BLS Signatures
Stanford University
USA
dabo@cs.stanford.edu
University of Waterloo
Waterloo, ON
Canada
sgorbunov@uwaterloo.ca
Carnegie Mellon University
USA
rsw@cs.stanford.edu
NTT Research and ENS, Paris
Boston, MA
USA
wee@di.ens.fr
Cloudflare, Inc.
San Francisco, CA
USA
caw@heapingbits.net
Algorand
Boston, MA
USA
zhenfei@algorand.com
Internet
CFRG
BLS is a digital signature scheme with aggregation properties.
Given set of signatures (signature_1, ..., signature_n) anyone can produce
an aggregated signature. Aggregation can also be done on secret keys and
public keys. Furthermore, the BLS signature scheme is deterministic, non-malleable,
and efficient. Its simplicity and cryptographic properties allows it
to be useful in a variety of use-cases, specifically when minimal
storage space or bandwidth are required.
Introduction
A signature scheme is a fundamental cryptographic primitive
that is used to protect authenticity and integrity of communication.
Only the holder of a secret key can sign messages, but anyone can
verify the signature using the associated public key.
Signature schemes are used in point-to-point secure communication
protocols, PKI, remote connections, etc.
Designing efficient and secure digital signature is very important
for these applications.
This document describes the BLS signature scheme. The scheme enjoys a variety
of important efficiency properties:
- The public key and the signatures are encoded as single group elements.
- Verification requires 2 pairing operations.
- A collection of signatures (signature_1, ..., signature_n) can be aggregated
into a single signature. Moreover, the aggregate signature can
be verified using only n+1 pairings (as opposed to 2n pairings, when verifying
n signatures separately).
Given the above properties,
the scheme enables many interesting applications.
The immediate applications include
- Authentication and integrity for Public Key Infrastructure (PKI) and blockchains.
- The usage is similar to classical digital signatures, such as ECDSA.
- Aggregating signature chains for PKI and Secure Border Gateway Protocol (SBGP).
- Concretely, in a PKI signature chain of depth n, we have n signatures by n
certificate authorities on n distinct certificates. Similarly, in SBGP,
each router receives a list of n signatures attesting to a path of length n
in the network. In both settings, using the BLS signature scheme would allow us
to aggregate the n signatures into a single signature.
- consensus protocols for blockchains.
- There, BLS signatures
are used for authenticating transactions as well as votes during the consensus
protocol, and the use of aggregation significantly reduces the bandwidth
and storage requirements.
Comparison with ECDSA
The following comparison assumes BLS signatures with curve BLS12-381, targeting
126 bits of security .
For 128 bits security, ECDSA takes 37 and 79 micro-seconds to sign and verify
a signature on a typical laptop. In comparison, for a similar level of security,
BLS takes 370 and 2700 micro-seconds to sign and verify
a signature.
In terms of sizes, ECDSA uses 32 bytes for public keys and 64 bytes for signatures;
while BLS uses 96 bytes for public keys, and 48 bytes for signatures.
Alternatively, BLS can also be instantiated with 48 bytes of public keys and 96 bytes
of signatures.
BLS also allows for signature aggregation. In other words, a single signature is
sufficient to authenticate multiple messages and public keys.
Organization of this document
This document is organized as follows:
- The remainder of this section defines terminology and the high-level API.
- defines primitive operations used in the BLS signature scheme.
These operations MUST NOT be used alone.
- defines three BLS Signature schemes giving slightly different
security and performance properties.
- defines the format for a ciphersuites and gives recommended ciphersuites.
- The appendices give test vectors, etc.
Terminology and definitions
The following terminology is used through this document:
- SK: The secret key for the signature scheme.
- PK: The public key for the signature scheme.
- message: The input to be signed by the signature scheme.
- signature: The digital signature output.
- aggregation: Given a list of signatures for a list of messages and public keys,
an aggregation algorithm generates one signature that authenticates the same
list of messages and public keys.
