| Internet-Draft | The BBS Signature Scheme | October 2022 |
| Looker, et al. | Expires 26 April 2023 | [Page] |
BBS is a digital signature scheme categorized as a form of short group signature that supports several unique properties. Notably, the scheme supports signing multiple messages whilst producing a single output digital signature. Through this capability, the possessor of a signature is able to generate proofs that selectively disclose subsets of the originally signed set of messages, whilst preserving the verifiable authenticity and integrity of the messages. Furthermore, these proofs are said to be zero-knowledge in nature as they do not reveal the underlying signature; instead, what they reveal is a proof of knowledge of the undisclosed signature.¶
This note is to be removed before publishing as an RFC.¶
Source for this draft and an issue tracker can be found at https://github.com/decentralized-identity/bbs-signature.¶
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-Drafts is at https://datatracker.ietf.org/drafts/current/.¶
Internet-Drafts are draft documents valid for a maximum of six months and may be updated, replaced, or obsoleted by other documents at any time. It is inappropriate to use Internet-Drafts as reference material or to cite them other than as "work in progress."¶
This Internet-Draft will expire on 26 April 2023.¶
Copyright (c) 2022 IETF Trust and the persons identified as the document authors. All rights reserved.¶
This document is subject to BCP 78 and the IETF Trust's Legal Provisions Relating to IETF Documents (https://trustee.ietf.org/license-info) in effect on the date of publication of this document. Please review these documents carefully, as they describe your rights and restrictions with respect to this document. Code Components extracted from this document must include Revised BSD License text as described in Section 4.e of the Trust Legal Provisions and are provided without warranty as described in the Revised BSD License.¶
A digital signature scheme is a fundamental cryptographic primitive that is used to provide data integrity and verifiable authenticity in various protocols. The core premise of digital signature technology is built upon asymmetric cryptography where-by the possessor of a private key is able to sign a message, where anyone in possession of the corresponding public key matching that of the private key is able to verify the signature.¶
The name BBS is derived from the authors of the original academic work of Dan Boneh, Xavier Boyen, and Hovav Shacham, where the scheme was first described.¶
Beyond the core properties of a digital signature scheme, BBS signatures provide multiple additional unique properties, three key ones are:¶
Selective Disclosure - The scheme allows a signer to sign multiple messages and produce a single -constant size- output signature. A holder/prover then possessing the messages and the signature can generate a proof whereby they can choose which messages to disclose, while revealing no-information about the undisclosed messages. The proof itself guarantees the integrity and authenticity of the disclosed messages (e.g. that they were originally signed by the signer).¶
Unlinkable Proofs - The proofs generated by the scheme are known as zero-knowledge, proofs-of-knowledge of the signature, meaning a verifying party in receipt of a proof is unable to determine which signature was used to generate the proof, removing a common source of correlation. In general, each proof generated is indistinguishable from random even for two proofs generated from the same signature.¶
Proof of Possession - The proofs generated by the scheme prove to a verifier that the party who generated the proof (holder/prover) was in possession of a signature without revealing it. The scheme also supports binding a presentation header to the generated proof. The presentation header can include arbitrary information such as a cryptographic nonce, an audience/domain identifier and or time based validity information.¶
Refer to the Appendix B for an elaboration on situations where these properties are useful¶
Below is a basic diagram describing the main entities involved in the scheme¶
(1) sign (3) ProofGen
+----- +-----
| | | |
| | | |
| \ / | \ /
+----------+ +-----------+
| | | |
| | | |
| | | |
| Signer |---(2)* Send signature + msgs----->| Holder/ |
| | | Prover |
| | | |
| | | |
+----------+ +-----------+
|
|
|
(4)* Send proof + disclosed msgs
|
|
\ /
+-----------+
| |
| |
| |
| Verifier |
| |
| |
| |
+-----------+
| / \
| |
| |
+-----
(5) ProofVerify
Note The protocols implied by the items annotated by an asterisk are out of scope for this specification¶
The following terminology is used throughout this document:¶
The following notation and primitives are used:¶
X containing all elements from and including the value at index a until and including the value at index b. Note when this syntax is applied to an octet string, each element in the array X is assumed to be a single byte.¶
Terms specific to pairing-friendly elliptic curves that are relevant to this document are restated below, originally defined in [I-D.irtf-cfrg-pairing-friendly-curves]¶
This document is organized as follows:¶
Scheme Definition defines the core operations and parameters for the BBS signature scheme.¶
Utility Operations defines utilities used by the BBS signature scheme.¶
Security Considerations describes a set of security considerations associated to the signature scheme.¶
Ciphersuites defines the format of a ciphersuite, alongside a concrete ciphersuite based on the BLS12-381 curve.¶
The keywords MUST, MUST NOT, REQUIRED, SHALL, SHALL NOT, SHOULD, SHOULD NOT, RECOMMENDED, MAY, and OPTIONAL, when they appear in this document, are to be interpreted as described in [RFC2119].¶
This section defines the BBS signature scheme, including the parameters required to define a concrete ciphersuite.¶
The schemes operations defined in this section depend on the following parameters:¶
A pairing-friendly elliptic curve, plus associated functionality given in Section 1.2.¶
A hash-to-curve suite as defined in [I-D.irtf-cfrg-hash-to-curve], using the aforementioned pairing-friendly curve. This defines the hash_to_curve and expand_message operations, used by this document.¶
PRF(n): a pseudo-random function similar to [RFC4868]. Returns n pseudo randomly generated bytes.¶
In definition of this signature scheme there are two possible variations based upon the sub-group selection, namely where public keys are defined in G2 and signatures in G1 OR the opposite where public keys are defined in G1 and signatures in G2. Some pairing cryptography based digital signature schemes such as [I-D.irtf-cfrg-bls-signature] elect to allow for both variations, because they optimize for different things. However, in the case of this scheme, due to the operations involved in both signature and proof generation being computational in-efficient when performed in G2 and in the pursuit of simplicity, the scheme is limited to a construction where public keys are in G2 and signatures in G1.¶
Throughout the operations of this signature scheme, each message that is signed is paired with a specific generator (point in G1). Specifically, if a generator H_1 is multiplied with msg_1 during signing, then H_1 MUST be multiplied with msg_1 in all other operations (signature verification, proof generation and proof verification).¶
Aside from the message generators, the scheme uses two additional generators: Q_1 and Q_2. The first (Q_1), is used for the blinding value (s) of the signature. The second generator (Q_2), is used to sign the signature's domain, which binds both the signature and generated proofs to a specific context and cryptographically protects any potential application-specific information (for example, messages that must always be disclosed etc.).¶
This operation generates a secret key (SK) deterministically from a secret octet string (IKM).¶
KeyGen uses an HKDF [RFC5869] instantiated with the hash function hash.¶
For security, IKM MUST be infeasible to guess, e.g. generated by a trusted source of randomness.¶
IKM MUST be at least 32 bytes long, but it MAY be longer.¶
Because KeyGen is deterministic, implementations MAY choose either to store the resulting SK or to store IKM and call KeyGen to derive SK when necessary.¶
KeyGen takes an optional parameter, key_info. This parameter MAY be used to derive multiple independent keys from the same IKM. By default, key_info is the empty string.¶
SK = KeyGen(IKM, key_info)
Inputs:
- IKM (REQUIRED), a secret octet string. See requirements above.
- key_info (OPTIONAL), an octet string. if this is not supplied, it
MUST default to an empty string.
Definitions:
- HKDF-Extract is as defined in [@!RFC5869], instantiated with hash function hash.
- HKDF-Expand is as defined in [@!RFC5869], instantiated with hash function hash.
- I2OSP and OS2IP are as defined in [@!RFC8017], Section 4.
- L is the integer given by ceil((3 * ceil(log2(r))) / 16).
- INITSALT is the ASCII string "BBS-SIG-KEYGEN-SALT-".
Outputs:
- SK, a uniformly random integer such that 0 < SK < r.
Procedure:
1. salt = INITSALT
2. SK = 0
3. while SK == 0:
4. salt = hash(salt)
5. PRK = HKDF-Extract(salt, IKM || I2OSP(0, 1))
6. OKM = HKDF-Expand(PRK, key_info || I2OSP(L, 2), L)
7. SK = OS2IP(OKM) mod r
8. return SK
¶
Note This operation is the RECOMMENDED way of generating a secret key, but its use is not required for compatibility, and implementations MAY use a different KeyGen procedure. For security, such an alternative MUST output a secret key that is statistically close to uniformly random in the range 0 < SK < r.¶
The operations in this section make use of a "Precomputations" set of steps. The "Precomputations" steps must be executed before the steps in the "Procedure" of each operation and include computations that can be cached and re-used multiple times (like creating the generators etc.) or procedural steps like de-structuring inputted arrays.¶
This operation computes a deterministic signature from a secret key (SK) and optionally over a header and or a vector of messages.¶
signature = Sign(SK, PK, header, messages)
Inputs:
- SK (REQUIRED), a non negative integer mod r outputted by the KeyGen
operation.
- PK (REQUIRED), an octet string of the form outputted by the SkToPk
operation provided the above SK as input.
- header (OPTIONAL), an octet string containing context and application
specific information. If not supplied, it defaults
to an empty string.
- messages (OPTIONAL), a vector of scalars. If not supplied, it defaults
to the empty array "()".
Parameters:
- ciphersuite_id, ASCII string. The unique ID of the ciphersuite.
- P1, fixed point of G1, defined by the ciphersuite.
Definitions:
- L, is the non-negative integer representing the number of messages to
be signed e.g length(messages). If no messages are supplied as an
input, the value of L MUST evaluate to zero (0).
Outputs:
- signature, a signature encoded as an octet string.
