The Asynchronous Remote Key Generation (ARKG) algorithm

Introduction Asymmetric cryptography, also called public key cryptography, is a fundamental component of much of modern information security. However, even the flexibility of asymmetric cryptosystems is not always enough for all applications. For the sake of privacy and forward secrecy it may be necessary to frequently generate new keys, but it is not always feasible for the holder of the private keys to be available whenever a new key pair is needed. For example, this is often the case when using a hardware security device to hold private keys, where the device may be detached or locked at the time a new key pair is needed. The Asynchronous Remote Key Generation (ARKG) algorithm enables the holder of private keys to delegate generation of public keys without giving access to the corresponding private keys. This enables a public key consumer to autonomously generate public keys whenever one is needed, while the private key holder can later derive the corresponding private key using a "key handle" generated along with the public key. The algorithm consists of three procedures: (1) the delegating party generates a seed pair and emits the public seed to a subordinate party, (2) the subordinate party uses the public seed to generate a public key and a key handle on behalf of the delegating party, and (3) the delegating party uses the key handle and the private seed to derive the private key corresponding to the public key generated by procedure (2). Procedure (1) is performed once, and procedures (2) and (3) may be repeated any number of times with the same seed pair. The required cryptographic primitives are a public key blinding scheme, a key encapsulation mechanism (KEM), a key derivation function (KDF) and a message authentication code (MAC) scheme. Both conventional primitives and quantum-resistant alternatives exist that meet these requirements. Some motivating use cases of ARKG include:

Efficient single-use signing keys. The European Union has proposed a digital identity system which, in order to protect users' privacy, needs a unique key pair for each authentication signature. In online usage the system could relatively easily create a key on demand, submit it to a certification authority to have a single-use certificate issued for that key, and then submit that certificate with an authentication signature to a third party to access a service. However, the proposed system also includes offline use cases: A user might for example need to use the system in a location with poor or no internet connectivity to present a digital driver's license or authorize a payment. For this, the system may need to pre-emptively generate a large amount of single-use certificates to be used offline. One candidate implementation under evaluation to provide signing and key management for this system is the W3C Web Authentication API [WebAuthn] (WebAuthn), which requires a user gesture whenever a WebAuthn operation is invoked. A WebAuthn-based implementation of the proposed digital identity system could use ARKG to pre-emptively generate key pairs for offline use without the need to prompt for a user gesture for each key pair generated.
Enhanced forward secrecy for encrypted messaging. For example, section 8.5.4 of RFC 9052 defines COSE representations for encrypted messages and notes that "Since COSE is designed for a store-and-forward environment rather than an online environment, [...] forward secrecy (see ) is not achievable. A static key will always be used for the receiver of the COSE object." Applications could work around this limitation by exchanging a large number of keys in advance, but that number limits how many messages can be sent before another such exchange is needed. This also requires the sender to allocate storage space for the keys, which may be challenging to support in constrained hardware. ARKG could enable the sender to generate ephemeral recipient public keys on demand. This may enhance forward secrecy if the sender keeps the ARKG public seed secret, since each recipient key pair is used to encrypt only one message.
Generating additional public keys as backup keys. For example, the W3C Web Authentication API [WebAuthn] (WebAuthn) generates a new key pair for each account on each web site. This makes it difficult for users to set up a backup authenticator, because each time a key pair is created for the primary authenticator, another key pair also needs to be created for the backup authenticator, which may be stored in a safe but inconvenient location. ARKG could enable the primary authenticator to also generate a public key for a paired backup authenticator whenever it generates a key pair for itself, allowing the user to set up the pairing once and then leave the backup authenticator in safe storage until the primary authenticator is lost.

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 when, and only when, they appear in all capitals, as shown here.

The Asynchronous Remote Key Generation (ARKG) algorithm The ARKG algorithm consists of three functions, each performed by one of two participants: the delegating party or the subordinate party. The delegating party generates an ARKG seed pair and emits the public seed to the subordinate party while keeping the private seed secret. The subordinate party can then use the public seed to generate derived public keys and key handles, and the delegating party can use the private seed and a key handle to derive the corresponding private key. The following subsections define some notation and the abstract instance parameters used to construct the three ARKG functions, followed by the definitions of the three ARKG functions.

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; 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.

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

BL: An asymmetric key blinding scheme , 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 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 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 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 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 [Design Rationale: Using a MAC].
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 [Design Rationale: Using a MAC].

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.

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 The key blinding scheme chosen for the ARKG instantiation. KEM The key encapsulation mechanism chosen for the ARKG instantiation. 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) ```

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 The key blinding scheme chosen for the ARKG instantiation. KEM The key encapsulation mechanism chosen for the ARKG instantiation. MAC The MAC scheme chosen for the ARKG instantiation. KDF The key derivation function chosen for the ARKG instantiation. 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. 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.

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 The key blinding scheme chosen for the ARKG instantiation. KEM The key encapsulation mechanism chosen for the ARKG instantiation. MAC The MAC scheme chosen for the ARKG instantiation. KDF The key derivation function chosen for the ARKG instantiation. 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. 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 [Design Rationale: Using a MAC].