- rogue key attack:
An attack in which a specially crafted public key (the "rogue" key) is used
to forge an aggregated signature.
specifies methods for securing against rogue key attacks.
The following notation and primitives are used:
- a || b denotes the concatenation of octet strings a and b.
- A pairing-friendly elliptic curve defines the following primitives
(see for detailed discussion):
- 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.
- P1, P2: distinguished points that generate G1 and G2, respectively.
- 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.
- For the above pairing-friendly curve, this document
writes 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.
- For each of E1 and E2 defined by the above pairing-friendly curve,
we assume that the pairing-friendly elliptic curve definition provides
several primitives, described below.
Note that these primitives are named generically.
When referring to one of these primitives for a specific group,
this document appends the name of the group, e.g.,
point_to_octets_E1, subgroup_check_E2, etc.
- point_to_octets(P) -> ostr: returns the canonical representation of
the point P as an octet string.
This operation is also known as serialization.
- octets_to_point(ostr) -> P: returns the point P corresponding to the
canonical representation ostr, or INVALID if ostr is not a valid output
of point_to_octets.
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., .
- I2OSP and OS2IP are the functions defined in , Section 4.
- hash_to_point(ostr) -> P: a cryptographic hash function that takes as input an
arbitrary octet string and returns a point on an elliptic curve.
Functions of this kind are defined in .
Each of the ciphersuites in specifies the hash_to_point
algorithm to be used.
API
The BLS signature scheme defines the following API:
- KeyGen(IKM) -> SK: a key generation algorithm that
takes as input an octet string comprising secret keying material,
and outputs a secret key SK.
- SkToPk(SK) -> PK: an algorithm that takes as input a secret key
and outputs the corresponding public key.
- Sign(SK, message) -> signature: a signing algorithm that generates a
deterministic signature given a secret key SK and a message.
- Verify(PK, message, signature) -> VALID or INVALID:
a verification algorithm that outputs VALID if signature is a valid
signature of message under public key PK, and INVALID otherwise.
- Aggregate((signature_1, ..., signature_n)) -> signature:
an aggregation algorithm that aggregates a collection of signatures
into a single signature.
- AggregateVerify((PK_1, ..., PK_n), (message_1, ..., message_n), signature) -> VALID or INVALID:
an aggregate verification algorithm that outputs VALID if signature
is a valid aggregated signature for a collection of public keys and messages,
and outputs INVALID otherwise.
Requirements
The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT",
"SHOULD", "SHOULD NOT", "RECOMMENDED", "MAY", and "OPTIONAL" in this
document are to be interpreted as described in .
Core operations
This section defines core operations used by the schemes defined in .
These operations MUST NOT be used except as described in that section.
Variants
Each core operation has two variants that trade off signature
and public key size:
- Minimal-signature-size: signatures are points in G1,
public keys are points in G2.
(Recall from that E1 has a more compact representation than E2.)
- Minimal-pubkey-size: public keys are points in G1,
signatures are points in G2.
Implementations using signature aggregation SHOULD use this approach,
since the size of (PK_1, ..., PK_n, signature) is dominated by
the public keys even for small n.
Parameters
The core operations in this section depend on several parameters:
- A signature variant, either minimal-signature-size or minimal-pubkey-size.
These are defined in .
- A pairing-friendly elliptic curve, plus associated functionality
given in .
- H, a hash function that MUST be a secure cryptographic hash function,
e.g., SHA-256 .
For security, H MUST output at least ceil(log2(r)) bits, where r is
the order of the subgroups G1 and G2 defined by the pairing-friendly
elliptic curve.
- hash_to_point, a function whose interface is described in .
When the signature variant is minimal-signature-size, this function
MUST output a point in G1.
When the signature variant is minimal-pubkey size, this function
MUST output a point in G2.
For security, this function MUST be either a random oracle encoding or a
nonuniform encoding, as defined in .
In addition, the following primitives are determined by the above parameters:
- P, an elliptic curve point.
When the signature variant is minimal-signature-size, P is the
distinguished point P2 that generates the group G2 (see ).
