The Asynchronous Remote Key Generation (ARKG) algorithm
draft-bradleylundberg-cfrg-arkg-07
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draft-bradleylundberg-cfrg-arkg-07
Crypto Forum E. Lundberg, Ed.
Internet-Draft J. Bradley
Intended status: Informational Yubico
Expires: 24 October 2025 22 April 2025
The Asynchronous Remote Key Generation (ARKG) algorithm
draft-bradleylundberg-cfrg-arkg-07
Abstract
Asynchronous Remote Key Generation (ARKG) is an abstract algorithm
that enables delegation of asymmetric public key generation without
giving access to the corresponding private keys. This capability
enables a variety of applications: a user agent can generate
pseudonymous public keys to prevent tracking; a message sender can
generate ephemeral recipient public keys to enhance forward secrecy;
two paired authentication devices can each have their own private
keys while each can register public keys on behalf of the other.
This document provides three main contributions: a specification of
the generic ARKG algorithm using abstract primitives; a set of
formulae for instantiating the abstract primitives using concrete
primitives; and an initial set of fully specified concrete ARKG
instances. We expect that additional instances will be defined in
the future.
About This Document
This note is to be removed before publishing as an RFC.
Status information for this document may be found at
https://datatracker.ietf.org/doc/draft-bradleylundberg-cfrg-arkg/.
Source for this draft and an issue tracker can be found at
https://github.com/Yubico/arkg-rfc.
Status of This Memo
This Internet-Draft is submitted in full conformance with the
provisions of BCP 78 and BCP 79.
Internet-Drafts are working documents of the Internet Engineering
Task Force (IETF). Note that other groups may also distribute
working documents as Internet-Drafts. The list of current Internet-
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This Internet-Draft will expire on 24 October 2025.
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Table of Contents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1. Requirements Language . . . . . . . . . . . . . . . . . . 5
1.2. Notation . . . . . . . . . . . . . . . . . . . . . . . . 5
2. The Asynchronous Remote Key Generation (ARKG) algorithm . . . 5
2.1. Instance parameters . . . . . . . . . . . . . . . . . . . 6
2.2. The function ARKG-Derive-Seed . . . . . . . . . . . . . . 8
2.2.1. Nondeterministic variants . . . . . . . . . . . . . . 8
2.3. The function ARKG-Derive-Public-Key . . . . . . . . . . . 9
2.3.1. Nondeterministic variants . . . . . . . . . . . . . . 10
2.4. The function ARKG-Derive-Private-Key . . . . . . . . . . 10
2.5. Using ctx values longer than 64 bytes . . . . . . . . . . 12
3. Generic ARKG instantiations . . . . . . . . . . . . . . . . . 12
3.1. Using elliptic curve addition for key blinding . . . . . 12
3.2. Using HMAC to adapt a KEM without ciphertext integrity . 14
3.3. Using ECDH as the KEM . . . . . . . . . . . . . . . . . . 16
3.4. Using X25519 or X448 as the KEM . . . . . . . . . . . . . 17
3.5. Using the same key for both key blinding and KEM . . . . 18
4. Concrete ARKG instantiations . . . . . . . . . . . . . . . . 18
4.1. ARKG-P256 . . . . . . . . . . . . . . . . . . . . . . . . 19
4.2. ARKG-P384 . . . . . . . . . . . . . . . . . . . . . . . . 19
4.3. ARKG-P521 . . . . . . . . . . . . . . . . . . . . . . . . 20
4.4. ARKG-P256k . . . . . . . . . . . . . . . . . . . . . . . 20
5. COSE bindings . . . . . . . . . . . . . . . . . . . . . . . . 21
5.1. COSE key type: ARKG public seed . . . . . . . . . . . . . 21
5.2. COSE key reference type: ARKG derived private key . . . . 23
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6. Security Considerations . . . . . . . . . . . . . . . . . . . 25
7. Privacy Considerations . . . . . . . . . . . . . . . . . . . 25
8. IANA Considerations . . . . . . . . . . . . . . . . . . . . . 25
8.1. COSE Key Types Registrations . . . . . . . . . . . . . . 25
8.2. COSE Key Type Parameters Registrations . . . . . . . . . 26
8.3. COSE Algorithms Registrations . . . . . . . . . . . . . . 27
9. Design rationale . . . . . . . . . . . . . . . . . . . . . . 28
9.1. Using a MAC . . . . . . . . . . . . . . . . . . . . . . . 28
9.2. Implementation Status . . . . . . . . . . . . . . . . . . 29
10. References . . . . . . . . . . . . . . . . . . . . . . . . . 29