Precomputations:
1. msg_1, ..., msg_L = messages[1], ..., messages[L]
2. (Q_1, Q_2, H_1, ..., H_L) = create_generators(L+2)
Procedure:
1. dom_array = (PK, L, Q_1, Q_2, H_1, ..., H_L, ciphersuite_id, header)
2. dom_for_hash = encode_for_hash(dom_array)
3. if dom_for_hash is INVALID, return INVALID
4. domain = hash_to_scalar(dom_for_hash, 1)
5. e_s_for_hash = encode_for_hash((SK, domain, msg_1, ..., msg_L))
6. if e_s_for_hash is INVALID, return INVALID
7. (e, s) = hash_to_scalar(e_s_for_hash, 2)
8. B = P1 + Q_1 * s + Q_2 * domain + H_1 * msg_1 + ... + H_L * msg_L
9. A = B * (1 / (SK + e))
10. signature_octets = signature_to_octets(A, e, s)
11. return signature_octets
¶
Note When computing step 9 of the above procedure there is an extremely small probability (around 2^(-r)) that the condition (SK + e) = 0 mod r will be met. How implementations evaluate the inverse of the scalar value 0 may vary, with some returning an error and others returning 0 as a result. If the returned value from the inverse operation 1/(SK + e) does evaluate to 0 the value of A will equal Identity_G1 thus an invalid signature. Implementations MAY elect to check (SK + e) = 0 mod r prior to step 9, and or A != Identity_G1 after step 9 to prevent the production of invalid signatures.¶
This operation checks that a signature is valid for a given header and vector of messages against a supplied public key (PK). The messages MUST be supplied in this operation in the same order they were supplied to Sign when creating the signature.¶
result = Verify(PK, signature, header, messages)
Inputs:
- PK (REQUIRED), an octet string of the form outputted by the SkToPk
operation.
- signature (REQUIRED), an octet string of the form outputted by the
Sign operation.
- header (OPTIONAL), an octet string containing context and application
specific information. If not supplied, it defaults
to an empty string.
- messages (OPTIONAL), a vector of scalars. If not supplied, it defaults
to the empty array "()".
Parameters:
- ciphersuite_id, ASCII string. The unique ID of the ciphersuite.
- P1, fixed point of G1, defined by the ciphersuite.
Definitions:
- L, is the non-negative integer representing the number of messages to
be signed e.g length(messages). If no messages are supplied as an
input, the value of L MUST evaluate to zero (0).
Outputs:
- result, either VALID or INVALID.
Precomputations:
1. (msg_1, ..., msg_L) = messages
2. (Q_1, Q_2, H_1, ..., H_L) = create_generators(L+2)
Procedure:
1. signature_result = octets_to_signature(signature)
2. if signature_result is INVALID, return INVALID
3. (A, e, s) = signature_result
4. W = octets_to_pubkey(PK)
5. if W is INVALID, return INVALID
6. dom_array = (PK, L, Q_1, Q_2, H_1, ..., H_L, ciphersuite_id, header)
7. dom_for_hash = encode_for_hash(dom_array)
8. if dom_for_hash is INVALID, return INVALID
9. domain = hash_to_scalar(dom_for_hash, 1)
10. B = P1 + Q_1 * s + Q_2 * domain + H_1 * msg_1 + ... + H_L * msg_L
11. if e(A, W + P2 * e) * e(B, -P2) != Identity_GT, return INVALID
12. return VALID
¶
This operation computes a zero-knowledge proof-of-knowledge of a signature, while optionally selectively disclosing from the original set of signed messages. The "prover" may also supply a presentation header, see Presentation header selection for more details.¶
The messages supplied in this operation MUST be in the same order as when supplied to Sign. To specify which of those messages will be disclosed, the prover can supply the list of indexes (disclosed_indexes) that the disclosed messages have in the array of signed messages. Each element in disclosed_indexes MUST be a non-negative integer, in the range from 1 to length(messages).¶
proof = ProofGen(PK, signature, header, ph, messages, disclosed_indexes)
Inputs:
- PK (REQUIRED), an octet string of the form outputted by the SkToPk
operation.
- signature (REQUIRED), an octet string of the form outputted by the
Sign operation.
- header (OPTIONAL), an octet string containing context and application
specific information. If not supplied, it defaults
to an empty string.
- ph (OPTIONAL), octet string containing the presentation header. If not
supplied, it defaults to an empty string.
- messages (OPTIONAL), a vector of scalars. If not supplied, it defaults
to the empty array "()".
- disclosed_indexes (OPTIONAL), vector of unsigned integers in ascending
order. Indexes of disclosed messages. If
not supplied, it defaults to the empty
array "()".
Parameters:
- ciphersuite_id, ASCII string. The unique ID of the ciphersuite.
- P1, fixed point of G1, defined by the ciphersuite.
Definitions:
- L, is the non-negative integer representing the number of messages,
i.e., L = length(messages). If no messages are supplied, the
value of L MUST evaluate to zero (0).
- R, is the non-negative integer representing the number of disclosed
(revealed) messages, i.e., R = length(disclosed_indexes). If no
messages are disclosed, R MUST evaluate to zero (0).
- U, is the non-negative integer representing the number of undisclosed
messages, i.e., U = L - R.
- prf_len = ceil(ceil(log2(r))/8), where r defined by the ciphersuite.
Outputs:
- proof, octet string; or INVALID.
Precomputations:
1. (i1, ..., iR) = disclosed_indexes
2. (j1, ..., jU) = range(1, L) \ disclosed_indexes
3. (msg_1, ..., msg_L) = messages
4. (msg_i1, ..., msg_iR) = (messages[i1], ..., messages[iR])
5. (msg_j1, ..., msg_jU) = (messages[j1], ..., messages[jU])
6. (Q_1, Q_2, MsgGenerators) = create_generators(L+2)
7. (H_1, ..., H_L) = MsgGenerators
8. (H_j1, ..., H_jU) = (MsgGenerators[j1], ..., MsgGenerators[jU])
Procedure:
1. signature_result = octets_to_signature(signature)
2. if signature_result is INVALID, return INVALID
3. (A, e, s) = signature_result
4. dom_array = (PK, L, Q_1, Q_2, H_1, ..., H_L, ciphersuite_id, header)
5. dom_for_hash = encode_for_hash(dom_array)
6. if dom_for_hash is INVALID, return INVALID
7. domain = hash_to_scalar(dom_for_hash, 1)
8. (r1, r2, e~, r2~, r3~, s~) = hash_to_scalar(PRF(prf_len), 6)
9. (m~_j1, ..., m~_jU) = hash_to_scalar(PRF(prf_len), U)
10. B = P1 + Q_1 * s + Q_2 * domain + H_1 * msg_1 + ... + H_L * msg_L
11. r3 = r1 ^ -1 mod r
12. A' = A * r1
13. Abar = A' * (-e) + B * r1
14. D = B * r1 + Q_1 * r2
15. s' = r2 * r3 + s mod r
16. C1 = A' * e~ + Q_1 * r2~
17. C2 = D * (-r3~) + Q_1 * s~ + H_j1 * m~_j1 + ... + H_jU * m~_jU
18. c_array = (A', Abar, D, C1, C2, R, i1, ..., iR,
msg_i1, ..., msg_iR, domain, ph)
19. c_for_hash = encode_for_hash(c_array)
20. if c_for_hash is INVALID, return INVALID
21. c = hash_to_scalar(c_for_hash, 1)
22. e^ = c * e + e~ mod r
23. r2^ = c * r2 + r2~ mod r
24. r3^ = c * r3 + r3~ mod r
25. s^ = c * s' + s~ mod r
26. for j in (j1, ..., jU): m^_j = c * msg_j + m~_j mod r
27. proof = (A', Abar, D, c, e^, r2^, r3^, s^, (m^_j1, ..., m^_jU))
28. return proof_to_octets(proof)
¶
This operation checks that a proof is valid for a header, vector of disclosed messages (along side their index corresponding to their original position when signed) and presentation header against a public key (PK).¶
The operation accepts the list of messages the prover indicated to be disclosed. Those messages MUST be in the same order as when supplied to Sign (as a subset of the signed messages list). The operation also requires the total number of signed messages (L). Lastly, it also accepts the indexes that the disclosed messages had in the original array of messages supplied to Sign (i.e., the disclosed_indexes list supplied to ProofGen). Every element in this list MUST be a non-negative integer in the range from 1 to L, in ascending order.¶
result = ProofVerify(PK, proof, L, header, ph,
disclosed_messages,
disclosed_indexes)
Inputs:
- PK (REQUIRED), an octet string of the form outputted by the SkToPk
operation.
- proof (REQUIRED), an octet string of the form outputted by the
ProofGen operation.
- L (REQUIRED), non-negative integer. The number of signed messages.
- header (OPTIONAL), an optional octet string containing context and
application specific information. If not supplied,
it defaults to an empty string.
- ph (OPTIONAL), octet string containing the presentation header. If not
supplied, it defaults to an empty string.
- disclosed_messages (OPTIONAL), a vector of scalars. If not supplied,
it defaults to the empty array "()".
- disclosed_indexes (OPTIONAL), vector of unsigned integers in ascending
order. Indexes of disclosed messages. If
not supplied, it defaults to the empty
array "()".
Parameters:
- ciphersuite_id, ASCII string. The unique ID of the ciphersuite.
- P1, fixed point of G1, defined by the ciphersuite.
Definitions:
- R, is the non-negative integer representing the number of disclosed
(revealed) messages, i.e., R = length(disclosed_indexes). If no
messages are disclosed, the value of R MUST evaluate to zero (0).
