Blind BBS Signatures

Introduction The BBS digital signature scheme, as defined in , can be extended to support blind signatures functionality. In a blind signatures setting, the user (called the Prover in the context of the BBS scheme) will request a signature on a list of messages, without revealing those messages to the Signer (who can optionally also include messages of their choosing to the signature). By allowing the Prover to acquire a valid signature over messages not known to the Signer, blind signatures address some limitations of their plain digital signature counterparts. In the BBS scheme, knowledge of a valid signature allows generation of BBS proofs. As a result, a signature compromise (by an eavesdropper, a phishing attack, a leakage of the Signer's logs etc.,) can lead to impersonation of the Prover by malicious actors (especially in cases involving "long-lived" signatures, as in digital credentials applications etc.,). Using Blind BBS Signatures on the other hand, the Prover can commit to a secret message (for example, a private key) before issuance, guaranteeing that no one will be able to generate a valid proof without knowledge of their secret. Furthermore, applications like Privacy Pass () may require a signature to be "scoped" to a specific audience or session (as to require "fresh" signatures for different sessions etc.,). However, simply sending an audience or session identifier to the Signer (to be included in the signature), will compromise the privacy guarantees that these applications try to enforce. Using blind signing, the Prover will be able to require signatures bound to those values, without having to reveal them to the Signer. The presented protocol, compared to the scheme defined in , introduces an additional communication step between the Prover and the Signer. The Prover will start by constructing a "hiding" commitment to the messages they want to get a signature on (i.e., a commitment which reveals no information about the committed values), together with a proof of correctness of that commitment. They will send the (commitment, proof) pair to the Signer, who, upon receiving the pair, will attempt to verify the commitment's proof of correctness. If successful, they will use it in generating a BBS signature over the messages committed by the Prover, including their own messages if any. This document, in addition to defining the operation for creating and verifying a commitment, also details a core signature generation operation, different from the one presented in , meant to handle the computation of the blind signature. The document will also define a new BBS Interface, which is needed to handle the different inputs, i.e., messages committed by the Prover or chosen by the Signer etc... The signature verification and proof generation core cryptographic operations however, will work as described in . To further facilitate deployment, both the exposed interface as well as the core cryptographic operation of proof verification will be the same as the one detailed in . Below is a basic diagram describing the main entities involved in the scheme.

Basic diagram capturing the main entities involved in using the scheme. (3) Blind Sign (1) Commit +----- +----- | | | | | | | | | \ / | \ / +----------+ +-----------+ | | | | | | | | | |<-(2)* Commitment + Proof of Correctness--| | | Signer | | Prover | | |-------(4)* Send signature + msgs-------->| | | | | | | | | | +----------+ +-----------+ | | | (5)* Send proof + disclosed msgs | | \ / +-----------+ | | | | | | | Verifier | | | | | | | +-----------+ | / \ | | | | +----- (6) ProofVerify Note The protocols implied by the items annotated by an asterisk are out of scope for this specification

Terminology Terminology defined by applies to this draft. Additionally, the following terminology is used throughout this document:

blind_signature: The blind digital signature output.
commitment: A point of G1, representing a Pedersen commitment () constructed over a vector of messages, as described e.g., in .
committed_messages: A list of messages committed by the Prover to a commitment.
commitment_proof: A zero knowledge proof of correctness of a commitment, consisting of a scalar value, a possibly empty set of scalars (of length equal to the number of committed_messages, see above) and another scalar, in that order.
secret_prover_blind: A random scalar used to blind (i.e., randomize) the commitment constructed by the prover.
signer_blind: A random scalar used by the signer to optionally re-blind the received commitment.

Notation Notation defined by applies to this draft. Additionally, the following notation and primitives are used:

list.append(elements): Append either a single element or a list of elements to the end of a list, maintaining the same order of the list's elements as well as the appended elements. For example, given list = [a, b, c] and elements = [d, a], the result of list.append(elements) will be [a, b, c, d, a].