10.1. Normative References . . . . . . . . . . . . . . . . . . 29
10.2. Informative References . . . . . . . . . . . . . . . . . 31
Appendix A. Acknowledgements . . . . . . . . . . . . . . . . . . 32
Appendix B. Test Vectors . . . . . . . . . . . . . . . . . . . . 32
B.1. ARKG-P256 . . . . . . . . . . . . . . . . . . . . . . . . 32
B.2. Other instances . . . . . . . . . . . . . . . . . . . . . 35
Appendix C. Document History . . . . . . . . . . . . . . . . . . 36
Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . 38
Authors' Addresses . . . . . . . . . . . . . . . . . . . . . . . 38
1. Introduction
Asynchronous Remote Key Generation (ARKG) introduces a mechanism to
generate public keys without access to the corresponding private
keys. Such a mechanism is useful for many scenarios when a new
public key is needed but the private key holder is not available to
perform the key generation. This may occur when private keys are
stored in a hardware security device, which may be unavailable or
locked at the time a new public key is needed.
Some motivating use cases of ARKG include:
* *Single-use asymmetric keys*: Envisioned for the European Union's
digital identity framework, which is set to use single-use
asymmetric keys to prevent colluding verifiers from using public
keys as correlation handles. Each digital identity credential
would thus be issued with a single-use proof-of-possession key,
used only once to present the credential to a verifier. ARKG
empowers both online and offline usage scenarios: for offline
scenarios, ARKG enables pre-generation of public keys for single-
use credentials without needing to access the hardware security
device that holds the private keys. For online scenarios, ARKG
gives the credential issuer assurance that all derived private
keys are bound to the same secure hardware element. In both
cases, application performance may be improved since public keys
can be generated in a general-purpose execution environment
instead of a secure enclave.
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* *Enhanced forward secrecy*: The use of ARKG can facilitate forward
secrecy in certain contexts. For instance, section 8.5.4 of RFC
9052 (https://www.rfc-editor.org/rfc/rfc9052.html#name-direct-key-
agreement) notes that "Since COSE is designed for a store-and-
forward environment rather than an online environment, [...]
forward secrecy (see [RFC4949]) is not achievable. A static key
will always be used for the receiver of the COSE object." As
opposed to workarounds like exchanging a large number of keys in
advance, ARKG enables the the sender to generate ephemeral
recipient public keys on demand.
* *Backup key generation*: For example, the W3C Web Authentication
API [WebAuthn] (WebAuthn) generates a new key pair for each
account on each web site. ARKG could allow for simultaneously
generating a backup public key when registering a new public key.
A primary authenticator could generate both a key pair for itself
and a public key for a paired backup authenticator. The backup
authenticator only needs to be paired with the primary
authenticator once, and can then be safely stored until it is
needed.
ARKG consists of three procedures:
* *Initialization*: The _delegating party_ generates a _seed pair_
and discloses the _public seed_ to a _subordinate party_, while
securely retaining the _private seed_.
* *Public key generation*: The subordinate party uses the public
seed to autonomously generate a new public key along with a unique
_key handle_ for the public key. This can be repeated any number
of times.
* *Private key derivation*: The delegating party uses a key handle
and the private seed to derive the private key corresponding to
the public key generated along with the key handle. This can be
repeated with any number of key handles.
Notably, ARKG can be built entirely using established cryptographic
primitives. The required primitives are a public key blinding scheme
and a key encapsulation mechanism (KEM), which may in turn use a key
derivation function (KDF) and a message authentication code (MAC)
scheme. Both conventional primitives and quantum-resistant
alternatives exist that meet these requirements. [Wilson]
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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. 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.