- U, is the non-negative integer representing the number of undisclosed
messages, i.e., U = L - R.
Outputs:
- result, either VALID or INVALID.
Precomputations:
1. (i1, ..., iR) = disclosed_indexes
2. (j1, ..., jU) = range(1, L) \ disclosed_indexes
3. (msg_i1, ..., msg_iR) = disclosed_messages
4. (Q_1, Q_2, MsgGenerators) = create_generators(L+2)
5. (H_1, ..., H_L) = MsgGenerators
6. (H_i1, ..., H_iR) = (MsgGenerators[i1], ..., MsgGenerators[iR])
7. (H_j1, ..., H_jU) = (MsgGenerators[j1], ..., MsgGenerators[jU])
Preconditions:
1. for i in (i1, ..., iR), if i < 1 or i > L, return INVALID
2. if length(disclosed_messages) != R, return INVALID
Procedure:
1. proof_result = octets_to_proof(proof)
2. if proof_result is INVALID, return INVALID
3. (A', Abar, D, c, e^, r2^, r3^, s^, (m^_j1,...,m^_jU)) = proof_result
4. W = octets_to_pubkey(PK)
5. if W is INVALID, return INVALID
6. dom_array = (PK, L, Q_1, Q_2, H_1, ..., H_L, ciphersuite_id, header)
7. dom_for_hash = encode_for_hash(dom_array)
8. if dom_for_hash is INVALID, return INVALID
9. domain = hash_to_scalar(dom_for_hash, 1)
10. C1 = (Abar - D) * c + A' * e^ + Q_1 * r2^
11. T = P1 + Q_2 * domain + H_i1 * msg_i1 + ... + H_iR * msg_iR
12. C2 = T * c - D * r3^ + Q_1 * s^ + H_j1 * m^_j1 + ... + H_jU * m^_jU
13. cv_array = (A', Abar, D, C1, C2, R, i1, ..., iR,
msg_i1, ..., msg_iR, domain, ph)
14. cv_for_hash = encode_for_hash(cv_array)
15. if cv_for_hash is INVALID, return INVALID
16. cv = hash_to_scalar(cv_for_hash, 1)
17. if c != cv, return INVALID
18. if A' == Identity_G1, return INVALID
19. if e(A', W) * e(Abar, -P2) != Identity_GT, return INVALID
20. return VALID
¶
This operation defines how to create a set of generators that form a part of the public parameters used by the BBS Signature scheme to accomplish operations such as Sign, Verify, ProofGen and ProofVerify. It takes one input, the number of generator points to create, which is determined in part by the number of signed messages.¶
As an optimization, implementations MAY cache the result of create_generators for a specific generator_seed (determined by the ciphersuite) and count. The values n and v MAY also be cached in order to efficiently extend a existing list of generator points.¶
generators = create_generators(count)
Inputs:
- count (REQUIRED), unsigned integer. Number of generators to create.
Parameters:
- hash_to_curve_suite, the hash to curve suite id defined by the
ciphersuite.
- hash_to_curve_g1, the hash_to_curve operation for the G1 subgroup,
defined by the suite specified by the
hash_to_curve_suite parameter.
- expand_message, the expand_message operation defined by the suite
specified by the hash_to_curve_suite parameter.
- generator_seed, octet string. A seed value selected by the
ciphersuite.
- P1, fixed point of G1, defined by the ciphersuite.
Definitions:
- seed_dst, octet string representing the domain seperation tag:
utf8(ciphersuite_id || "SIG_GENERATOR_SEED_"), where
ciphersuite_id is defined by the ciphersuite.
- generator_dst, octet string representing the domain seperation tag:
utf8(ciphersuite_id || "SIG_GENERATOR_DST_"), where
ciphersuite_id is defined by the ciphersuite.
- seed_len = ceil((ceil(log2(r)) + k)/8), where r and k are defined by
the ciphersuite.
Outputs:
- generators, an array of generators.
Procedure:
1. v = expand_message(generator_seed, seed_dst, seed_len)
2. n = 1
3. for i in range(1, count):
4. v = expand_message(v || I2OSP(n, 4), seed_dst, seed_len)
5. n = n + 1
6. generator_i = Identity_G1
7. candidate = hash_to_curve_g1(v, generator_dst)
8. if candidate in (generator_1, ..., generator_i):
9. go back to step 4
10. generator_i = candidate
11. return (generator_1, ..., generator_count)
¶
There are multiple ways in which messages can be mapped to their respective scalar values, which is their required form to be used with the Sign, Verify, ProofGen and ProofVerify operations.¶
This operation takes an input message and maps it to a scalar value via a cryptographic hash function for the given curve. The operation takes also as an optional input a domain separation tag (dst). If a dst is not supplied, its value MUST default to the octet string returned from utf8(ciphersuite_id || "MAP_MSG_TO_SCALAR_AS_HASH_"), where ciphersuite_id is the ASCII string representing the unique ID of the ciphersuite.¶
result = MapMessageToScalarAsHash(msg, dst)
Inputs:
- msg (REQUIRED), octet string.
- dst (OPTIONAL), an octet string representing a domain separation tag.
If not supplied, it default to the octet string
utf8(ciphersuite_id || "MAP_MSG_TO_SCALAR_AS_HASH_")
where ciphersuite_id is defined by the ciphersuite.
Outputs:
- result, a scalar value.
Procedure:
1. msg_for_hash = encode_for_hash(msg)
2. if msg_for_hash is INVALID, return INVALID
3. if length(dst) > 255, return INVALID
4. return hash_to_scalar(msg_for_hash, 1, dst)
¶
This operation describes how to hash an arbitrary octet string to n scalar values in the multiplicative group of integers mod r (i.e., values in the range [1, r-1]). This procedure acts as a helper function, used internally in various places within the operations described in the spec. To map a message to a scalar that would be passed as input to the Sign, Verify, ProofGen and ProofVerify functions, one must use MapMessageToScalarAsHash instead.¶
This operation makes use of expand_message defined in [I-D.irtf-cfrg-hash-to-curve], in a similar way used by the hash_to_field operation of Section 5 from the same document (with the additional checks for getting a scalar that is 0). If an implementer wants to use hash_to_field instead, they MUST use the multiplicative group of integers mod r (Fr), as the target group (F). Note however, that the hash_to_curve document, makes use of hash_to_field with the target group being the multiplicative group of integers mod p (Fp). For this reason, we don't directly use hash_to_field here, rather we define a similar operation (hash_to_scalar), making direct use of the expand_message function, that will be defined by the hash-to-curve suite used (i.e., either expand_message_xmd or expand_message_xof). If someone also has a hash_to_field implementation available, with the target group been Fr, they can use this instead (adding the check for a scalar been 0).¶
The operation takes as input an octet string representing the message to hash (msg), the number of the scalars to return (count) as well as an optional domain separation tag (dst). If a dst is not supplied, its value MUST default to the octet string returned from utf8(ciphersuit_id || "H2S_"), where ciphersuite_id is the ASCII string representing the unique ID of the ciphersuite.¶
scalars = hash_to_scalar(msg_octets, count, dst)
Inputs:
- msg_octets (REQUIRED), octet string. The message to be hashed.
- count (REQUIRED), an integer greater or equal to 1. The number of
scalars to output.
- dst (OPTIONAL), an octet string representing a domain separation tag.
If not supplied, it defaults to the octet string given
by utf8(ciphersuite_id || "H2S_"), where
ciphersuite_id is defined by the ciphersuite.
Parameters:
- hash_to_curve_suite, the hash to curve suite id defined by the
ciphersuite.
- expand_message, the expand_message operation defined by the suite
specified by the hash_to_curve_suite parameter.
Definitions:
- expand_len = ceil((ceil(log2(r))+k)/8), where r and k are defined by
the ciphersuite.
Outputs:
- scalars, an array of non-zero scalars mod r.
Procedure:
1. len_in_bytes = count * expand_len
2. t = 0
3. msg_prime = msg_octets || I2OSP(t, 1) || I2OSP(count, 4)
4. uniform_bytes = expand_message(msg_prime, dst, len_in_bytes)
5. for i in (1, ..., count):
6. tv = uniform_bytes[(i-1)*expand_len..i*expand_len-1]
7. scalar_i = OS2IP(tv) mod r
8. if 0 in (scalar_1, ..., scalar_count):
9. t = t + 1
10. go back to step 3
11. return (scalar_1, ..., scalar_count)
¶
This operation describes how to decode an octet string, validate it and return the underlying components that make up the signature.¶
signature = octets_to_signature(signature_octets)
Inputs:
- signature_octets (REQUIRED), octet string of the form output from
signature_to_octets operation.
Outputs:
signature, a signature in the form (A, e, s), where A is a point in G1
and e and s are non-zero scalars mod r.
Procedure:
1. expected_len = octet_point_length + 2 * octet_scalar_length
2. if length(signature_octets) != expected_len, return INVALID
3. A_octets = signature_octets[0..(octet_point_length - 1)]
4. A = octets_to_point_g1(A_octets)
5. if A is INVALID, return INVALID
6. if A == Identity_G1, return INVALID
7. index = octet_point_length
8. end_index = index + octet_scalar_length - 1
9. e = OS2IP(signature_octets[index..end_index])
10. if e = 0 OR e >= r, return INVALID
11. index += octet_scalar_length
12. end_index = index + octet_scalar_length - 1
13. s = OS2IP(signature_octets[index..end_index])
14. if s = 0 OR s >= r, return INVALID
15. return (A, e, s)
¶
This operation describes how to encode a signature to an octet string.¶
Note this operation deliberately does not perform the relevant checks on the inputs A, e and s
because its assumed these are done prior to its invocation, e.g as is the case with the Sign operation.¶
signature_octets = signature_to_octets(signature)
Inputs:
- signature (REQUIRED), a valid signature, in the form (A, e, s), where
A a point in G1 and e, s non-zero scalars mod r.