Generic ARKG instantiations This section defines generic formulae for instantiating the individual ARKG parameters, which can be used to define concrete ARKG instantiations. TODO: IANA registry? COSE/JOSE?

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 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.
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.
N is the order of crv.
G is the generator of crv.

``` BL-Generate-Keypair() -> (pk, sk) BL-Blind-Public-Key(pk, tau) -> pk_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 = 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? ``` ]]>

Using ECDH as the KEM Instantiations of ARKG can use ECDH 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.
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.
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.
ECDH(pk, sk) represents the compact output of ECDH 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) KEM-Encaps(pk) -> (k, c) (pk', sk') = KEM-Generate-Keypair() KEM-Decaps(sk, c) -> k

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

Using HMAC as the MAC Let Hash be a cryptographic hash function. Then the MAC parameter of ARKG may be instantiated using HMAC as follows: ``` MAC-Tag(k, m) -> t MAC-Verify(k, m, t) -> { 0, 1 }

Using HKDF as the KDF Let Hash be a cryptographic hash function. Then the KDF parameter of ARKG may be instantiated using HKDF as follows: ``` KDF(info, ikm, L) -> okm

Concrete ARKG instantiations This section defines an initial set of concrete ARKG instantiations. TODO: IANA registry? COSE/JOSE?

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

BL: Elliptic curve arithmetic as described in section [Using elliptic curve arithmetic for key blinding] with the parameter:
- crv: The NIST curve secp256r1 [SEC2].
KEM: ECDH as described in section [Using ECDH as the KEM] with the parameter:
- crv: The NIST curve secp256r1 [SEC2].
MAC: HMAC as described in section [Using HMAC as the MAC] with the parameter:
- Hash: SHA-256 [FIPS 180-4].
KDF: HKDF as described in section [Using HKDF as the KDF] with the parameter:
- Hash: SHA-256 [FIPS 180-4].
L_bl: 32
L_mac: 32

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

BL: Elliptic curve arithmetic as described in section [Using elliptic curve arithmetic for key blinding] with the parameter:
- crv: The NIST curve secp384r1 [SEC2].
KEM: ECDH as described in section [Using ECDH as the KEM] with the parameter:
- crv: The NIST curve secp384r1 [SEC2].
MAC: HMAC as described in section [Using HMAC as the MAC] with the parameter:
- Hash: SHA-384 [FIPS 180-4].
KDF: HKDF as described in section [Using HKDF as the KDF] with the parameter:
- Hash: SHA-384 [FIPS 180-4].
L_bl: 48
L_mac: 48

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

BL: Elliptic curve arithmetic as described in section [Using elliptic curve arithmetic for key blinding] with the parameter:
- crv: The NIST curve secp521r1 [SEC2].
KEM: ECDH as described in section [Using ECDH as the KEM] with the parameter:
- crv: The NIST curve secp521r1 [SEC2].
MAC: HMAC as described in section [Using HMAC as the MAC] with the parameter:
- Hash: SHA-512 [FIPS 180-4].
KDF: HKDF as described in section [Using HKDF as the KDF] with the parameter:
- Hash: SHA-512 [FIPS 180-4].
L_bl: 64
L_mac: 64

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

BL: Elliptic curve arithmetic as described in section [Using elliptic curve arithmetic for key blinding] with the parameter:
- crv: The SECG curve secp256k1 [SEC2].
KEM: ECDH as described in section [Using ECDH as the KEM] with the parameter:
- crv: The SECG curve secp256k1 [SEC2].
MAC: HMAC as described in section [Using HMAC as the MAC] with the parameter:
- Hash: SHA-256 [FIPS 180-4].
KDF: HKDF as described in section [Using HKDF as the KDF] with the parameter:
- Hash: SHA-256 [FIPS 180-4].
L_bl: 32
L_mac: 32

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

BL: Elliptic curve arithmetic as described in section [Using elliptic curve arithmetic for key blinding] with the parameter:
- crv: The curve Ed25519 [REF?].
KEM: ECDH as described in section [Using ECDH as the KEM] with the parameter:
- crv: The curve X25519 [REF?].
MAC: HMAC as described in section [Using HMAC as the MAC] with the parameter:
- Hash: SHA-256 [FIPS 180-4].
KDF: HKDF as described in section [Using HKDF as the KDF] with the parameter:
- Hash: SHA-256 [FIPS 180-4].
L_bl: 32
L_mac: 32

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

BL: Elliptic curve arithmetic as described in section [Using elliptic curve arithmetic for key blinding] with the parameter:
- crv: The curve X25519 [REF?].
KEM: ECDH as described in section [Using ECDH as the KEM] with the parameter:
- crv: The curve X25519 [REF?].
MAC: HMAC as described in section [Using HMAC as the MAC] with the parameter:
- Hash: SHA-256 [FIPS 180-4].
KDF: HKDF as described in section [Using HKDF as the KDF] with the parameter:
- Hash: SHA-256 [FIPS 180-4].
L_bl: 32
L_mac: 32

COSE bindings TODO?: Define COSE representations for interoperability: - ARKG public seed (for interoperability between different implementers of ARKG-Generate-Seed and ARKG-Derive-Public-Key) - ARKG key handle (for interoperability between different implementers of ARKG-Derive-Public-Key and ARKG-Derive-Secret-Key)