This construction of ARKG is fully deterministic, extracting input
entropy as explicit parameters, as opposed to the internal random
sampling typically used in the academic literature [Frymann2020]
[Wilson] [Clermont]. Implementations MAY choose to instead implement
the ARKG-Derive-Seed and KEM-Encaps functions as nondeterministic
procedures omitting their respective ikm parameters and sampling
random entropy internally; this choice does not affect
interoperability.
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The following subsections define the abstract instance parameters
used to construct the three ARKG functions, followed by the
definitions of the three ARKG functions.
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:
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- 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.
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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)
2.2.1. Nondeterministic variants
Applications that do not need a deterministic interface MAY choose to
instead implement ARKG-Derive-Seed, KEM-Derive-Key-Pair and BL-
Derive-Key-Pair as nondeterministic procedures omitting their
respective ikm parameters and sampling random entropy internally;
this choice does not affect interoperability.
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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.
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See Section 2.5 for guidance on using ctx arguments longer than 64
bytes.
2.3.1. Nondeterministic variants
Applications that do not need a deterministic interface MAY choose to
instead implement ARKG-Derive-Public-Key and KEM-Encaps as
nondeterministic procedures omitting their respective ikm parameter
and sampling random entropy internally; this choice does not affect
interoperability.
BL-Blind-Public-Key must always be deterministic for compatibility
with ARKG-Derive-Private-Key.
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.
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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.
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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. Generic ARKG instantiations
This section defines generic formulae for instantiating the
individual ARKG parameters, which can be used to define concrete ARKG
instantiations.
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].
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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
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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
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(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:
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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 above parameters define the following intermediate value:
* DST_aug: 'ARKG-ECDH.' || DST_ext.
The KEM parameter of ARKG may be instantiated as described in section
Section 3.2 with the parameters:
* Hash: Hash.
* DST_ext: DST_aug.
* 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.
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Sub-Kem-Derive-Key-Pair(ikm) -> (pk, sk)
sk = hash_to_field(ikm, 1) with the parameters:
DST: 'ARKG-KEM-ECDH-KG.' || DST_aug
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.
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* 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. Concrete ARKG instantiations
This section defines an initial set of concrete ARKG instantiations.
TODO: IANA registry? COSE/JOSE?
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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'.
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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].
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- 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. COSE bindings
This section proposes additions to COSE [RFC9052] to support ARKG use
cases. These consist of new key type definitions to represent ARKG
public seeds and references [I-D.lundberg-cose-2p-algs] to private
keys derived using ARKG.
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.
+=======+=======+==========+===========+============================+
| Name | Label | Value | Required? | Description |
| | | type | | |
+=======+=======+==========+===========+============================+
| pkbl | -1 | COSE_Key | Required | BL key of ARKG public |
| | | | | seed |
+-------+-------+----------+-----------+----------------------------+
| pkkem | -2 | COSE_Key | Required | KEM key of ARKG |
| | | | | public seed |
+-------+-------+----------+-----------+----------------------------+
| dkalg | -3 | int / | Optional | alg parameter of |
| | | tstr | | public and private |
| | | | | keys derived from |
| | | | | this ARKG public seed |
+-------+-------+----------+-----------+----------------------------+
Table 1: COSE key type parameters for the ARKG-pub key type.
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.
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+============+==================+==============================+
| Name | Value | Description |
+============+==================+==============================+
| ARKG-P256 | TBD (placeholder | The ARKG instance defined in |
| | -65700) | Section 4.1 of this document |
+------------+------------------+------------------------------+
| ARKG-P384 | TBD (placeholder | The ARKG instance defined in |
| | -65701) | Section 4.2 of this document |
+------------+------------------+------------------------------+
| ARKG-P521 | TBD (placeholder | The ARKG instance defined in |
| | -65702) | Section 4.3 of this document |
+------------+------------------+------------------------------+
| ARKG-P256k | TBD (placeholder | The ARKG instance defined in |
| | -65703) | Section 4.4 of this document |
+------------+------------------+------------------------------+
Table 2: COSE algorithm identifiers to represent ARKG instances.
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:
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{
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.
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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.