Outputs:
- signature_octets, octet string.
Procedure:
1. (A, e, s) = signature
2. A_octets = point_to_octets_g1(A)
3. e_octets = I2OSP(e, octet_scalar_length)
4. s_octets = I2OSP(s, octet_scalar_length)
5. return (A_octets || e_octets || s_octets)
¶
This operation describes how to decode an octet string representing a proof, validate it and return the underlying components that make up the proof value.¶
The proof value outputted by this operation consists of the following components, in that order:¶
proof = octets_to_proof(proof_octets)
Inputs:
- proof_octets (REQUIRED), octet string of the form outputted from the
proof_to_octets operation.
Parameters:
- r (REQUIRED), non-negative integer. The prime order of the G1 and
G2 groups, defined by the ciphersuite.
- octet_scalar_length (REQUIRED), non-negative integer. The length of
a scalar octet representation, defined
by the ciphersuite.
- octet_point_length (REQUIRED), non-negative integer. The length of
a point in G1 octet representation,
defined by the ciphersuite.
Outputs:
- proof, a proof value in the form described above or INVALID
Procedure:
1. proof_len_floor = 3 * octet_point_length + 5 * octet_scalar_length
2. if length(proof_octets) < proof_len_floor, return INVALID
// Points (i.e., (A', Abar, D) in ProofGen) de-serialization.
3. index = 0
4. for i in range(0, 2):
5. end_index = index + octet_point_length - 1
6. A_i = octets_to_point_g1(proof_octets[index..end_index])
7. if A_i is INVALID or Identity_G1, return INVALID
8. index += octet_point_length
// Scalars (i.e., (c, e^, r2^, r3^, s^, (m^_j1, ..., m^_jU)) in
// ProofGen) de-serialization.
9. j = 0
10. while index < length(proof_octets):
11. end_index = index + octet_scalar_length - 1
12. s_j = OS2IP(proof_octets[index..end_index])
13. if s_j = 0 or if s_j >= r, return INVALID
14. index += octet_scalar_length
15. j += 1
16. if index != length(proof_octets), return INVALID
17. msg_commitments = ()
18. If j > 5, set msg_commitments = (s_5, ..., s_(j-1))
19. return (A_0, A_1, A_2, s_0, s_1, s_2, s_3, s_4, msg_commitments)
¶
This operation describes how to encode a proof, as computed at step 25 in ProofGen, to an octet string. The input to the operation MUST be a valid proof.¶
The inputed proof value must consist of the following components, in that order:¶
A', Abar, D, in ProofGen)¶
c, e^, r2^, r3^, s^, in ProofGen).¶
m^_j1, ..., m^_jU, in ProofGen, where U the number of undisclosed messages).¶
proof_octets = proof_to_octets(proof)
Inputs:
- proof (REQUIRED), a BBS proof in the form calculated by ProofGen in
step 25 (see above).
Parameters:
- octet_scalar_length (REQUIRED), non-negative integer. The length of
a scalar octet representation, defined
by the ciphersuite.
Outputs:
- proof_octets, octet string.
Procedure:
1. (A', Abar, D, c, e^, r2^, r3^, s^, (m^_1, ..., m^_U)) = proof
2. Let proof_octets be an empty octet string.
// Points Serialization.
3. for point in (A', Abar, D):
4. point_octets = point_to_octets_g1(point)
5. proof_octets = proof_octets || point_octets
// Scalar Serialization.
6. for scalar in (c, e^, r2^, r3^, s^, m^_1, ..., m^_U):
7. scalar_octets = I2OSP(scalar, octet_scalar_length)
8. proof_octets = proof_octets || scalar_octets
9. return proof_octets
¶
This operation describes how to decode an octet string representing a public key, validates it and returns the corresponding point in G2. Steps 2 to 5 check if the public key is valid. As an optimization, implementations MAY cache the result of those steps, to avoid unnecessarily repeating validation for known public keys.¶
W = octets_to_pubkey(PK)
Inputs:
- PK, octet string. A public key in the form ouputted by the SkToPK
operation
Outputs:
- W, a valid point in G2 or INVALID
Procedure:
1. W = octets_to_point_g2(PK)
2. If W is INVALID, return INVALID
3. if subgroup_check(W) is INVALID, return INVALID
4. If W == Identity_G2, return INVALID
5. return W
¶
This document uses the hash_to_scalar function to hash elements to scalars in the multiplicative group mod r (see Section 5.3). To avoid ambiguity, elements passed to that operation, must first be encoded appropriately using encode_for_hash. The following procedure describes how to encode each element accordingly by serializing it to an appropriate format depending on its type and concatenating the results. Specifically,¶
point_to_octets_g* implementation for a particular ciphersuite.¶
I2OSP with an output length of 8 bytes.¶
After encoding, octet strings will be prepended with a value representing the length of their binary representation in the form of the number of bytes. This length must be encoded to octets using I2OSP with output length of 8 bytes. The combined value (encoded value + length prefix) binary representation is then encoded as a single octet string. For example, the string 0x14d will be encoded as 0x0000000000000002014d. If the length of the octet string is larger than 2^64 - 1, the octet string must be rejected. Similarly, ASCII strings, after encoded to octets (using utf8), will also be appended with the length of their octet-string representation.¶
An exception to the above rule is octet strings that represent a public key or ASCII strings that represent a ciphersuite ID, since those have a constant, well defined by each ciphersuite, length.¶
Optional input/parameters to operations that feature in a call to hash_to_scalar, that are not supplied to the operation should default to an empty octet string. For example, if X is an optional input/parameter that is not supplied, whilst A and B are required, then the procedural step of hash(A || X || B) MUST be evaluated to hash(A || "" || B).¶
The above is further described in the following procedure.¶
result = encode_for_hash(input_array)
Inputs:
- input_array, an array of elements to be hashed. All elements of this
array that are octet strings MUST be multiples of 8 bits.
Parameters:
- octet_scalar_length, non-negative integer. The length of a scalar
octet representation, defined by the ciphersuite.
Outputs:
- result, an octet string or INVALID.
Procedure:
1. let octets_to_hash be an empty octet string.
2. for el in input_array:
3. if el is an ASCII string: el = utf8(el)
4. if el is an octet string representing a public key:
5. el_octs = el
6. else if el is representing a utf8 encoded Ciphersuite ID:
7. el_octs = el
8. else if el is an octet string:
9. if length(el) > 2^64 - 1, return INVALID
10. el_octs = I2OSP(length(el), 8) || el
11. else if el is a Point in G1: el_octs = point_to_octets_g1(el)
12. else if el is a Point in G2: el_octs = point_to_octets_g2(el)
10. else if el is a Scalar: el_octs = I2OSP(el, octet_scalar_length)
11. else if el is a non-negative integer: el_octs = I2OSP(el, 8)
12. else: return INVALID
13. octets_to_hash = octets_to_hash || el_octs
14. return octets_to_hash
¶
It is RECOMENDED for any operation in Core Operations involving public keys, that they deserialize the public key first using the OctetsToPublicKey operation, even if they only require the octet-string representation of the public key. If the octets_to_pubkey procedure (see the OctetsToPublicKey section) returns INVALID, the calling operation should also return INVALID and abort. An example of where this recommendation applies is the Sign operation. An example of where an explicit invocation to the octets_to_pubkey operation is already defined and therefore required is the Verify operation.¶
This document makes use of octet_to_point_g* to parse octet strings to elliptic curve points (either in G1 or G2). It is assumed (even if not explicitly described) that the result of this operation will not be INVALID. If octet_to_point_g* returns INVALID, then the calling operation should immediately return INVALID as well and abort the operation. Note that the only place where the output is assumed to be VALID implicitly is in the EncodingForHash section.¶
Some existing implementations skip the subgroup_check invocation in Verify, whose purpose is ensuring that the signature is an element of a prime-order subgroup. This check is REQUIRED of conforming implementations, for two reasons.¶
For most pairing-friendly elliptic curves used in practice, the pairing operation e Section 1.2 is undefined when its input points are not in the prime-order subgroups of E1 and E2. The resulting behavior is unpredictable, and may enable forgeries.¶
Even if the pairing operation behaves properly on inputs that are outside the correct subgroups, skipping the subgroup check breaks the strong unforgeability property [ADR02].¶
Implementations of the signing algorithm SHOULD protect the secret key from side-channel attacks. One method for protecting against certain side-channel attacks is ensuring that the implementation executes exactly the same sequence of instructions and performs exactly the same memory accesses, for any value of the secret key. In other words, implementations on the underlying pairing-friendly elliptic curve SHOULD run in constant time.¶
The IKM input to KeyGen MUST be infeasible to guess and MUST be kept secret. One possibility is to generate IKM from a trusted source of randomness. Guidelines on constructing such a source are outside the scope of this document.¶
Secret keys MAY be generated using other methods; in this case they MUST be infeasible to guess and MUST be indistinguishable from uniformly random modulo r.¶