When the signature variant is minimal-pubkey-size, P is the
distinguished point P1 that generates the group G1.
- r, the order of the subgroups G1 and G2 defined by the pairing-friendly curve.
- pairing, a function that invokes the function e of ,
with argument order depending on signature variant:
- For minimal-signature-size:
pairing(U, V) := e(U, V)
- For minimal-pubkey-size:
pairing(U, V) := e(V, U)
- point_to_pubkey and point_to_signature, functions that invoke the
appropriate serialization routine () depending on
signature variant:
- For minimal-signature-size:
point_to_pubkey(P) := point_to_octets_E2(P)
point_to_signature(P) := point_to_octets_E1(P)
- For minimal-pubkey-size:
point_to_pubkey(P) := point_to_octets_E1(P)
point_to_signature(P) := point_to_octets_E2(P)
- pubkey_to_point and signature_to_point, functions that invoke the
appropriate deserialization routine () depending on
signature variant:
- For minimal-signature-size:
pubkey_to_point(ostr) := octets_to_point_E2(ostr)
signature_to_point(ostr) := octets_to_point_E1(ostr)
- For minimal-pubkey-size:
pubkey_to_point(ostr) := octets_to_point_E1(ostr)
signature_to_point(ostr) := octets_to_point_E2(ostr)
- pubkey_subgroup_check and signature_subgroup_check, functions
that invoke the appropriate subgroup check routine ()
depending on signature variant:
- For minimal-signature-size:
pubkey_subgroup_check(P) := subgroup_check_E2(P)
signature_subgroup_check(P) := subgroup_check_E1(P)
- For minimal-pubkey-size:
pubkey_subgroup_check(P) := subgroup_check_E1(P)
signature_subgroup_check(P) := subgroup_check_E2(P)
KeyGen
The KeyGen procedure described in this section generates a secret key SK
deterministically from a secret octet string IKM.
SK is guaranteed to be nonzero, as required by KeyValidate ().
KeyGen uses HKDF instantiated with the hash function H.
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.
KeyGen takes two parameters. The first parameter, salt, is required;
see below for further discussion of this value.
The second parameter, key_info, is optional; it MAY be used to
derive multiple independent keys from the same IKM.
By default, key_info is the empty string.
SK = KeyGen(IKM)
Inputs:
- IKM, a secret octet string. See requirements above.
Outputs:
- SK, a uniformly random integer such that 1 <= SK < r.
Parameters:
- salt, a required octet string.
- key_info, an optional octet string.
If key_info is not supplied, it defaults to the empty string.
Definitions:
- HKDF-Extract is as defined in RFC5869, instantiated with hash H.
- HKDF-Expand is as defined in RFC5869, instantiated with hash H.
- I2OSP and OS2IP are as defined in RFC8017, Section 4.
- L is the integer given by ceil((3 * ceil(log2(r))) / 16).
Procedure:
1. while True:
2. PRK = HKDF-Extract(salt, IKM || I2OSP(0, 1))
3. OKM = HKDF-Expand(PRK, key_info || I2OSP(L, 2), L)
4. SK = OS2IP(OKM) mod r
5. if SK != 0:
6. return SK
7. salt = H(salt)
KeyGen is the RECOMMENDED way of generating secret keys, but its use is not
required for compatibility, and implementations MAY use a different KeyGen
procedure. For security, such an alternative KeyGen procedure MUST output SK
that is statistically close to uniformly random in the range 1 <= SK < r.
For compatibility with prior versions of this document, implementations
SHOULD allow applications to choose the salt value.
Setting salt to the value H("BLS-SIG-KEYGEN-SALT-") (i.e., the hash of an
ASCII string comprising 20 octets) results in a KeyGen algorithm that is
compatible with version 4 of this document.
Setting salt to the value "BLS-SIG-KEYGEN-SALT-" (i.e., an ASCII string
comprising 20 octets) results in a KeyGen algorithm that is compatible with
versions of this document prior to number 4.
See for more information on choosing a salt value.