+======+=======+============+===========+===========================+
| Name | Label | Value | Required? | Description |
| | | type | | |
+======+=======+============+===========+===========================+
| kh | -1 | bstr | Required | kh argument to ARKG- |
| | | | | Derive-Private-Key |
+------+-------+------------+-----------+---------------------------+
| ctx | -2 | bstr | Required | ctx argument to ARKG- |
| | | | | Derive-Private-Key |
+------+-------+------------+-----------+---------------------------+
| inst | -3 | int / | Optional | COSE algorithm identifier |
| | | tstr | | of ARKG instance |
+------+-------+------------+-----------+---------------------------+
Table 3: COSE key type parameters for the Ref-ARKG-derived type.
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:
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{
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'
6. Security Considerations
TODO
7. Privacy Considerations
TODO
8. IANA Considerations
8.1. COSE Key Types Registrations
This section registers the following values in the IANA "COSE Key
Types" registry [IANA.cose].
* Name: ARKG-pub
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- 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
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- 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)
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- 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. Design rationale
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,
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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 KDF_1
parameter in [Frymann2020], and the use of HMAC and HKDF in
Section 3.2 corresponds to the MAC and KDF_2 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. References
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>.
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[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>.
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[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>.
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[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>.
Appendix A. Acknowledgements
ARKG was first proposed under this name by Frymann et al.
[Frymann2020], who analyzed a proposed extension to W3C Web
Authentication by Lundberg and Nilsson [WebAuthn-Recovery], which was
in turn inspired by a similar construction by Wuille [BIP32] used to
create privacy-preserving Bitcoin addresses. Frymann et al.
[Frymann2020] generalized the constructions by Lundberg, Nilsson and
Wuille from elliptic curves to any discrete logarithm (DL) problem,
and also proved the security of arbitrary asymmetric protocols
composed with ARKG. Further generalizations to include quantum-
resistant instantiations were developed independently by Clermont
[Clermont], Frymann et al. [Frymann2023] and Wilson [Wilson].
This document adopts the construction proposed by Wilson [Wilson],
modified by the inclusion of a MAC in the key handles as done in the
original construction by Frymann et al. [Frymann2020].
The authors would like to thank all of these authors for their
research and development work that led to the creation of this
document.
Appendix B. Test Vectors
This section lists test vectors for validating implementations.
Test vectors are listed in CDDL [RFC8610] syntax using parameter and
output names defined in Section 2. Elliptic curve points are encoded
using the Elliptic-Curve-Point-to-Octet-String procedure defined in
section 2.3.3 of [SEC1], without point compression.
B.1. ARKG-P256
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; 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'04c38bbdd7286196733fa177e43b73cfd3d6d72cd11cc0bb2c9236cf85a42dcff5
dfa339c1e07dfcdfda8d7be2a5a3c7382991f387dfe332b1dd8da6e0622cfb35'
sk_bl = 0xd959500a78ccf850ce46c80a8c5043c9a2e33844232b3829df37d05b3069f455