BBS proofs are nondeterministic, meaning care must be taken against attacks arising from using bad randomness, for example, the nonce reuse attack on ECDSA [HDWH12]. It is RECOMMENDED that the presentation header used in this specification contain a nonce chosen at random from a trusted source of randomness, see the Section 5.6 for additional considerations.¶
When a trusted source of randomness is used, signatures and proofs are much harder to forge or break due to the use of multiple nonces.¶
The signature proofs of knowledge generated in this specification are created using a specified presentation header. A verifier-specified cryptographically random value (e.g., a nonce) featuring in the presentation header provides strong protections against replay attacks, and is RECOMMENDED in most use cases. In some settings, proofs can be generated in a non-interactive fashion, in which case verifiers MUST be able to verify the uniqueness of the presentation header values.¶
The security analysis models hash_to_curve_g1 as random oracles. It is crucial that these functions are implemented using a cryptographically secure hash function. For this purpose, implementations MUST meet the requirements of [I-D.irtf-cfrg-hash-to-curve].¶
In addition, ciphersuites MUST specify unique domain separation tags for hash_to_curve. Some guidance around defining this can be found in Section 6.¶
BBS signatures can be implemented on any pairing-friendly curve. However care MUST be taken when selecting one that is appropriate, this specification defines a ciphersuite for using the BLS12-381 curve in Section 6 which as a curve achieves around 117 bits of security according to a recent NCC ZCash cryptography review [ZCASH-REVIEW].¶
The proof, as returned by ProofGen, is a zero-knowledge proof-of-knowledge [CDL16]. This guarantees that no information will be revealed about the signature itself or the undisclosed messages, from the output of ProofGen. Note that the security proofs in [CDL16] work on type 3 pairing setting. This means that G1 should be different from G2 and with no efficient isomorphism between them.¶
This section defines the format for a BBS ciphersuite. It also gives concrete ciphersuites based on the BLS12-381 pairing-friendly elliptic curve [I-D.irtf-cfrg-pairing-friendly-curves].¶
The following section defines the format of the unique identifier for the ciphersuite denoted ciphersuite_id. The REQUIRED format for this string is¶
"BBS_" || H2C_SUITE_ID || ADD_INFO¶
H2C_SUITE_ID is the suite ID of the hash-to-curve suite used to define the hashtocurve function.¶
ADD_INFO is an optional string indicating any additional information used to uniquely qualify the ciphersuite. When present this value MUST only contain ASCII characters with codes between 0x21 and 0x7e (inclusive) and MUST end with an underscore (ASCII code: 0x5f), other than the last character the string MUST not contain any other underscores (ASCII code: 0x5f).¶
The parameters that each ciphersuite needs to define are generally divided into three main categories; the basic parameters (a hash function etc.,), the serialization operations (point_to_octets_g1 etc.,) and the generator parameters. See below for more details.¶
Basic parameters:¶
hash: a cryptographic hash function.¶
octet_scalar_length: Number of bytes to represent a scalar value, in the multiplicative group of integers mod r, encoded as an octet string. It is RECOMMENDED this value be set to ceil(log2(r)/8).¶
octet_point_length: Number of bytes to represent a point encoded as an octet string outputted by the point_to_octets_g* function. It is RECOMMENDED that this value is set to ceil(log2(p)/8).¶
hash_to_curve_suite: The hash-to-curve ciphersuite id, in the form defined in [I-D.irtf-cfrg-hash-to-curve]. This defines the hash_to_curve_g1 (the hash_to_curve operation for the G1 subgroup, see the Notation section) and the expand_message (either expand_message_xmd or expand_message_xof) operations used in this document.¶
P1: A fixed point in the G1 subgroup.¶
Serialization functions:¶
point_to_octets_g1: a function that returns the canonical representation of the point P for the G1 subgroup as an octet string.¶
point_to_octets_g2: a function that returns the canonical representation of the point P for the G2 subgroup as an octet string.¶
octets_to_point_g1:
a function that returns the point P in the subgroup G1 corresponding to the canonical representation ostr, or INVALID if ostr is not a valid output of point_to_octets_g1.¶
octets_to_point_g2:
a function that returns the point P in the subgroup G2 corresponding to the canonical representation ostr, or INVALID if ostr is not a valid output of point_to_octets_g2.¶
Generator parameters:¶
The following two ciphersuites are based on the BLS12-381 elliptic curves defined in Section 4.2.1 of [I-D.irtf-cfrg-pairing-friendly-curves]. The targeted security level of both suites in bits is k = 128.¶
The first ciphersuite makes use of an extendable output function, and most specifically of SHAKE-256, as defined in Section 6.2 of [SHA3]. It also uses the hash-to-curve suite defined by this document in Appendix A.1, which also makes use of the SHAKE-256 function.¶
The second ciphersuite uses SHA-256, as defined in Section 6.2 of [SHA2] and the BLS12-381 G1 hash-to-curve suite defined in Section 8.8.1 of the [I-D.irtf-cfrg-hash-to-curve] document.¶
Note that these two ciphersuites differ only in the hash function (SHAKE-256 vs SHA-256) and in the hash-to-curve suites used. The hash-to-curve suites differ in the expand_message variant and underlying hash function. More concretely, the BLS12-381-SHAKE-256 ciphersuite makes use of expand_message_xof with SHAKE-256, while BLS12-381-SHA-256 makes use of expand_message_xmd with SHA-256. Curve parameters are common between the two ciphersuites.¶
Basic parameters:¶
ciphersuite_id: "BBS_BLS12381G1_XOF:SHAKE-256_SSWU_RO_"¶
octet_scalar_length: 32, based on the RECOMMENDED approach of ceil(log2(r)/8).¶
octet_point_length: 48, based on the RECOMMENDED approach of ceil(log2(p)/8).¶
hash_to_curve_suite: "BLS12381G1_XOF:SHAKE-256_SSWU_RO_" as defined in Appendix A.1 for the G1 subgroup.¶
P1: The G1 point returned from the create_generators procedure, with generator_seed = "BBS_BLS12381G1_XOF:SHAKE-256_SSWU_RO_BP_MESSAGE_GENERATOR_SEED". More specifically,¶
P1 = {{ $generatorFixtures.bls12-381-shake-256.generators.BP }}
¶
Serialization functions:¶
point_to_octets_g1: follows the format documented in Appendix C section 1 of [I-D.irtf-cfrg-pairing-friendly-curves] for the G1 subgroup, using compression (i.e., setting C_bit = 1).¶
point_to_octets_g2: follows the format documented in Appendix C section 1 of [I-D.irtf-cfrg-pairing-friendly-curves] for the G2 subgroup, using compression (i.e., setting C_bit = 1).¶
octets_to_point_g1: follows the format documented in Appendix C section 2 of [I-D.irtf-cfrg-pairing-friendly-curves] for the G1 subgroup.¶
octets_to_point_g2: follows the format documented in Appendix C section 2 of [I-D.irtf-cfrg-pairing-friendly-curves] for the G2 subgroup.¶
Generator parameters:¶
Basic parameters:¶
Ciphersuite_ID: "BBS_BLS12381G1_XMD:SHA-256_SSWU_RO_"¶
octet_scalar_length: 32, based on the RECOMMENDED approach of ceil(log2(r)/8).¶
octet_point_length: 48, based on the RECOMMENDED approach of ceil(log2(p)/8).¶
hash_to_curve_suite: "BLS12381G1_XMD:SHA-256_SSWU_RO_" as defined in Section 8.8.1 of the [I-D.irtf-cfrg-hash-to-curve] for the G1 subgroup.¶
P1: The G1 point returned from the create_generators procedure, with generator_seed = "BBS_BLS12381G1_XMD:SHA-256_SSWU_RO_BP_MESSAGE_GENERATOR_SEED". More specifically,¶
P1 = {{ $generatorFixtures.bls12-381-sha-256.generators.BP }}
¶
Serialization functions:¶
point_to_octets_g1: follows the format documented in Appendix C section 1 of [I-D.irtf-cfrg-pairing-friendly-curves] for the G1 subgroup, using compression (i.e., setting C_bit = 1).¶
point_to_octets_g2: follows the format documented in Appendix C section 1 of [I-D.irtf-cfrg-pairing-friendly-curves] for the G2 subgroup, using compression (i.e., setting C_bit = 1).¶
octets_to_point_g1: follows the format documented in Appendix C section 2 of [I-D.irtf-cfrg-pairing-friendly-curves] for the G1 subgroup.¶
octets_to_point_g2: follows the format documented in Appendix C section 2 of [I-D.irtf-cfrg-pairing-friendly-curves] for the G2 subgroup.¶
Generator parameters:¶
The following section details a basic set of test vectors that can be used to confirm an implementations correctness¶
NOTE All binary data below is represented as octet strings encoded in hexadecimal format¶
NOTE These fixtures are a work in progress and subject to change¶
The following key pair will be used for the test vectors of both ciphersuites. Note that it is made based on the BLS12-381-SHA-356 ciphersuite, meaning that it uses SHA-256 as a hash function. Although KeyGen is not REQUIRED for ciphersuite compatibility, it is RECOMMENDED that implementations will NOT re-use keys across different ciphersuites (even if they are based on the same curve).¶
NOTE: this is work in progress and in the future, we may add different key pairs per ciphersuite for the test vectors.¶
Following the procedure defined in Section 3.3.1 with an input IKM value as follows¶
{{ $keyPair.seed }}
¶
Outputs the following SK value¶
{{ $keyPair.keyPair.secretKey }}
¶
