The BBS Signature Scheme

1.1. Terminology

The following terminology is used throughout this document:¶

SK

The secret key for the signature scheme.¶

PK

The public key for the signature scheme.¶

L

The total number of signed messages.¶

R

The number of message indexes that are disclosed (revealed) in a proof-of-knowledge of a signature.¶

U

The number of message indexes that are undisclosed in a proof-of-knowledge of a signature.¶

msg

An input message to be signed by the signature scheme.¶

generator

A valid point on the selected subgroup of the curve being used that is employed to commit a value.¶

signature

The digital signature output.¶

nonce

A cryptographic nonce¶

presentation_header (ph)

A payload generated and bound to the context of a specific spk.¶

nizk

A non-interactive zero-knowledge proof from fiat-shamir heuristic.¶

dst

The domain separation tag.¶

I2OSP

As defined by Section 4 of [RFC8017]¶

OS2IP

As defined by Section 4 of [RFC8017].¶

1.2. Notation

The following notation and primitives are used:¶

a || b

Denotes the concatenation of octet strings a and b.¶

I \ J

For sets I and J, denotes the difference of the two sets i.e., all the elements of I that do not appear in J, in the same order as they were in I.¶

X[a..b]

Denotes a slice of the array 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.¶

range(a, b)

For integers a and b, with a <= b, denotes the ascending ordered list of all integers between a and b inclusive (i.e., the integers "i" such that a <= i <= b).¶

utf8(ascii_string)

Encoding the inputted ASCII string to an octet string using UTF-8 character encoding.¶

length(input)

Takes as input either an array or an octet string. If the input is an array, returns the number of elements of the array. If the input is an octet string, returns the number of bytes of the inputted octet string.¶

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]¶

E1, E2

elliptic curve groups defined over finite fields. This document assumes that E1 has a more compact representation than E2, i.e., because E1 is defined over a smaller field than E2.¶

G1, G2

subgroups of E1 and E2 (respectively) having prime order r.¶

GT

a subgroup, of prime order r, of the multiplicative group of a field extension.¶

e

G1 x G2 -> GT: a non-degenerate bilinear map.¶

r

The prime order of the G1 and G2 subgroups.¶

P1, P2

points on G1 and G2 respectively. For a pairing-friendly curve, this document denotes operations in E1 and E2 in additive notation, i.e., P + Q denotes point addition and x * P denotes scalar multiplication. Operations in GT are written in multiplicative notation, i.e., a * b is field multiplication.¶

Identity_G1, Identity_G2, Identity_GT

The identity element for the G1, G2, and GT subgroups respectively.¶

hash_to_curve_g1(ostr, dst) -> P

A cryptographic hash function that takes an arbitrary octet string as input and returns a point in G1, using the hash_to_curve operation defined in [I-D.irtf-cfrg-hash-to-curve] and the inputted dst as the domain separation tag for that operation (more specifically, the inputted dst will become the DST parameter for the hash_to_field operation, called by hash_to_curve).¶

point_to_octets_g1(P) -> ostr, point_to_octets_g2(P) -> ostr

returns the canonical representation of the point P for the respective subgroup as an octet string. This operation is also known as serialization.¶

octets_to_point_g1(ostr) -> P, octets_to_point_g2(ostr) -> P

returns the point P for the respective subgroup corresponding to the canonical representation ostr, or INVALID if ostr is not a valid output of the respective point_to_octets_g* function. This operation is also known as deserialization.¶

subgroup_check(P) -> VALID or INVALID

returns VALID when the point P is an element of the subgroup of order r, and INVALID otherwise. This function can always be implemented by checking that r * P is equal to the identity element. In some cases, faster checks may also exist, e.g., [Bowe19].¶

1.3. Organization of this document

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.¶

3.1. Parameters

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.¶

3.2. Considerations

3.3. Key Generation Operations

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

PK = SkToPk(SK)

Inputs:

- SK (REQUIRED), a secret integer such that 0 < SK < r.

Outputs:

- PK, a public key encoded as an octet string.