Conventions The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT", "SHOULD", "SHOULD NOT", "RECOMMENDED", "NOT RECOMMENDED", "MAY", and "OPTIONAL" in this document are to be interpreted as described in BCP 14 when, and only when, they appear in all capitals, as shown here.

BBS Signature Scheme Operations This document makes use of various operations defined by the BBS Signature Scheme document . For clarity, whenever an operation will be used defined in , it will be prefixed by "BBS." (e.g., "BBS.CoreProofGen" etc.). More specifically, the operations used are the following:

BBS.CoreVerify: Refers to the CoreVerify operation defined in Section 3.6.2 of .
BBS.CoreProofGen: Refers to the CoreProofGen operation defined in Section 3.6.3 of .
BBS.create_generators: Refers to the create_generators operation defined in Section 4.1.1 of .
BBS.messages_to_scalars: Refers to the messages_to_scalars operation defined in Section 4.1.2 of .
BBS.get_random_scalars: Refers to the get_random_scalars operation defined in Section 4.2.1 of .
BBS.hash_to_scalar: Refers to the hash_to_scalar operation defined in Section 4.2.2 of .

Scheme Definition

Commitment Operations

Commitment Computation This operation is used by the Prover to create a commitment to a set of messages (committed_messages), that they intend to include in the blind signature. Note that this operation returns both the serialized combination of the commitment and its proof of correctness (commitment_with_proof), as well as the random scalar used to blind the commitment (secret_prover_blind). (commitment_with_proof, secret_prover_blind) = commit( committed_messages, api_id) Inputs: - committed_messages (OPTIONAL), a vector of octet strings. If not supplied it defaults to the empty array ("()"). - api_id (OPTIONAL), octet string. If not supplied it defaults to the empty octet string (""). Outputs: - (commitment_with_proof, secret_prover_blind), a tuple comprising from an octet string and a random scalar in that order. Procedure: 1. committed_message_scalars = BBS.messages_to_scalars( committed_messages, api_id) 2. blind_generators = BBS.create_generators( length(committed_message_scalars) + 1, "BLIND_" || api_id) 3. return CoreCommit(committed_message_scalars, blind_generators, api_id)

Commitment Validation and Deserialization The following is a helper operation used by the BlindSign procedure () to validate an optional commitment. The commitment input to BlindSign is optional. If a commitment is not supplied, or if it is the Identity_G1, the following operation will return the Identity_G1 as the "default" commitment point, which will be ignored by all computations during BlindSign. commit = deserialize_and_validate_commit(commitment_with_proof, blind_generators, api_id) Inputs: - commitment_with_proof (OPTIONAL), octet string. If it is not supplied it defaults to the empty octet string (""). - blind_generators (OPTIONAL), vector of points of G1. If it is not supplied it defaults to the empty set ("()"). - api_id (OPTIONAL), octet string. If not supplied it defaults to the empty octet string (""). Outputs: - commit, a point of G1; or INVALID. Procedure: 1. if commitment_with_proof is the empty string (""), return Identity_G1 2. com_res = octets_to_commitment_with_proof(commitment_with_proof) 3. if com_res is INVALID, return INVALID 4. (commit, commit_proof) = com_res 5. if length(commit_proof[1]) + 1 != length(blind_generators), return INVALID 6. validation_res = CoreCommitVerify(commit, commit_proof, blind_generators, api_id) 7. if validation_res is INVALID, return INVALID 8. commitment

Blind BBS Signatures Interface The following section defines a BBS Interface for blind BBS signatures. The identifier of the Interface is defined as ciphersuite_id || BLIND_H2G_HM2S_, where ciphersuite_id the unique identifier of the BBS ciphersuite used, as is defined in Section 6 of ). Each BBS Interface MUST define operations to map the input messages to scalar values and to create the generator set, required by the core operations. The input messages to the defined Interface will be mapped to scalars using the messages_to_scalars operation defined in Section 4.1.2 of . The generators will be created using the create_generators operation defined in Section 4.1.1 of . Other than the BlindSign operation defined in , which uses the FinalizeBlindSign procedure, defined in , all other interface operations defined in this section use the core operations defined in Section 3.6 of .