Following the procedure defined in Section 3.3.2 with an input SK value as above produces the following PK value¶
{{ $keyPair.keyPair.publicKey }}
¶
The following messages are used by the test vectors of both ciphersuites (unless otherwise stated).¶
{{ $messages[0] }}
{{ $messages[1] }}
{{ $messages[2] }}
{{ $messages[3] }}
{{ $messages[4] }}
{{ $messages[5] }}
{{ $messages[6] }}
{{ $messages[7] }}
{{ $messages[8] }}
{{ $messages[9] }}
¶
Test vectors of the BLS12-381-SHAKE-256 ciphersuite. Further fixtures are available in additional BLS12-381-SHAKE-256 test vectors.¶
The messages in Section 7.2 must be mapped to scalars before passed to the Sign, Verify, ProofGen and ProofVerify operations. For the purpose of the test vectors presented in this document we are using the MapMessageToScalarAsHash operation to map each message to a scalar. For the BLS12-381-SHAKE-256 ciphersuite, on input each message in Section 7.2 and the following default dst¶
{{ $MapMessageToScalarFixtures.bls12-381-shake-256.MapMessageToScalarAsHash.dst }}
¶
The output scalars, encoded to octets using I2OSP and represented in big endian order, are the following,¶
{{ $MapMessageToScalarFixtures.bls12-381-shake-256.MapMessageToScalarAsHash.cases[0].scalar }}
{{ $MapMessageToScalarFixtures.bls12-381-shake-256.MapMessageToScalarAsHash.cases[1].scalar }}
{{ $MapMessageToScalarFixtures.bls12-381-shake-256.MapMessageToScalarAsHash.cases[2].scalar }}
{{ $MapMessageToScalarFixtures.bls12-381-shake-256.MapMessageToScalarAsHash.cases[3].scalar }}
{{ $MapMessageToScalarFixtures.bls12-381-shake-256.MapMessageToScalarAsHash.cases[4].scalar }}
{{ $MapMessageToScalarFixtures.bls12-381-shake-256.MapMessageToScalarAsHash.cases[5].scalar }}
{{ $MapMessageToScalarFixtures.bls12-381-shake-256.MapMessageToScalarAsHash.cases[6].scalar }}
{{ $MapMessageToScalarFixtures.bls12-381-shake-256.MapMessageToScalarAsHash.cases[7].scalar }}
{{ $MapMessageToScalarFixtures.bls12-381-shake-256.MapMessageToScalarAsHash.cases[8].scalar }}
{{ $MapMessageToScalarFixtures.bls12-381-shake-256.MapMessageToScalarAsHash.cases[9].scalar }}
¶
Note that in both the following test vectors, as well as the additional BLS12-381-SHAKE-256 test vectors in Appendix C.1, when we are referring to a message that will be passed to one of the Sign, Verify, ProofGen or ProofVerify operations, we assume that it will first be mapped into one of the above scalars, using the MapMessageToScalarAsHash operation.¶
Following the procedure defined in Section 4.1 with an input count value of 12, for the BLS12-381-SHAKE-256 suite, outputs the following values (note that the first 2 correspond to Q_1 and Q_2, while the next 10, to the message generators H_1, ..., H_10).¶
{{ $generatorFixtures.bls12-381-shake-256.generators.Q1 }}
{{ $generatorFixtures.bls12-381-shake-256.generators.Q2 }}
{{ $generatorFixtures.bls12-381-shake-256.generators.MsgGenerators[0] }}
{{ $generatorFixtures.bls12-381-shake-256.generators.MsgGenerators[1] }}
{{ $generatorFixtures.bls12-381-shake-256.generators.MsgGenerators[2] }}
{{ $generatorFixtures.bls12-381-shake-256.generators.MsgGenerators[3] }}
{{ $generatorFixtures.bls12-381-shake-256.generators.MsgGenerators[4] }}
{{ $generatorFixtures.bls12-381-shake-256.generators.MsgGenerators[5] }}
{{ $generatorFixtures.bls12-381-shake-256.generators.MsgGenerators[6] }}
{{ $generatorFixtures.bls12-381-shake-256.generators.MsgGenerators[7] }}
{{ $generatorFixtures.bls12-381-shake-256.generators.MsgGenerators[8] }}
{{ $generatorFixtures.bls12-381-shake-256.generators.MsgGenerators[9] }}
¶
Using the following header¶
{{ $signatureFixtures.bls12-381-sha-256.signature001.header }}
¶
And the following message (the first message defined in Section 7.2)¶
{{ $signatureFixtures.bls12-381-sha-256.signature001.messages[0] }}
¶
After it is mapped to the first scalar in Section 7.3.1, along with the SK value as defined in Section 7.1 as inputs into the Sign operations, yields the following output signature¶
{{ $signatureFixtures.bls12-381-shake-256.signature001.signature }}
¶
Using the following header¶
{{ $signatureFixtures.bls12-381-shake-256.signature004.header }}
¶
And the messages defined in Section 7.2 (Note the ordering of the messages MUST be preserved), after they are mapped to the scalars in Section 7.3.1, along with the SK value as defined in Section 7.1 as inputs into the Sign operations, yields the following output signature¶
{{ $signatureFixtures.bls12-381-shake-256.signature004.signature }}
¶
Test vectors of the BLS12-381-SHA-256 ciphersuite. Further fixtures are available in additional BLS12-381-SHA-256 test vectors.¶
Similarly to how messages are mapped to scalars in BLS12381-SHAKE-256 Test Vectors, we are using the MapMessageToScalarAsHash operation to map each message to a scalar. For the BLS12-381-SHA-256 ciphersuite, on input each message in Section 7.2 and the following default dst¶
{{ $MapMessageToScalarFixtures.bls12-381-sha-256.MapMessageToScalarAsHash.dst }}
¶
The output scalars, encoded to octets using I2OSP and represented in big endian order, are the following,¶
{{ $MapMessageToScalarFixtures.bls12-381-sha-256.MapMessageToScalarAsHash.cases[0].scalar }}
{{ $MapMessageToScalarFixtures.bls12-381-sha-256.MapMessageToScalarAsHash.cases[1].scalar }}
{{ $MapMessageToScalarFixtures.bls12-381-sha-256.MapMessageToScalarAsHash.cases[2].scalar }}
{{ $MapMessageToScalarFixtures.bls12-381-sha-256.MapMessageToScalarAsHash.cases[3].scalar }}
{{ $MapMessageToScalarFixtures.bls12-381-sha-256.MapMessageToScalarAsHash.cases[4].scalar }}
{{ $MapMessageToScalarFixtures.bls12-381-sha-256.MapMessageToScalarAsHash.cases[5].scalar }}
{{ $MapMessageToScalarFixtures.bls12-381-sha-256.MapMessageToScalarAsHash.cases[6].scalar }}
{{ $MapMessageToScalarFixtures.bls12-381-sha-256.MapMessageToScalarAsHash.cases[7].scalar }}
{{ $MapMessageToScalarFixtures.bls12-381-sha-256.MapMessageToScalarAsHash.cases[8].scalar }}
{{ $MapMessageToScalarFixtures.bls12-381-sha-256.MapMessageToScalarAsHash.cases[9].scalar }}
¶
Note that in both the following test vectors, as well as the additional BLS12-381-SHA-256 test vectors in Appendix C.2, when we are referring to a message that will be passed to one of the Sign, Verify, ProofGen or ProofVerify operations, we assume that it will first be mapped into one of the above scalars, using the MapMessageToScalarAsHash operation.¶
Following the procedure defined in Section 4.1 with an input count value of 12, for the BLS12-381-SHA-256 suite, outputs the following values (note that the first 2 correspond to Q_1 and Q_2, while the next 10, to the message generators H_1, ..., H_10).¶
{{ $generatorFixtures.bls12-381-sha-256.generators.Q1 }}
{{ $generatorFixtures.bls12-381-sha-256.generators.Q2 }}
{{ $generatorFixtures.bls12-381-sha-256.generators.MsgGenerators[0] }}
{{ $generatorFixtures.bls12-381-sha-256.generators.MsgGenerators[1] }}
{{ $generatorFixtures.bls12-381-sha-256.generators.MsgGenerators[2] }}
{{ $generatorFixtures.bls12-381-sha-256.generators.MsgGenerators[3] }}
{{ $generatorFixtures.bls12-381-sha-256.generators.MsgGenerators[4] }}
{{ $generatorFixtures.bls12-381-sha-256.generators.MsgGenerators[5] }}
{{ $generatorFixtures.bls12-381-sha-256.generators.MsgGenerators[6] }}
{{ $generatorFixtures.bls12-381-sha-256.generators.MsgGenerators[7] }}
{{ $generatorFixtures.bls12-381-sha-256.generators.MsgGenerators[8] }}
{{ $generatorFixtures.bls12-381-sha-256.generators.MsgGenerators[9] }}
¶
Using the following header¶
{{ $signatureFixtures.bls12-381-sha-256.signature001.header }}
¶
And the following message (the first message defined in Section 7.2)¶
{{ $signatureFixtures.bls12-381-sha-256.signature001.messages[0] }}
¶
After it is mapped to the first scalar in Section 7.4.1, along with the SK value as defined in Section 7.1 as inputs into the Sign operations, yields the following output signature¶
{{ $signatureFixtures.bls12-381-shake-256.signature001.signature }}
¶
Using the following header¶
{{ $signatureFixtures.bls12-381-shake-256.signature004.header }}
¶
And the messages defined in Section 7.2 (Note the ordering of the messages MUST be preserved), after they are mapped to the scalars in Section 7.4.1, along with the SK value as defined in Section 7.1 as inputs into the Sign operations, yields the following output signature¶
{{ $signatureFixtures.bls12-381-shake-256.signature004.signature }}
¶
This document does not make any requests of IANA.¶
The authors would like to acknowledge the significant amount of academic work that preceeded the development of this document. In particular the original work of [BBS04] which was subsequently developed in [ASM06] and in [CDL16]. This last academic work is the one mostly used by this document.¶
The current state of this document is the product of the work of the Decentralized Identity Foundation Applied Cryptography Working group, which includes numerous active participants. In particular, the following individuals contributed ideas, feedback and wording that influenced this specification:¶
Orie Steele, Christian Paquin, Alessandro Guggino and Tomislav Markovski¶
The following defines a hash_to_curve suite [I-D.irtf-cfrg-hash-to-curve] for the BLS12-381 curve for both the G1 and G2 subgroups using the extendable output function (xof) of SHAKE-256 as per the guidance defined in section 8.9 of [I-D.irtf-cfrg-hash-to-curve].¶
Note the notation used in the below definitions is sourced from [I-D.irtf-cfrg-hash-to-curve].¶