Procedure:

1. W = SK * P2
2. return point_to_octets_g2(W)

3.4. Core Operations

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.¶

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.
- generator_seed, ASCII string. The generators seed 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(generator_seed, 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

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.
- generator_seed, ASCII string. The generators seed 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(generator_seed, 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

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.
- generator_seed, ASCII string. The generators seed 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(generator_seed, 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)

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.
- generator_seed, ASCII string. The generators seed 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(generator_seed, 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

4.1. Generator point computation

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.

Definitions:

- seed_dst, the octet string representing the ASCII encoded characters:
           "BBS_" || hash_to_curve_suite || "SIG_GENERATOR_SEED_".
- generator_dst, the octet string representing:
                 "BBS_" || hash_to_curve_suite || "SIG_GENERATOR_DST_",
                 in the ASCII characters encoding.
- 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 (P1, generator_1, ..., generator_i):
9.        go back to step 4
10.    generator_i = candidate
11. return (generator_1, ..., generator_count)

4.2. MapMessageToScalar

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.¶

result = MapMessageToScalarAsHash(msg, dst)

Inputs:

- msg (REQUIRED), octet string.
- dst (REQUIRED), octet string. Domain separation tag; note this is not
                  defined as a function argument as per
                  [@!I-D.irtf-cfrg-hash-to-curve] but as a parameter.

Outputs:

- result, a scalar value.

Procedure:

1. If length(dst) > 2^8 - 1 or length(msg) > 2^64 - 1, return INVALID
2. dst_prime = I2OSP(length(dst), 1) || dst
3. msg_prime = I2OSP(length(msg), 8) || msg
4. result = hash_to_scalar(msg_prime || dst_prime, 1)
5. return result

4.3. Hash to Scalar

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). Note that, 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). However, the hash_to_curve ciphersuites used by this document, make use of hash_to_field with the target group being the multiplicative group of integers mod p (Fp). For completeness, we define here the operation making use of the expand_message function, that will be defined by the hash-to-curve suite used. 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).¶

scalars = hash_to_scalar(msg_octets, count)

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.

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:

- h2s_dst, the octet string representing the ASCII encoded characters:
           "BBS_" || hash_to_curve_suite || "HASH_TO_SCALAR_".
- 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, h2s_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)

4.4. Serialization

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)

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)

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)

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

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

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: el_octs = el
5.      else if el is an octet string:
6.          if length(el) > 2^64 - 1, return INVALID
7.          el_octs = I2OSP(length(el), 8) || el
8.      else if el is a Point in G1: el_octs = point_to_octets_g1(el)
9.      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

5.1. Validating public keys

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.¶

5.2. Point de-serialization

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.¶

5.3. Skipping membership checks

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.¶

1. 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.¶

2. 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].¶

5.4. Side channel attacks

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.¶

5.5. Randomness considerations

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.¶

5.6. Presentation header selection

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.¶

5.7. Implementing hash_to_curve_g1

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.¶

5.8. Choice of underlying curve

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].¶

5.9. Security of proofs generated by ProofGen

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.¶

6.1. Ciphersuite Format

  "BBS_" || H2C_SUITE_ID || ADD_INFO

6.2. BLS12-381 Ciphersuite

The following ciphersuite is based on the BLS12-381 elliptic curve defined in Section 4.2.1 of [I-D.irtf-cfrg-pairing-friendly-curves]. The targeted security level of the suite in bits is k = 128. The 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.¶

Basic parameters:¶

• Ciphersuite_ID: "BBS_BLS12381G1_XOF:SHAKE-256_SSWU_RO_"¶

• hash: SHAKE-256 as defined in [SHA3].¶

• 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_R0_" as defined in Appendix A.1 for the G1 subgroup.¶

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:¶

• generator_seed: A global seed value of "BBS_BLS12381G1_XOF:SHAKE-256_SSWU_RO_MESSAGE_GENERATOR_SEED" which is used by the create_generators operation to compute the required set of message generators.¶