sk_kem = 0x74e0a4cd81ca2d24246ff75bfd6d4fb7f9dfc938372627feb2c2348f8b1493b5
; Derive-Public-Key outputs:
pk_prime = h'04572a111ce5cfd2a67d56a0f7c684184b16ccd212490dc9c5b579df749647d107
dac2a1b197cc10d2376559ad6df6bc107318d5cfb90def9f4a1f5347e086c2cd'
;kh = (implementation defined)
; Derive-Private-Key outputs:
sk_prime = 0x775d7fe9a6dfba43ce671cb38afca3d272c4d14aff97bd67559eb500a092e5e7
; 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'04c38bbdd7286196733fa177e43b73cfd3d6d72cd11cc0bb2c9236cf85a42dcff5
dfa339c1e07dfcdfda8d7be2a5a3c7382991f387dfe332b1dd8da6e0622cfb35'
sk_bl = 0xd959500a78ccf850ce46c80a8c5043c9a2e33844232b3829df37d05b3069f455
sk_kem = 0x74e0a4cd81ca2d24246ff75bfd6d4fb7f9dfc938372627feb2c2348f8b1493b5
; Derive-Public-Key outputs:
pk_prime = h'040e983f44cafa9036066857d1831b58cc2227677489df07d1ae0801259ddc0a6a
a77f98712ecf662773ef73b6414d752bab57288cdce1299f73e606306bf77c54'
;kh = (implementation defined)
; Derive-Private-Key outputs:
sk_prime = 0x298271791090c1a0f3ef346a974b8daeab2876f2943207b2cddfe4ddff7a6295
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; 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'04c38bbdd7286196733fa177e43b73cfd3d6d72cd11cc0bb2c9236cf85a42dcff5
dfa339c1e07dfcdfda8d7be2a5a3c7382991f387dfe332b1dd8da6e0622cfb35'
sk_bl = 0xd959500a78ccf850ce46c80a8c5043c9a2e33844232b3829df37d05b3069f455
sk_kem = 0x74e0a4cd81ca2d24246ff75bfd6d4fb7f9dfc938372627feb2c2348f8b1493b5
; Derive-Public-Key outputs:
pk_prime = h'04b79b65d6bbb419ff97006a1bd52e3f4ad53042173992423e06e52987a037cb61
dd82b126b162e4e7e8dc5c9fd86e82769d402a1968c7c547ef53ae4f96e10b0e'
;kh = (implementation defined)
; Derive-Private-Key outputs:
sk_prime = 0x2a97f4232f9abba32fbfc28c6686f8afd2d851c2a95a3ed2f0a384b9ad55068d
; 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'04c38bbdd7286196733fa177e43b73cfd3d6d72cd11cc0bb2c9236cf85a42dcff5
dfa339c1e07dfcdfda8d7be2a5a3c7382991f387dfe332b1dd8da6e0622cfb35'
sk_bl = 0xd959500a78ccf850ce46c80a8c5043c9a2e33844232b3829df37d05b3069f455
sk_kem = 0x74e0a4cd81ca2d24246ff75bfd6d4fb7f9dfc938372627feb2c2348f8b1493b5
; Derive-Public-Key outputs:
pk_prime = h'04dfd47f9357efc0146e243c2cab4601c250b792111d6a364587a728d5624cfaf1
6e62dbf37ebc132537038f5daa2ff6cd38f229fd3063c618b4333cea35af6e85'
;kh = (implementation defined)
; Derive-Private-Key outputs:
sk_prime = 0xe5e0fab3367300dc45904128a3f8991a9d9059b585aac29e6f4e7cb45f59fce0
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; 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'04c38bbdd7286196733fa177e43b73cfd3d6d72cd11cc0bb2c9236cf85a42dcff5
dfa339c1e07dfcdfda8d7be2a5a3c7382991f387dfe332b1dd8da6e0622cfb35'
sk_bl = 0xd959500a78ccf850ce46c80a8c5043c9a2e33844232b3829df37d05b3069f455
sk_kem = 0x74e0a4cd81ca2d24246ff75bfd6d4fb7f9dfc938372627feb2c2348f8b1493b5
; Derive-Public-Key outputs:
pk_prime = h'04cc85763fae2c8f38964ddc1f3dd9eebe2d2cb5c2842b0a622939b608f9cef967
aafa50b9b24d6ae5a273f5b5d03b6a1ce8abd4f4dbaf487c417ef7380d1481b5'
;kh = (implementation defined)
; Derive-Private-Key outputs:
sk_prime = 0x1a60b7fe69b315fe1262c46711af990d47228471ef5a296f6aa26ba6a5a1a6ec
; 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'04c38bbdd7286196733fa177e43b73cfd3d6d72cd11cc0bb2c9236cf85a42dcff5
dfa339c1e07dfcdfda8d7be2a5a3c7382991f387dfe332b1dd8da6e0622cfb35'
sk_bl = 0xd959500a78ccf850ce46c80a8c5043c9a2e33844232b3829df37d05b3069f455
sk_kem = 0x74e0a4cd81ca2d24246ff75bfd6d4fb7f9dfc938372627feb2c2348f8b1493b5
; Derive-Public-Key outputs:
pk_prime = h'04056c654c97ea460ebfae997c8e12314184e45183aefdb8ce9547afca1faaee70
da6c6433c8fe71c32284cf0a015eb463ea2fc81438f6698684525f011d7ae83c'
;kh = (implementation defined)
; Derive-Private-Key outputs:
sk_prime = 0xbb609831741d9232ecd7d58770c503992ca78d34361a865a0d6715861955526f
B.2. Other instances
TODO
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Appendix C. Document History
-07
* Fixed hash_to_field DST in Sub-Kem-Derive-Key-Pair in section
"Using ECDH as the KEM" to agree with test vectors.