Blind Signature Generation This operation returns a BBS blind signature from a secret key (SK), over a header, a set of messages and optionally a commitment value (see ). If supplied, the commitment value must be accompanied by its proof of correctness (commitment_with_proof, as outputted by the Commit operation defined in ). The issuer can also further randomize the supplied commitment, by supplying a random scalar (signer_blind), that MUST be computed as, signer_blind = BBS.get_random_scalars(1) If the signer_blind input is not supplied, it will default to the zero scalar (0). The BlindSign operation makes use of the FinalizeBlindSign procedure defined in . blind_signature = BlindSign(SK, PK, commitment_with_proof, header, messages, signer_blind) Inputs: - SK (REQUIRED), a secret key in the form outputted by the KeyGen operation. - PK (REQUIRED), an octet string of the form outputted by SkToPk provided the above SK as input. - commitment_with_proof (OPTIONAL), an octet string, representing a serialized commitment and commitment_proof, as the first element outputted by the Commit operation. If not supplied, it defaults to the empty string (""). - 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 octet strings. If not supplied, it defaults to the empty array ("()"). - signer_blind (OPTIONAL), a random scalar value. If not supplied it defaults to zero ("0"). Parameters: - api_id, the octet string ciphersuite_id || "BLIND_H2G_HM2S_", where ciphersuite_id is defined by the ciphersuite and "BLIND_H2G_HM2S_"is an ASCII string composed of 15 bytes. - (octet_point_length, octet_scalar_length), defined by the ciphersuite. Outputs: - blind_signature, a blind signature encoded as an octet string; or INVALID. Deserialization: 1. L = length(messages) // calculate the number of blind generators used by the commitment, // if any. 2. M = length(commitment_with_proof) 3. if M != 0, M = M - octet_point_length - octet_scalar_length 4. M = M / octet_scalar_length 5. if M < 0, return INVALID Procedure: 1. generators = BBS.create_generators(L + 1, api_id) 2. blind_generators = BBS.create_generators(M, "BLIND_" || api_id) 3. commit = deserialize_and_validate_commit(commitment_with_proof, blind_generators, api_id) 4. if commit is INVALID, return INVALID 5. (msg_1, ..., msg_L) = BBS.messages_to_scalars(messages, api_id) 6. B = commit + P1 + Q_1 * msg_1 + ... + Q_L * msg_L 6. res = B_calculate(messages, generators, commit) 7. if res is INVALID, return INVALID 8. (B) = res 4. blind_sig = FinalizeBlindSign(SK, PK, B, generators, blind_generators, header, api_id) 5. if blind_sig is INVALID, return INVALID 6. return blind_sig

Blind Signature Verification This operation validates a blind BBS signature (signature), given the Signer's public key (PK), a header (header), a set of, known to the Signer, messages (messages) and if used, a set of committed messages (committed_messages), the secret_prover_blind as returned by the Commit operation () and a blind factor supplied by the Signer (signer_blind). This operation makes use of the CoreVerify operation as defined in Section 3.6.2 of . result = Verify(PK, signature, header, messages, committed_messages, secret_prover_blind) 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 octet strings. If not supplied, it defaults to the empty array "()". - committed_messages (OPTIONAL), a vector of octet strings. If not supplied, it defaults to the empty array "()". - secret_prover_blind (OPTIONAL), a scalar value. If not supplied it defaults to zero "0". Parameters: - api_id, the octet string ciphersuite_id || "BLIND_H2G_HM2S_", where ciphersuite_id is defined by the ciphersuite and "BLIND_H2G_HM2S_"is an ASCII string composed of 15 bytes. Outputs: - result: either VALID or INVALID Procedure: 1. message_scalars = BBS.messages_to_scalars(messages, api_id) 2. committed_message_scalars = () 4. committed_message_scalars.append(secret_prover_blind) 5. committed_message_scalars.append(BBS.messages_to_scalars( committed_messages, api_id)) 6. generators = BBS.create_generators(length(message_scalars) + 1, api_id) 7. blind_generators = BBS.create_generators(length(committed_message_scalars), "BLIND_" || api_id) 8. res = BBS.CoreVerify( PK, signature, generators.append(blind_generators), header, message_scalars.append(committed_message_scalars), api_id) 9. return res