SkToPk
The SkToPk algorithm takes a secret key SK and outputs the corresponding
public key PK.
discusses requirements for SK.
PK = SkToPk(SK)
Inputs:
- SK, a secret integer such that 1 <= SK < r.
Outputs:
- PK, a public key encoded as an octet string.
Procedure:
1. xP = SK * P
2. PK = point_to_pubkey(xP)
3. return PK
KeyValidate
The KeyValidate algorithm ensures that a public key is valid.
In particular, it ensures that a public key represents a valid,
non-identity point that is in the correct subgroup.
See for further discussion.
As an optimization, implementations MAY cache the result of KeyValidate
in order to avoid unnecessarily repeating validation for known keys.
result = KeyValidate(PK)
Inputs:
- PK, a public key in the format output by SkToPk.
Outputs:
- result, either VALID or INVALID
Procedure:
1. xP = pubkey_to_point(PK)
2. If xP is INVALID, return INVALID
3. If xP is the identity element, return INVALID
4. If pubkey_subgroup_check(xP) is INVALID, return INVALID
5. return VALID
CoreSign
The CoreSign algorithm computes a signature from SK, a secret key,
and message, an octet string.
signature = CoreSign(SK, message)
Inputs:
- SK, a secret key in the format output by KeyGen.
- message, an octet string.
Outputs:
- signature, an octet string.
Procedure:
1. Q = hash_to_point(message)
2. R = SK * Q
3. signature = point_to_signature(R)
4. return signature
CoreVerify
The CoreVerify algorithm checks that a signature is valid for
the octet string message under the public key PK.
result = CoreVerify(PK, message, signature)
Inputs:
- PK, a public key in the format output by SkToPk.
- message, an octet string.
- signature, an octet string in the format output by CoreSign.
Outputs:
- result, either VALID or INVALID.
Procedure:
1. R = signature_to_point(signature)
2. If R is INVALID, return INVALID
3. If signature_subgroup_check(R) is INVALID, return INVALID
4. If KeyValidate(PK) is INVALID, return INVALID
5. xP = pubkey_to_point(PK)
6. Q = hash_to_point(message)
7. C1 = pairing(Q, xP)
8. C2 = pairing(R, P)
9. If C1 == C2, return VALID, else return INVALID
Aggregate
The Aggregate algorithm aggregates multiple signatures into one.
signature = Aggregate((signature_1, ..., signature_n))
Inputs:
- signature_1, ..., signature_n, octet strings output by
either CoreSign or Aggregate.
Outputs:
- signature, an octet string encoding a aggregated signature
that combines all inputs; or INVALID.
Precondition: n >= 1, otherwise return INVALID.
Procedure:
1. aggregate = signature_to_point(signature_1)
2. If aggregate is INVALID, return INVALID
3. for i in 2, ..., n:
4. next = signature_to_point(signature_i)
5. If next is INVALID, return INVALID
6. aggregate = aggregate + next
7. signature = point_to_signature(aggregate)
8. return signature
CoreAggregateVerify
The CoreAggregateVerify algorithm checks an aggregated signature
over several (PK, message) pairs.
result = CoreAggregateVerify((PK_1, ..., PK_n),
(message_1, ... message_n),
signature)
Inputs:
- PK_1, ..., PK_n, public keys in the format output by SkToPk.
- message_1, ..., message_n, octet strings.
- signature, an octet string output by Aggregate.
Outputs:
- result, either VALID or INVALID.
Precondition: n >= 1, otherwise return INVALID.
Procedure:
1. R = signature_to_point(signature)
2. If R is INVALID, return INVALID
3. If signature_subgroup_check(R) is INVALID, return INVALID
4. C1 = 1 (the identity element in GT)
5. for i in 1, ..., n:
6. If KeyValidate(PK_i) is INVALID, return INVALID
7. xP = pubkey_to_point(PK_i)
8. Q = hash_to_point(message_i)
9. C1 = C1 * pairing(Q, xP)
10. C2 = pairing(R, P)
11. If C1 == C2, return VALID, else return INVALID
BLS Signatures
This section defines three signature schemes: basic, message augmentation,
and proof of possession.
