkg-mac" || 0x00 || info, k, L_mac)

    If MAC-Verify(mk, c || info, tag) = 0:
        Abort with an error.

    tau = KDF("arkg-blind" || 0x00 || info, k, L_bl)
    sk' = BL-Blind-Secret-Key(sk_bl, tau)

Errors in this procedure are typically unrecoverable. For example, KDF might return an invalid tau or mk, or the tag may be invalid. 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 a correct and honest implementation of ARKG-Derive-Secret-Key. See also Section 9.1.¶

3.1. Using elliptic curve arithmetic for key blinding

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

Let crv be an elliptic curve. Then the BL parameter of ARKG may be instantiated as follows:¶

• Elliptic curve points are encoded to and from octet strings using the procedures defined in sections 2.3.3 and 2.3.4 of SEC 1 [SEC1].¶

• Elliptic curve scalar values are encoded to and from octet strings using the procedures defined in sections 2.3.7 and 2.3.8 of SEC 1 [SEC1].¶

• N is the order of crv.¶

• G is the generator of crv.¶

BL-Generate-Keypair() -> (pk, sk)

    sk = Random(1, N)
    pk_tmp = G^sk
    If pk_tmp equals the point at infinity, abort with an error.
    pk = pk_tmp

    TODO: Also reject G?


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

    If tau = 0 or tau >= N, abort with an error.
    pk_tau_tmp = pk * (G^tau)
    If pk_tau_tmp equals the point at infinity, abort with an error.
    pk_tau = pk_tau_tmp

    TODO: Also reject G?


BL-Blind-Secret-Key(sk, tau) -> sk_tau

    If tau = 0 or tau >= N, abort with an error.
    sk_tau_tmp = sk + tau
    If sk_tau_tmp = 0, abort with an error.
    sk_tau = sk_tau_tmp

    TODO: Also reject 1?

3.2. Using ECDH as the KEM

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

Let crv be an elliptic curve used for ECDH. Then the KEM parameter of ARKG may be instantiated as follows:¶

• Elliptic curve points are encoded to and from octet strings using the procedures defined in sections 2.3.3 and 2.3.4 of SEC 1 [SEC1].¶

• Elliptic curve coordinate field elements are encoded to and from octet strings using the procedures defined in sections 2.3.5 and 2.3.6 of SEC 1 [SEC1].¶

• Elliptic curve scalar values are encoded to and from octet strings using the procedures defined in sections 2.3.7 and 2.3.8 of SEC 1 [SEC1].¶

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

• N is the order of crv.¶

• G is the generator of crv.¶

KEM-Generate-Keypair() -> (pk, sk)

    sk = Random(1, N)
    pk_tmp = G^sk
    If pk_tmp equals the point at infinity, abort with an error.
    pk = pk_tmp

    TODO: Also reject G?


KEM-Encaps(pk) -> (k, c)
    (pk', sk') = KEM-Generate-Keypair()

    k = ECDH(pk, sk')
    c = pk'


KEM-Decaps(sk, c) -> k

    pk' = c
    k = ECDH(pk', sk)

3.3. Using both elliptic curve arithmetic for key blinding and ECDH as the KEM

If elliptic curve arithmetic is used for key blinding and ECDH is used as the KEM, as described in the previous sections, then both of them MAY use the same curve or MAY use different curves. If both use the same curve, then it is also possible to use the same public key as both the key blinding public key and the KEM public key. [Frymann2020]¶

TODO: Caveats? I think I read in some paper or thesis about specific drawbacks of using the same key for both.¶

3.4. Using HMAC as the MAC

Let Hash be a cryptographic hash function. Then the MAC parameter of ARKG may be instantiated using HMAC [RFC2104] as follows:¶

MAC-Tag(k, m) -> t

    t = HMAC-Hash(K=k, text=m)


MAC-Verify(k, m, t) -> { 0, 1 }

    t' = HMAC-Hash(K=k, text=m)
    If t = t':
        return 1
    Else:
        return 0

3.5. Using HKDF as the KDF

Let Hash be a cryptographic hash function. Then the KDF parameter of ARKG may be instantiated using HKDF [RFC5869] as follows:¶

KDF(info, ikm, L) -> okm

    PRK = HKDF-Extract with the arguments:
        Hash: Hash
        salt: not set
        IKM: ikm

    okm = HKDF-Expand with the arguments:
        Hash: Hash
        PRK: PRK
        info: info
        L: L

4.1. ARKG-P256-ECDH-P256-HMAC-SHA256-HKDF-SHA256

The identifier ARKG-P256-ECDH-P256-HMAC-SHA256-HKDF-SHA256 represents the following ARKG instance:¶

• BL: Elliptic curve arithmetic as described in Section 3.1 with the parameter:¶

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

• KEM: ECDH as described in Section 3.2 with the parameter:¶

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

• MAC: HMAC as described in Section 3.4 with the parameter:¶

  • Hash: SHA-256 [FIPS 180-4].¶

• KDF: HKDF as described in Section 3.5 with the parameter:¶

  • Hash: SHA-256 [FIPS 180-4].¶

• L_bl: 32¶

• L_mac: 32¶

4.2. ARKG-P384-ECDH-P384-HMAC-SHA384-HKDF-SHA384

The identifier ARKG-P384-ECDH-P384-HMAC-SHA384-HKDF-SHA384 represents the following ARKG instance:¶