Proof Generation This operation creates a BBS proof, which is a zero-knowledge, proof-of-knowledge, of a BBS signature, while optionally disclosing any subset of the signed messages. Note that in contrast to the ProofGen operation of (see Section 3.5.3), the ProofGen operation defined in this section accepts 2 different lists of messages and disclosed indexes, one for the messages known to the Signer (messages) and the corresponding disclosed indexes (disclosed_indexes) and one for the messages committed by the Prover (committed_messages) and the corresponding disclosed indexes (disclosed_commitment_indexes). Furthermore, the operation also expects the secret_prover_blind (as returned from the Commit operation defined in ) and signer_blind (as inputted in the BlindSign operation defined in ) values. If the BBS signature is generated using a commitment value, then the secret_prover_blind returned by the Commit operation used to generate the commitment should be provided to the ProofGen operation (otherwise the resulting proof will be invalid). This operation makes use of the CoreProofGen operation as defined in Section 3.6.3 of . proof = BlindProofGen(PK, signature, header, ph, messages, committed_messages, disclosed_indexes, disclosed_commitment_indexes, secret_prover_blind) 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), an octet string containing the presentation header. If not supplied, it defaults to an empty string. - messages (OPTIONAL), a vector of octet strings. If not supplied, it defaults to the empty array "()". - committed_messages (OPTIONAL), a vector of octet strings. 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 "()". - disclosed_commitment_indexes (OPTIONAL), vector of unsigned integers in ascending order. Indexes of disclosed committed messages. If not supplied, it defaults to the empty array "()". - secret_prover_blind (OPTIONAL), a scalar value. If not supplied it defaults to zero "0". Parameters: - api_id, the octet string ciphersuite_id || "BLIND_H2G_HM2S_", where ciphersuite_id is defined by the ciphersuite and "BLIND_H2G_HM2S_"is an ASCII string composed of 15 bytes. Outputs: - proof, an octet string; or INVALID. Deserialization: 1. L = length(messages) 2. M = length(committed_messages) 3. if length(disclosed_indexes) > L, return INVALID 4. for i in disclosed_indexes, if i < 0 or i >= L, return INVALID 5. if length(disclosed_commitment_indexes) > M, return INVALID 6. for j in disclosed_commitment_indexes, if i < 0 or i >= M, return INVALID Procedure: 1. message_scalars = BBS.messages_to_scalars(messages, api_id) 2. committed_message_scalars = () 3. committed_message_scalars.append(secret_prover_blind) 4. committed_message_scalars.append(BBS.messages_to_scalars( committed_messages, api_id)) 5. generators = BBS.create_generators(length(message_scalars) + 1, api_id) 6. blind_generators = BBS.create_generators(length(committed_message_scalars) + 1, "BLIND_" || api_id) 7. indexes = () 8. indexes.append(disclosed_indexes) 9. for j in disclosed_commitment_indexes: indexes.append(j + L + 1) 10. proof = BBS.CoreProofGen( PK, signature, generators.append(blind_generators), header, ph, message_scalars.append(committed_message_scalars), indexes, api_id) 11. return proof