-06
* Changed DST construction in section "Using ECDH as the KEM" to
include the "ARKG-ECDH." prefix everywhere in the formula.
Previously the prefix was added in the argument to the "Using HMAC
to adapt a KEM without ciphertext integrity" formula but not in
the Sub-Kem functions defined in "Using ECDH as the KEM".
-05
* Deleted concrete instances ARKG-curve25519ADD-X25519, ARKG-
curve448ADD-X448, ARKG-edwards25519ADD-X25519 and ARKG-
edwards448ADD-X448 since implementations with a non-prime order
generator, including EdDSA, are incompatible with the additive
blinding scheme defined in section "Using elliptic curve addition
for key blinding".
* Remodeled procedures to be fully deterministic:
- BL-Generate-Keypair() replaced with BL-Derive-Key-Pair(ikm).
- KEM-Generate-Keypair() replaced with KEM-Derive-Key-Pair(ikm).
- ARKG-Generate-Seed() replaced with ARKG-Derive-Seed(ikm_bl,
ikm_kem).
- Parameter ikm added to ARKG-Derive-Public-Key.
- Instance parameter hash-to-crv-suite added to generic formula
"Using ECDH as the KEM", affecting concrete instances ARKG-
P256ADD-ECDH, ARKG-P384ADD-ECDH, ARKG-P521ADD-ECDH and ARKG-
P256kADD-ECDH.
- Section "Deterministic key generation" deleted.
* Flipped order of (pk_bl, pk_kem) and (sk_bl, sk_kem) parameter and
return value tuples for consistent ordering between BL and KEM
throughout document.
* info parameter renamed to ctx.
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* ctx length limited to at most 64 bytes.
* Encoding of ctx in ARKG-Derive-Public-Key and ARKG-Derive-Private-
Key now embeds the length of ctx.
* Renamed concrete instances and corresponding DST_ext values:
- ARKG-P256ADD-ECDH to ARKG-P256
- ARKG-P384ADD-ECDH to ARKG-P384
- ARKG-P521ADD-ECDH to ARKG-P521
- ARKG-P256kADD-ECDH to ARKG-P256k
* Added ARKG-P256 test vectors.
-04
* Extracted COSE_Key_Ref definition and COSE algorithm registrations
to draft-lundberg-cose-two-party-signing-algs.
* Redefined alg (3) parameter and added dkalg (-3) in ARKG-pub
COSE_Key.
* Defined alg (3) and inst (-3) parameters of Ref-ARKG-derived COSE
key type.
-03
* Renamed section "Using HMAC to adapt a KEM without {integrity
protection => ciphertext integrity}".
* Fixed info argument to HMAC in section "Using HMAC to adapt a KEM
without ciphertext integrity".
* Added reference to Shoup for definition of key encapsulation
mechanism.
* Added CDDL definition of COSE_Key_Ref.
* Editorial fixes to references.
* Renamed proposed COSE Key Types.
-02
* Rewritten introduction.
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* Renamed ARKG-Derive-Secret-Key to ARKG-Derive-Private-Key.
* Overhauled EC instantiations to use hash_to_field and account for
non-prime order curve key generation.
* Eliminated top-level MAC and KDF instance parameters.
* Added info parameter to instance parameter functions.
* Added requirement of KEM ciphertext integrity and generic formula
for augmenting any KEM using HMAC.
* Added curve/edwards25519/448 instances.
* Added proposal for COSE bindings and key reference types.
-01
* Editorial Fixes to formatting and references.
-00
* Initial Version
Contributors
Dain Nilsson
Yubico
Peter Altmann
Agency for Digital Government
Sweden
Michael B. Jones
Self-Issued Consulting
United States
URI: https://self-issued.info/
Sander Dijkhuis
Cleverbase
Netherlands
Authors' Addresses
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Emil Lundberg (editor)
Yubico
Gävlegatan 22
Stockholm
Sweden
Email: emil@emlun.se
John Bradley
Yubico
Email: ve7jtb@ve7jtb.com
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