These schemes differ in the ways that they defend against rogue key
attacks ().
All of the schemes in this section are built on a set of core operations
defined in .
Thus, defining a scheme requires fixing a set of parameters as
defined in .
All three schemes expose the KeyGen, SkToPk, and Aggregate operations
that are defined in .
The sections below define the other API functions ()
for each scheme.
Basic scheme
In a basic scheme, rogue key attacks are handled by requiring
all messages signed by an aggregate signature to be distinct.
This requirement is enforced in the definition of AggregateVerify.
The Sign and Verify functions are identical to CoreSign and
CoreVerify (), respectively.
AggregateVerify is defined below.
AggregateVerify
This function first ensures that all messages are distinct, and then
invokes CoreAggregateVerify.
result = AggregateVerify((PK_1, ..., PK_n),
(message_1, ..., message_n),
signature)
Inputs:
- PK_1, ..., PK_n, public keys in the format output by SkToPk.
- message_1, ..., message_n, octet strings.
- signature, an octet string output by Aggregate.
Outputs:
- result, either VALID or INVALID.
Precondition: n >= 1, otherwise return INVALID.
Procedure:
1. If any two input messages are equal, return INVALID.
2. return CoreAggregateVerify((PK_1, ..., PK_n),
(message_1, ..., message_n),
signature)
Message augmentation
In a message augmentation scheme, signatures are generated
over the concatenation of the public key and the message,
ensuring that messages signed by different public keys are
distinct.
Sign
To match the API for Sign defined in , this function
recomputes the public key corresponding to the input SK.
Implementations MAY instead implement an interface that takes
the public key as an input.
Note that the point P and the point_to_pubkey function are
defined in .
signature = Sign(SK, message)
Inputs:
- SK, a secret key in the format output by KeyGen.
- message, an octet string.
Outputs:
- signature, an octet string.
Procedure:
1. PK = SkToPk(SK)
2. return CoreSign(SK, PK || message)
Verify
result = Verify(PK, message, signature)
Inputs:
- PK, a public key in the format output by SkToPk.
- message, an octet string.
- signature, an octet string in the format output by CoreSign.
Outputs:
- result, either VALID or INVALID.
Procedure:
1. return CoreVerify(PK, PK || message, signature)
AggregateVerify
result = AggregateVerify((PK_1, ..., PK_n),
(message_1, ..., message_n),
signature)
Inputs:
- PK_1, ..., PK_n, public keys in the format output by SkToPk.
- message_1, ..., message_n, octet strings.
- signature, an octet string output by Aggregate.
Outputs:
- result, either VALID or INVALID.
Precondition: n >= 1, otherwise return INVALID.
Procedure:
1. for i in 1, ..., n:
2. mprime_i = PK_i || message_i
3. return CoreAggregateVerify((PK_1, ..., PK_n),
(mprime_1, ..., mprime_n),
signature)
Proof of possession
A proof of possession scheme uses a separate public key
validation step, called a proof of possession, to defend against
rogue key attacks.
This enables an optimization to aggregate signature verification
for the case that all signatures are on the same message.
The Sign, Verify, and AggregateVerify functions
are identical to CoreSign, CoreVerify, and CoreAggregateVerify
(), respectively.
In addition, a proof of possession scheme defines three functions beyond
the standard API ():
- PopProve(SK) -> proof: an algorithm that generates a proof of possession
for the public key corresponding to secret key SK.
- PopVerify(PK, proof) -> VALID or INVALID:
an algorithm that outputs VALID if proof is valid for PK, and INVALID otherwise.
- FastAggregateVerify((PK_1, ..., PK_n), message, signature) -> VALID or INVALID:
a verification algorithm for the aggregate of multiple signatures on
the same message.
This function is faster than AggregateVerify.
All public keys used by Verify, AggregateVerify, and FastAggregateVerify
MUST be accompanied by a proof of possession, and the result of evaluating
PopVerify on each public key and its proof MUST be VALID.