Proof Verification The ProofVerify operation validates a BBS proof, given the Signer's public key (PK), a header and presentation header values, two arrays of disclosed messages (the ones known to the Signer and the ones committed by the prover) and two corresponding arrays of indexes those messages had in the original vectors of signed messages. In addition, the BlindProofVerify operation defined in this section accepts the integer L, representing the total number of signed messages known by the Signer. This operation makes use of the CoreProofVerify operation as defined in Section 3.6.4 of . result = BlindProofVerify(PK, proof, header, ph, L, disclosed_messages, disclosed_committed_messages, disclosed_indexes, disclosed_committed_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. - header (OPTIONAL), an optional octet string containing context and application specific information. If not supplied, it defaults to the empty octet string (""). - ph (OPTIONAL), an octet string containing the presentation header. If not supplied, it defaults to the empty octet string (""). - L (OPTIONAL), an integer, representing the total number of Signer known messages if not supplied it defaults to 0. - disclosed_messages (OPTIONAL), a vector of octet strings. 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: - api_id, the octet string ciphersuite_id || "H2G_HM2S_", where ciphersuite_id is defined by the ciphersuite and "H2G_HM2S_"is an ASCII string comprised of 9 bytes. - (octet_point_length, octet_scalar_length), defined by the ciphersuite. Outputs: - result, either VALID or INVALID. Deserialization: 1. proof_len_floor = 2 * octet_point_length + 3 * octet_scalar_length 2. if length(proof) < proof_len_floor, return INVALID 3. U = floor((length(proof) - proof_len_floor) / octet_scalar_length) 4. total_no_messages = length(disclosed_indexes) + length(disclosed_committed_indexes) + U 5. M = total_no_messages - L Procedure: 1. generators = BBS.create_generators(L + 1, api_id) 2. blind_generators = BBS.create_generators(M + 1, "BLIND_" || api_id) 3. disclosed_message_scalars = messages_to_scalars( disclosed_messages, api_id) 4. disclosed_committed_message_scalars = messages_to_scalars( disclosed_committed_messages, api_id) 5. message_scalars = disclosed_message_scalars.append( disclosed_committed_message_scalars) 6. indexes = () 7. indexes.append(disclosed_indexes) 8. for j in disclosed_commitment_indexes: indexes.append(j + L + 1) 9. result = BBS.CoreProofVerify(PK, proof, generators.append(blind_generators), header, ph, message_scalars, indexes, api_id) 10. return result

Core Operations

Core Commitment Computation commit_with_proof = CoreCommit(blind_generators, committed_messages, api_id) Inputs: - blind_generators (REQUIRED), vector of pseudo-random points in G1. - committed_messages (OPTIONAL), a vector of scalars. If not supplied, it defaults to the empty array ("()"). - api_id (OPTIONAL), an octet string. If not supplied it defaults to the empty octet string (""). Deserialization: 1. M = length(committed_messages) 2. if length(blind_generators) != M + 1, return INVALID 3. (Q_2, J_1, ..., J_M) = blind_generators Procedure: 1. (secret_prover_blind, s~, m~_1, ..., m~_M) = BBS.get_random_scalars(M + 2) 2. C = Q_2 * secret_prover_blind + J_1 * msg_1 + ... + J_M * msg_M 3. Cbar = Q_2 * s~ + J_1 * m~_1 + ... + J_M * m~_M 4. challenge = calculate_blind_challenge(C, Cbar, blind_generators, api_id) 5. s^ = s~ + secret_prover_blind * challenge 6. for m in (1, 2, ..., M): m^_i = m~_1 + msg_i * challenge 7. proof = (s^, (m^_1, ..., m^_M), challenge) 8. commit_with_proof = commitment_with_proof_to_octets(C, proof) 9. return (commit_with_proof, secret_prover_blind)