The suite of BLS12381G1_XOF:SHAKE-256_SSWU_RO_ is defined as follows:¶
* encoding type: hash_to_curve (Section 3 of
[@!I-D.irtf-cfrg-hash-to-curve])
* E: y^2 = x^3 + 4
* p: 0x1a0111ea397fe69a4b1ba7b6434bacd764774b84f38512bf6730d2a0f6b0f624
1eabfffeb153ffffb9feffffffffaaab
* r: 0x73eda753299d7d483339d80809a1d80553bda402fffe5bfeffffffff00000001
* m: 1
* k: 128
* expand_message: expand_message_xof (Section 5.3.2 of
[@!I-D.irtf-cfrg-hash-to-curve])
* hash: SHAKE-256
* L: 64
* f: Simplified SWU for AB == 0 (Section 6.6.3 of
[@!I-D.irtf-cfrg-hash-to-curve])
* Z: 11
* E': y'^2 = x'^3 + A' * x' + B', where
- A' = 0x144698a3b8e9433d693a02c96d4982b0ea985383ee66a8d8e8981aef
d881ac98936f8da0e0f97f5cf428082d584c1d
- B' = 0x12e2908d11688030018b12e8753eee3b2016c1f0f24f4070a0b9c14f
cef35ef55a23215a316ceaa5d1cc48e98e172be0
* iso_map: the 11-isogeny map from E' to E given in Appendix E.2 of
[@!I-D.irtf-cfrg-hash-to-curve]
* h_eff: 0xd201000000010001
¶
Note that the h_eff values for this suite are copied from that defined for the BLS12381G1_XMD:SHA-256_SSWU_RO_ suite defined in section 8.8.1 of [I-D.irtf-cfrg-hash-to-curve].¶
An optimized example implementation of the Simplified SWU mapping to the curve E' isogenous to BLS12-381 G1 is given in Appendix F.2 [I-D.irtf-cfrg-hash-to-curve].¶
In the most general sense BBS signatures can be used in any application where a cryptographically secured token is required but correlation caused by usage of the token is un-desirable.¶
For example in protocols like OAuth2.0 the most commonly used form of the access token leverages the JWT format alongside conventional cryptographic primitives such as traditional digital signatures or HMACs. These access tokens are then used by a relying party to prove authority to a resource server during a request. However, because the access token is most commonly sent by value as it was issued by the authorization server (e.g in a bearer style scheme), the access token can act as a source of strong correlation for the relying party. Relevant prior art can be found here.¶
BBS Signatures due to their unique properties removes this source of correlation but maintains the same set of guarantees required by a resource server to validate an access token back to its relevant authority (note that an approach to signing JSON tokens with BBS that may be of relevance is the JWP format and serialization). In the context of a protocol like OAuth2.0 the access token issued by the authorization server would feature a BBS Signature, however instead of the relying party providing this access token as issued, in their request to a resource server, they generate a unique proof from the original access token and include that in the request instead, thus removing this vector of correlation.¶
Bearer based security tokens such as JWT based access tokens used in the OAuth2.0 protocol are a highly popular format for expressing authorization grants. However their usage has several security limitations. Notably a bearer based authorization scheme often has to rely on a secure transport between the authorized party (client) and the resource server to mitigate the potential for a MITM attack or a malicious interception of the access token. The scheme also has to assume a degree of trust in the resource server it is presenting an access token to, particularly when the access token grants more than just access to the target resource server, because in a bearer based authorization scheme, anyone who possesses the access token has authority to what it grants. Bearer based access tokens also suffer from the threat of replay attacks.¶
Improved schemes around authorization protocols often involve adding a layer of proof of cryptographic key possession to the presentation of an access token, which mitigates the deficiencies highlighted above as well as providing a way to detect a replay attack. However, approaches that involve proof of cryptographic key possession such as DPoP (https://datatracker.ietf.org/doc/html/draft-ietf-oauth-dpop-04) suffer from an increase in protocol complexity. A party requesting authorization must pre-generate appropriate key material, share the public portion of this with the authorization server alongside proving possession of the private portion of the key material. The authorization server must also be-able to accommodate receiving this information and validating it.¶
BBS Signatures ofter an alternative model that solves the same problems that proof of cryptographic key possession schemes do for bearer based schemes, but in a way that doesn't introduce new up-front protocol complexity. In the context of a protocol like OAuth2.0 the access token issued by the authorization server would feature a BBS Signature, however instead of the client providing this access token as issued, in their request to a resource server, they generate a unique proof from the original access token and include that in the request instead. Because the access token is not shared in a request to a resource server, attacks such as MITM are mitigated. A resource server also obtains the ability to detect a replay attack by ensuring the proof presented is unique.¶
BBS signatures when applied to the problem space of identity credentials can help to enhance user privacy. For example a digital drivers license that is cryptographically signed with a BBS signature, allows the holder or subject of the license to disclose different claims from their drivers license to different parties. Furthermore, the unlinkable presentations property of proofs generated by the scheme remove an important possible source of correlation for the holder across multiple presentations.¶
NOTE These fixtures are a work in progress and subject to change¶
Using the following header¶
{{ $signatureFixtures.bls12-381-shake-256.signature002.header }}
¶
And the following message (the first message defined in Section 7.2)¶
{{ $signatureFixtures.bls12-381-shake-256.signature002.messages[0] }}
¶
After is mapped to the first scalar in Section 7.3.1, and with the following signature¶
{{ $signatureFixtures.bls12-381-shake-256.signature002.signature }}
¶
Along with the PK value as defined in Section 7.1 as inputs into the Verify operation should fail signature validation due to the message value being different from what was signed¶
Using the following header¶
{{ $signatureFixtures.bls12-381-shake-256.signature003.header }}
¶
And the following messages (the two first messages defined in Section 7.2)¶
{{ $signatureFixtures.bls12-381-shake-256.signature003.messages[0] }}
{{ $signatureFixtures.bls12-381-shake-256.signature003.messages[1] }}
¶
After they are mapped to the first 2 scalars in Section 7.3.1, and with the following signature (which is a signature to only the first of the above two messages)¶
{{ $signatureFixtures.bls12-381-shake-256.signature003.signature }}
¶
Along with the PK value as defined in Section 7.1 as inputs into the Verify operation should fail signature validation due to an additional message being supplied that was not signed.¶
Using the following header¶
{{ $signatureFixtures.bls12-381-shake-256.signature005.header }}
¶
And the following messages (the two first messages defined in Section 7.2)¶
{{ $signatureFixtures.bls12-381-shake-256.signature005.messages[0] }}
{{ $signatureFixtures.bls12-381-shake-256.signature005.messages[1] }}
¶
After they are mapped to the first 2 scalars in Section 7.3.1, and with the following signature (which is a signature on all the messages defined in Section 7.2)¶
{{ $signatureFixtures.bls12-381-shake-256.signature005.signature }}
¶
Along with the PK value as defined in Section 7.1 as inputs into the Verify operation should fail signature validation due to missing messages that were originally present during the signing.¶
Using the following header¶
{{ $signatureFixtures.bls12-381-shake-256.signature006.header }}
¶
And the following messages (re-ordering of the messages defined in Section 7.2)¶
{{ $signatureFixtures.bls12-381-shake-256.signature006.messages[0] }}
{{ $signatureFixtures.bls12-381-shake-256.signature006.messages[1] }}
{{ $signatureFixtures.bls12-381-shake-256.signature006.messages[2] }}
{{ $signatureFixtures.bls12-381-shake-256.signature006.messages[3] }}
{{ $signatureFixtures.bls12-381-shake-256.signature006.messages[4] }}
{{ $signatureFixtures.bls12-381-shake-256.signature006.messages[5] }}
{{ $signatureFixtures.bls12-381-shake-256.signature006.messages[6] }}
{{ $signatureFixtures.bls12-381-shake-256.signature006.messages[7] }}
{{ $signatureFixtures.bls12-381-shake-256.signature006.messages[8] }}
{{ $signatureFixtures.bls12-381-shake-256.signature006.messages[9] }}
¶
After they are mapped to the corresponding scalars in Section 7.3.1, and with the following signature¶
{{ $signatureFixtures.bls12-381-shake-256.signature006.signature }}
¶
Along with the PK value as defined in Section 7.1 as inputs into the Verify operation should fail signature validation due to messages being re-ordered from the order in which they were signed¶
Using the following header¶
{{ $signatureFixtures.bls12-381-shake-256.signature007.header }}
¶
And the messages as defined in Section 7.2, mapped to the scalars in Section 7.3.1 and with the following signature¶
{{ $signatureFixtures.bls12-381-shake-256.signature007.signature }}
¶