• BL: Elliptic curve arithmetic as described in Section 3.1 with the parameter:¶

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

• KEM: ECDH as described in Section 3.2 with the parameter:¶

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

• MAC: HMAC as described in Section 3.4 with the parameter:¶

  • Hash: SHA-384 [FIPS 180-4].¶

• KDF: HKDF as described in Section 3.5 with the parameter:¶

  • Hash: SHA-384 [FIPS 180-4].¶

• L_bl: 48¶

• L_mac: 48¶

4.3. ARKG-P521-ECDH-P521-HMAC-SHA512-HKDF-SHA512

The identifier ARKG-P521-ECDH-P521-HMAC-SHA512-HKDF-SHA512 represents the following ARKG instance:¶

• BL: Elliptic curve arithmetic as described in Section 3.1 with the parameter:¶

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

• KEM: ECDH as described in Section 3.2 with the parameter:¶

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

• MAC: HMAC as described in Section 3.4 with the parameter:¶

  • Hash: SHA-512 [FIPS 180-4].¶

• KDF: HKDF as described in Section 3.5 with the parameter:¶

  • Hash: SHA-512 [FIPS 180-4].¶

• L_bl: 64¶

• L_mac: 64¶

4.4. ARKG-P256k-ECDH-P256k-HMAC-SHA256-HKDF-SHA256

The identifier ARKG-P256k-ECDH-P256k-HMAC-SHA256-HKDF-SHA256 represents the following ARKG instance:¶

• BL: Elliptic curve arithmetic as described in Section 3.1 with the parameter:¶

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

• KEM: ECDH as described in Section 3.2 with the parameter:¶

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

• MAC: HMAC as described in Section 3.4 with the parameter:¶

  • Hash: SHA-256 [FIPS 180-4].¶

• KDF: HKDF as described in Section 3.5 with the parameter:¶

  • Hash: SHA-256 [FIPS 180-4].¶

• L_bl: 32¶

• L_mac: 32¶

4.5. ARKG-Ed25519-X25519-HMAC-SHA256-HKDF-SHA256

The identifier ARKG-Ed25519-X25519-HMAC-SHA256-HKDF-SHA256 represents the following ARKG instance:¶

• BL: Elliptic curve arithmetic as described in Section 3.1 with the parameter:¶

  • crv: The curve Ed25519 [REF?].¶

• KEM: ECDH as described in Section 3.2 with the parameter:¶

  • crv: The curve X25519 [REF?].¶

• MAC: HMAC as described in Section 3.4 with the parameter:¶

  • Hash: SHA-256 [FIPS 180-4].¶

• KDF: HKDF as described in Section 3.5 with the parameter:¶

  • Hash: SHA-256 [FIPS 180-4].¶

• L_bl: 32¶

• L_mac: 32¶

4.6. ARKG-X25519-X25519-HMAC-SHA256-HKDF-SHA256

The identifier ARKG-X25519-X25519-HMAC-SHA256-HKDF-SHA256 represents the following ARKG instance:¶

• BL: Elliptic curve arithmetic as described in Section 3.1 with the parameter:¶

  • crv: The curve X25519 [REF?].¶

• KEM: ECDH [RFC6090] as described in Section 3.2 with the parameter:¶

  • crv: The curve X25519 [REF?].¶

• MAC: HMAC as described in Section 3.4 with the parameter:¶

  • Hash: SHA-256 [FIPS 180-4].¶

• KDF: HKDF as described in Section 3.5 with the parameter:¶

  • Hash: SHA-256 [FIPS 180-4].¶

• L_bl: 32¶

• L_mac: 32¶

9.1. Using a MAC

The ARKG construction by Wilson [Wilson] omits the MAC and instead encodes application context in the PRF labels, arguing 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] for two purposes.¶

The first is so that the delegating party can distinguish between key handles addressed to it and those addressed to other delegating parties. We anticipate use cases where a private key usage request may contain key handles for several delegating parties eligible to fulfill the request, and the delegate party to be used can be chosen opportunistically depending on which are available at the time. Without the MAC, choosing the wrong key handle would cause the ARKG-Derive-Secret-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. The MAC also allows ARKG key handles to be transmitted via heterogeneous data channels, possibly including a mix of ARKG key handles and similar values used for other algorithms.¶

The second purpose is so that the delegating party can be assured that no errors should happen during the execution of ARKG-Derive-Secret-Key, such as out-of-range or invalid key values. For example, key generation in ARKG-Derive-Public-Key might be done by randomly testing candidates [NIST.SP.800-56Ar3] and retrying ARKG-Derive-Public-Key until a valid candidate is found. A MAC enables ARKG-Derive-Secret-Key to assume that the first candidate from a given pseudo-random seed will be successful, and otherwise return an explicit error rejecting the key handle as invalid. ARKG-Derive-Public-Key is likely to run on powerful general-purpose hardware, such as a laptop, smartphone or server, while ARKG-Derive-Secret-Key might run on more constrained hardware such as a cryptographic smart card, which benefits greatly from such optimizations.¶

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 KEM as group exponentiation and instantiating BL as group multiplication to blind public keys and modular integer addition to blind secret keys. The MAC and KDF parameters correspond trivially to the MAC and KDF parameters in [Frymann2020], where KDF₁(k) = KDF(k, l₁) and KDF₂(k) = KDF(k, l₂) with fixed labels l₁ and l₂. 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¶