Core Commitment Verification This operation is used by the Signer to verify the correctness of a commitment_proof for a supplied commitment, over a list of points of G1 called the blind_generators, used to compute that commitment. result = CoreCommitVerify(commitment, commitment_proof, blind_generators, api_id) Inputs: - commitment (REQUIRED), a commitment (see (#terminology)). - commitment_proof (REQUIRED), a commitment_proof (see (#terminology)). - blind_generators (REQUIRED), vector of pseudo-random points in G1. - api_id (OPTIONAL), octet string. If not supplied it defaults to the empty octet string (""). Outputs: - result: either VALID or INVALID Deserialization: 1. (s^, commitments, cp) = commitment_proof 2. M = length(commitments) 3. (m^_1, ..., m^_M) = commitments 4. if length(blind_generators) != M + 1, return INVALID 5. (Q_2, J_1, ..., J_M) = blind_generators Procedure: 1. Cbar = Q_2 * s^ + J_1 * m^_1 + ... + J_M * m^_M + commitment * (-cp) 2. cv = calculate_blind_challenge(commitment, Cbar, blind_generators, api_id) 3. if cv != cp, return INVALID 4. return VALID

Finalize Blind Sign This operation computes a blind BBS signature, from a secret key (SK), a set of generators (points of G1), a supplied commitment with its proof of correctness (commitment_with_proof), a header (header) and a set of messages (messages). The operation also accepts a random scalar (signer_blind) and the identifier of the BBS Interface, calling this core operation. blind_signature = FinalizeBlindSign(SK, PK, B, generators, blind_generators, header, api_id) Inputs: - SK (REQUIRED), a secret key in the form outputted by the KeyGen operation. - PK (REQUIRED), an octet string of the form outputted by SkToPk provided the above SK as input. - B (REQUIRED), a point of G1, different than Identity_G1. - generators (REQUIRED), vector of pseudo-random points in G1. - blind_generators (OPTIONAL), vector of pseudo-random points in G1. If not supplied it defaults to the empty array. - header (OPTIONAL), an octet string containing context and application specific information. If not supplied, it defaults to an empty string. - api_id (OPTIONAL), an octet string. If not supplied it defaults to the empty octet string (""). Outputs: - blind_signature, a blind signature encoded as an octet string; or INVALID. Definitions: 1. signature_dst, an octet string representing the domain separation tag: api_id || "H2S_" where "H2S_" is an ASCII string composed of 4 bytes. Deserialization: 1. L = length(generators) - 1 2. M = length(blind_generators) 3. if L <= 0 or M <=0, return INVALID 4. (Q_1, H_1, ..., H_L) = generators 5. (J_1, ..., J_M) = blind_generators Procedure: 1. domain = calculate_domain(PK, Q_1, (H_1, ..., H_L, J_1, ..., J_M), header, api_id) 2. e_octs = serialize((SK, B, domain)) 3. e = BBS.hash_to_scalar(e_octs, signature_dst) 4. A = B * (1 / (SK + e)) 5. return signature_to_octets((A, e))

Utilities

Blind Challenge Calculation challenge = calculate_blind_challenge(C, Cbar, generators, api_id) Inputs: - C (REQUIRED), a point of G1. - Cbar (REQUIRED), a point of G1. - generators (REQUIRED), an array of points from G1, of length at least 1. - api_id (OPTIONAL), octet string. If not supplied it defaults to the empty octet string (""). Definition: - blind_challenge_dst, an octet string representing the domain separation tag: api_id || "H2S_" where ciphersuite_id is defined by the ciphersuite and "H2S_" is an ASCII string composed of 4 bytes. Deserialization: 1. if length(generators) == 0, return INVALID 2. M = length(generators) - 1 Procedure: 1. c_arr = (M) 2. c_arr.append(generators) 3. c_octs = serialize(c_arr.append(C, Cbar)) 4. return BBS.hash_to_scalar(c_octs, blind_challenge_dst)

Serialize

Commitment with Proof to Octets commitment_octets = commitment_with_proof_to_octets(commitment, proof) Inputs: - commitment (REQUIRED), a point of G1. - proof (REQUIRED), a vector comprising of a scalar, a possibly empty vector of scalars and another scalar in that order. Outputs: - commitment_octets, an octet string or INVALID. Procedure: 1. commitment_octs = serialize(commitment) 2. if commitment_octs is INVALID, return INVALID 3. proof_octs = serialize(proof) 4. if proof_octs is INVALID, return INVALID 5. return commitment_octs || proof_octs