BBS_BLS12381G1_XOF:SHAKE-256_SSWU_RO_MESSAGE_GENERATOR_SEED

BBS_BLS12381G1_XOF:SHAKE-256_SSWU_RO_

{{ $generators[0] }}

{{ $generators[1] }}

{{ $generators[2] }}

{{ $generators[3] }}

{{ $generators[4] }}

{{ $generators[5] }}

{{ $generators[6] }}

{{ $generators[7] }}

{{ $generators[8] }}

{{ $generators[9] }}

{{ $keyPair.seed }}

{{ $keyPair.keyPair.secretKey }}

{{ $keyPair.keyPair.publicKey }}

{{ $signatureFixtures.signature001.messages[0] }}

{{ $signatureFixtures.signature001.signature }}

{{ $signatureFixtures.signature004.messages[0] }}

{{ $signatureFixtures.signature004.messages[1] }}

{{ $signatureFixtures.signature004.messages[2] }}

{{ $signatureFixtures.signature004.messages[3] }}

{{ $signatureFixtures.signature004.messages[4] }}

{{ $signatureFixtures.signature004.messages[5] }}

{{ $signatureFixtures.signature004.messages[6] }}

{{ $signatureFixtures.signature004.messages[7] }}

{{ $signatureFixtures.signature004.messages[8] }}

{{ $signatureFixtures.signature004.messages[9] }}

{{ $signatureFixtures.signature004.signature }}

A.1. BLS12-381 G1

The suite of BLS12381G1_XOF:SHAKE-256_SSWU_R0_ 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].¶

B.1. Non-correlating Security Token

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.¶

B.2. Improved Bearer Security Token

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.¶

B.3. Selectively Disclosure Enabled Identity Credentials

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.¶

C.1. Modified Message Signature

Using the following message¶

{{ $signatureFixtures.signature002.messages[0] }}

And the following signature¶

{{ $signatureFixtures.signature002.signature }}

Along with the PK value as defined in Section 6.2.1.2 as inputs into the Verify operation should fail signature validation due to the message value being different from what was signed¶

C.2. Extra Unsigned Message Signature

Using the following messages¶

{{ $signatureFixtures.signature003.messages[0] }}

{{ $signatureFixtures.signature003.messages[1] }}

And the following signature¶

{{ $signatureFixtures.signature002.signature }}

Along with the PK value as defined in Section 6.2.1.2 as inputs into the Verify operation should fail signature validation due to an additional message being supplied that was not signed¶

C.3. Missing Message Signature

Using the following messages¶

{{ $signatureFixtures.signature005.messages[0] }}

{{ $signatureFixtures.signature005.messages[1] }}

And the following signature¶

{{ $signatureFixtures.signature005.signature }}

Along with the PK value as defined in Section 6.2.1.2 as inputs into the Verify operation should fail signature validation due to missing messages that were originally present during the signing¶

C.4. Reordered Message Signature

Using the following messages¶

{{ $signatureFixtures.signature006.messages[0] }}

{{ $signatureFixtures.signature006.messages[1] }}

{{ $signatureFixtures.signature006.messages[2] }}

{{ $signatureFixtures.signature006.messages[3] }}

{{ $signatureFixtures.signature006.messages[4] }}

{{ $signatureFixtures.signature006.messages[5] }}

{{ $signatureFixtures.signature006.messages[6] }}

{{ $signatureFixtures.signature006.messages[7] }}

{{ $signatureFixtures.signature006.messages[8] }}

{{ $signatureFixtures.signature006.messages[9] }}

And the following signature¶

{{ $signatureFixtures.signature006.signature }}

Along with the PK value as defined in Section 6.2.1.2 as inputs into the Verify operation should fail signature validation due to messages being re-ordered from the order in which they were signed¶

C.5. Wrong Public Key Signature

Using the following messages¶

{{ $signatureFixtures.signature007.messages[0] }}

{{ $signatureFixtures.signature007.messages[1] }}

{{ $signatureFixtures.signature007.messages[2] }}

{{ $signatureFixtures.signature007.messages[3] }}

{{ $signatureFixtures.signature007.messages[4] }}

{{ $signatureFixtures.signature007.messages[5] }}

{{ $signatureFixtures.signature007.messages[6] }}

{{ $signatureFixtures.signature007.messages[7] }}

{{ $signatureFixtures.signature007.messages[8] }}

{{ $signatureFixtures.signature007.messages[9] }}

And the following signature¶

{{ $signatureFixtures.signature007.signature }}

Along with the PK value as defined in Section 6.2.1.2 as inputs into the Verify operation should fail signature validation due to public key used to verify is in-correct¶