Along with the PK value as defined in Section 7.1 as inputs into the Verify operation should fail signature validation due to public key used to verify is in-correct¶
Using the following header¶
{{ $signatureFixtures.bls12-381-shake-256.signature008.header }}
¶
And the messages as defined in Section 7.2, mapped to the scalars in Section 7.3.1 and with the following signature¶
{{ $signatureFixtures.bls12-381-shake-256.signature008.signature }}
¶
Along with the PK value as defined in Section 7.1 as inputs into the Verify operation should fail signature validation due to header value being modified from what was originally signed¶
Using the following input message,¶
{{ $H2sFixture.bls12-381-shake-256.h2s001.message }}
¶
And the default dst defined in hash-to-scalar, i.e.,¶
{{ $H2sFixture.bls12-381-shake-256.h2s001.dst }}
¶
With an output count of 1, we get the following scalar, encoded with I2OSP and represented in big endian order,¶
{{ $H2sFixture.bls12-381-shake-256.h2s001.scalars[0] }}
¶
With the same input message and dst but with an output count of 10 we get the following scalars (again encoded with I2OSP and represented in big endian order),¶
{{ $H2sFixture.bls12-381-shake-256.h2s002.scalars[0] }}
{{ $H2sFixture.bls12-381-shake-256.h2s002.scalars[1] }}
{{ $H2sFixture.bls12-381-shake-256.h2s002.scalars[2] }}
{{ $H2sFixture.bls12-381-shake-256.h2s002.scalars[3] }}
{{ $H2sFixture.bls12-381-shake-256.h2s002.scalars[4] }}
{{ $H2sFixture.bls12-381-shake-256.h2s002.scalars[5] }}
{{ $H2sFixture.bls12-381-shake-256.h2s002.scalars[6] }}
{{ $H2sFixture.bls12-381-shake-256.h2s002.scalars[7] }}
{{ $H2sFixture.bls12-381-shake-256.h2s002.scalars[8] }}
{{ $H2sFixture.bls12-381-shake-256.h2s002.scalars[9] }}
¶
Using the following header¶
{{ $signatureFixtures.bls12-381-sha-256.signature002.header }}
¶
And the following message (the first message defined in Section 7.2)¶
{{ $signatureFixtures.bls12-381-sha-256.signature002.messages[0] }}
¶
After is maped to the first scalar in Section 7.4.1, and with the following signature¶
{{ $signatureFixtures.bls12-381-sha-256.signature002.signature }}
¶
Along with the PK value as defined in Section 7.1 as inputs into the Verify operation should fail signature validation due to the message value being different from what was signed.¶
Using the following header¶
{{ $signatureFixtures.bls12-381-sha-256.signature003.header }}
¶
And the following messages (the two first messages defined in Section 7.2)¶
{{ $signatureFixtures.bls12-381-sha-256.signature003.messages[0] }}
{{ $signatureFixtures.bls12-381-sha-256.signature003.messages[1] }}
¶
After they are mapped to the first 2 scalars in Section 7.4.1, and with the following signature (which is a signature to only the first of the above two messages)¶
{{ $signatureFixtures.bls12-381-sha-256.signature003.signature }}
¶
Along with the PK value as defined in Section 7.1 as inputs into the Verify operation should fail signature validation due to an additional message being supplied that was not signed.¶
Using the following header¶
{{ $signatureFixtures.bls12-381-sha-256.signature005.header }}
¶
And the following messages (the two first messages defined in Section 7.2)¶
{{ $signatureFixtures.bls12-381-sha-256.signature005.messages[0] }}
{{ $signatureFixtures.bls12-381-sha-256.signature005.messages[1] }}
¶
After they are mapped to the first 2 scalars in Section 7.4.1, and with the following signature (which is a signature on all the messages defined in Section 7.2)¶
{{ $signatureFixtures.bls12-381-sha-256.signature005.signature }}
¶
Along with the PK value as defined in Section 7.1 as inputs into the Verify operation should fail signature validation due to missing messages that were originally present during the signing.¶
Using the following header¶
{{ $signatureFixtures.bls12-381-sha-256.signature006.header }}
¶
And the following messages (re-ordering of the messages defined in Section 7.2)¶
{{ $signatureFixtures.bls12-381-sha-256.signature006.messages[0] }}
{{ $signatureFixtures.bls12-381-sha-256.signature006.messages[1] }}
{{ $signatureFixtures.bls12-381-sha-256.signature006.messages[2] }}
{{ $signatureFixtures.bls12-381-sha-256.signature006.messages[3] }}
{{ $signatureFixtures.bls12-381-sha-256.signature006.messages[4] }}
{{ $signatureFixtures.bls12-381-sha-256.signature006.messages[5] }}
{{ $signatureFixtures.bls12-381-sha-256.signature006.messages[6] }}
{{ $signatureFixtures.bls12-381-sha-256.signature006.messages[7] }}
{{ $signatureFixtures.bls12-381-sha-256.signature006.messages[8] }}
{{ $signatureFixtures.bls12-381-sha-256.signature006.messages[9] }}
¶
After they are mapped to the corresponding scalars in Section 7.4.1, and with the following signature¶
{{ $signatureFixtures.bls12-381-sha-256.signature006.signature }}
¶
Along with the PK value as defined in Section 7.1 as inputs into the Verify operation should fail signature validation due to messages being re-ordered from the order in which they were signed.¶
Using the following header¶
{{ $signatureFixtures.bls12-381-sha-256.signature007.header }}
¶
And the messages as defined in Section 7.2, mapped to the scalars in Section 7.4.1 and with the following signature¶
{{ $signatureFixtures.bls12-381-sha-256.signature007.signature }}
¶
Along with the PK value as defined in Section 7.1 as inputs into the Verify operation should fail signature validation due to public key used to verify is in-correct.¶
Using the following header¶
{{ $signatureFixtures.bls12-381-sha-256.signature008.header }}
¶
And the messages as defined in Section 7.2, mapped to the scalars in Section 7.4.1 and with the following signature¶
{{ $signatureFixtures.bls12-381-sha-256.signature008.signature }}
¶
Along with the PK value as defined in Section 7.1 as inputs into the Verify operation should fail signature validation due to header value being modified from what was originally signed.¶
Using the following input message,¶
{{ $H2sFixture.bls12-381-sha-256.h2s001.message }}
¶
And the default dst defined in hash-to-scalar, i.e.,¶
{{ $H2sFixture.bls12-381-sha-256.h2s001.dst }}
¶
With an output count of 1, we get the following scalar, encoded with I2OSP and represented in big endian order,¶
{{ $H2sFixture.bls12-381-sha-256.h2s001.scalars[0] }}
¶
With the same input message and dst but with an output count of 10 we get the following scalars (again encoded with I2OSP and represented in big endian order),¶
{{ $H2sFixture.bls12-381-sha-256.h2s002.scalars[0] }}
{{ $H2sFixture.bls12-381-sha-256.h2s002.scalars[1] }}
{{ $H2sFixture.bls12-381-sha-256.h2s002.scalars[2] }}
{{ $H2sFixture.bls12-381-sha-256.h2s002.scalars[3] }}
{{ $H2sFixture.bls12-381-sha-256.h2s002.scalars[4] }}
{{ $H2sFixture.bls12-381-sha-256.h2s002.scalars[5] }}
{{ $H2sFixture.bls12-381-sha-256.h2s002.scalars[6] }}
{{ $H2sFixture.bls12-381-sha-256.h2s002.scalars[7] }}
{{ $H2sFixture.bls12-381-sha-256.h2s002.scalars[8] }}
{{ $H2sFixture.bls12-381-sha-256.h2s002.scalars[9] }}
¶
The following section provides an explanation of how the ProofGen and ProofVerify operations work.¶
Let the prover be in possession of a BBS signature (A, e, s) on messages msg_1, ..., msg_L and a domain value (see Sign). Let A = B * (1/(e + SK)) where SK the signer's secret key and,¶
B = P1 + Q_1 * s + Q_2 * domain + H_1 * msg_1 + ... + H_L * msg_L¶
Let (i1, ..., iR) be the indexes of generators corresponding to messages the prover wants to disclose and (j1, ..., jU) be the indexes corresponding to undisclosed messages (i.e., (j1, ..., jU) = range(1, L) \ (i1, ..., iR)). To prove knowledge of a signature on the disclosed messages, work as follows,¶
Hide the signature by randomizing it. To randomize the signature (A, e, s), take uniformly random r1, r2 in [1, r-1], and calculate,¶
1. A' = A * r1, 2. Abar = A' * (-e) + B * r1 3. D = B * r1 + Q_1 * r2.¶
Also set,¶
4. r3 = r1 ^ -1 mod r 5. s' = r2 * r3 + s mod r.¶
The values (A', Abar, D) will be part of the proof and are used to prove possession of a BBS signature, without revealing the signature itself. Note that; e(A', PK) = e(Abar, P2) where PK the signer's public key and P2 the base element in G2 (used to create the signer's PK, see SkToPk). This also serves to bind the proof to the signer's PK.¶
Set the following,¶
1. C1 = Abar - D 2. C2 = P1 + Q_2 * domain + H_i1 * msg_i1 + ... + H_iR * msg_iR¶
Create a non-interactive zero-knowledge proof-of-knowledge (nizk) of the values e, r2, r3, s' and msg_j1, ..., msg_jU (the undisclosed messages) so that both of the following equalities hold,¶
EQ1. C1 = A' * (-e) - Q_1 * r2 EQ2. C2 = Q_1 * s' - D * r3 + H_j1 * msg_j1 + ... + H_jU * msg_jU.¶
Note that the verifier will know the elements in the left side of the above equations (i.e., C1 and C2) but not in the right side (i.e., s', r3 and the undisclosed messages: msg_j1, ..., msg_jU). However, using the nizk, the prover can convince the verifier that they (the prover) know the elements that satisfy those equations, without disclosing them. Then, if both EQ1 and EQ2 hold, and e(A', PK) = e(Abar, P2), an extractor can return a valid BBS signature from the signer's SK, on the disclosed messages. The proof returned is (A', Abar, D, nizk). To validate the proof, a verifier checks that e(A', PK) = e(Abar, P2) and verifies the nizk. Validating the proof, will guarantee the authenticity and integrity of the disclosed messages, as well as ownership of the undisclosed messages and of the signature.¶