Octet to Commitment with Proof commitment = octets_to_commitment_with_proof(commitment_octs) Inputs: - commitment_octs (REQUIRED), an octet string in the form outputted from the commitment_to_octets operation. Parameters: - (octet_point_length, octet_scalar_length), defined by the ciphersuite. Outputs: - commitment, a commitment in the form (C, proof), where C a point of G1 and a proof vector comprising of a scalar, a possibly empty vector of scalars and another scalar in that order. Procedure: 1. commit_len_floor = octet_point_length + 2 * octet_scalar_length 2. if length(commitment_octs) < commit_len_floor, return INVALID 3. C_octets = commitment_octs[0..(octet_point_length - 1)] 4. C = octets_to_point_g1(C_octets) 5. if C is INVALID, return INVALID 6. if C == Identity_G1, return INVALID 7. j = 0 8. index = octet_point_length 9. while index < length(commitment_octs): 10. end_index = index + octet_scalar_length - 1 11. s_j = OS2IP(commitment_octets[index..end_index]) 12. if s_j = 0 or if s_j >= r, return INVALID 13. index += octet_scalar_length 14. j += 1 15. if index != length(commitment_octs), return INVALID 16. if j < 2, return INVALID 17. msg_commitment = () 18. if j >= 3, set msg_commitment = (s_2, ..., s_(j-1)) 19. return (C, (s_0, msg_commitments, s_j))

Security Considerations Security considerations detailed in Section 6 of apply to this draft as well.

Prover Blind Factor The random scalar value secret_prover_blind calculated and returned by the Commit operation is responsible for "hiding" the committed messages (otherwise, in many practical applications, the Signer may be able to retrieve them). Furthermore, it guarantees that the entity generating the BBS proof (see BlindProofGen defined in ) has knowledge of that factor. As a result, the secret_prover_blind MUST remain private by the Prover and it MUST be generated using a cryptographically secure pseudo-random number generator. See Section 6.7 of on recommendations and requirements for implementing the BBS.get_random_scalars operation (which is used to calculate the secret_prover_blind value).

Key Binding One natural use case for the blind signatures extension of the BBS scheme is key binding. In the context of BBS Signatures, key binding guarantees that only entities in control of a specific private key can compute BBS proofs. This can be achieved by committing to the private key prior to issuance, resulting in a BBS signature that includes that key as one of the signed messages. Creating a BBS proof from that signature will then require knowledge of that key (similar to any signed message). The Prover MUST NOT disclose that key as part of a proof generation procedure. Note also that the secret_prover_blind value returned by the Commit operation defined in (see ), has a similar property, i.e., it's knowledge is required to generate a proof from a blind signature. Many applications however, requiring key binding, mandate that the same private key is used among multiple signatures, whereas the secret_prover_blind is uniquely generated for each blind signature issuance request. In those cases, a commitment to a private key must be used, as described above.

Commitment Randomization A commitment is "randomized" using the secret_prover_blind random value. The Signer MAY elect to re-randomize a commitment by using it's own randomness. This can be helpful for applications that need to guarantee the uniqueness of each commitment (and of the resulting signatures) supplied by (untrusted) Provers. Examples include voting systems, where each unique signature will provide a single vote. To re-randomize a commitment, the Signer can provide the signer_blind input to the BlindSign operation defined in . If used, the signer_blind MUST be computed using the BBS.get_random_scalars operation. In contrast with the secret_prover_blind value however, the signer_blind doesn't need to be secret. The Signer will need to return it to the Prover, who requires it to verify the signature and generate the proofs.

Ciphersuites This document uses the BBS_BLS12381G1_XOF:SHAKE-256_SSWU_RO_ and BBS_BLS12381G1_XMD:SHA-256_SSWU_RO_ defined in Section 7.2.1 and Section 7.2.2 correspondingly, of .

Test Vectors TBD

IANA Considerations This document does not make any requests of IANA.