Parameters
In addition to the parameters required to instantiate the core operations
(), a proof of possession scheme requires one further parameter:
- hash_pubkey_to_point(PK) -> P: a cryptographic hash function that takes as
input a public key and outputs a point in the same subgroup as the
hash_to_point algorithm used to instantiate the core operations.
For security, this function MUST be domain separated from the hash_to_point function.
In addition, this function MUST be either a random oracle encoding or a
nonuniform encoding, as defined in .
The RECOMMENDED way of instantiating hash_pubkey_to_point is to use
the same hash-to-curve function as hash_to_point, with a
different domain separation tag (see , Section 3.1).
discusses the RECOMMENDED way to construct the
domain separation tag.
PopProve
This function recomputes the public key coresponding to the input SK.
Implementations MAY instead implement an interface that takes the
public key as input.
Note that the point P and the point_to_pubkey and point_to_signature
functions are defined in .
The hash_pubkey_to_point function is defined in .
proof = PopProve(SK)
Inputs:
- SK, a secret key in the format output by KeyGen.
Outputs:
- proof, an octet string.
Procedure:
1. PK = SkToPk(SK)
2. Q = hash_pubkey_to_point(PK)
3. R = SK * Q
4. proof = point_to_signature(R)
5. return proof
PopVerify
PopVerify uses several functions defined in .
The hash_pubkey_to_point function is defined in .
As an optimization, implementations MAY cache the result of PopVerify
in order to avoid unnecessarily repeating validation for known keys.
result = PopVerify(PK, proof)
Inputs:
- PK, a public key in the format output by SkToPk.
- proof, an octet string in the format output by PopProve.
Outputs:
- result, either VALID or INVALID
Procedure:
1. R = signature_to_point(proof)
2. If R is INVALID, return INVALID
3. If signature_subgroup_check(R) is INVALID, return INVALID
4. If KeyValidate(PK) is INVALID, return INVALID
5. xP = pubkey_to_point(PK)
6. Q = hash_pubkey_to_point(PK)
7. C1 = pairing(Q, xP)
8. C2 = pairing(R, P)
9. If C1 == C2, return VALID, else return INVALID
FastAggregateVerify
FastAggregateVerify uses several functions defined in .
All public keys passed as arguments to this algorithm MUST have a
corresponding proof of possession, and the result of evaluating
PopVerify on each public key and its proof MUST be VALID.
The caller is responsible for ensuring that this precondition is met.
If it is violated, this scheme provides no security against aggregate
signature forgery.
result = FastAggregateVerify((PK_1, ..., PK_n), message, signature)
Inputs:
- PK_1, ..., PK_n, public keys in the format output by SkToPk.
- message, an octet string.
- signature, an octet string output by Aggregate.
Outputs:
- result, either VALID or INVALID.
Preconditions:
- n >= 1, otherwise return INVALID.
- The caller MUST know a proof of possession for all PK_i, and the
result of evaluating PopVerify on PK_i and this proof MUST be VALID.
See discussion above.
Procedure:
1. aggregate = pubkey_to_point(PK_1)
2. for i in 2, ..., n:
3. next = pubkey_to_point(PK_i)
4. aggregate = aggregate + next
5. PK = point_to_pubkey(aggregate)
6. return CoreVerify(PK, message, signature)
Ciphersuites
This section defines the format for a BLS ciphersuite.
It also gives concrete ciphersuites based on the BLS12-381 pairing-friendly
elliptic curve .
Ciphersuite format
A ciphersuite specifies all parameters from ,
a scheme from , and any parameters the scheme requires.
In particular, a ciphersuite comprises:
- ID: the ciphersuite ID, an ASCII string. The REQUIRED format for
this string is
"BLS_SIG_" || H2C_SUITE_ID || SC_TAG || "_"
- Strings in double quotes are ASCII-encoded literals.
- H2C_SUITE_ID is the suite ID of the hash-to-curve suite
used to define the hash_to_point and hash_pubkey_to_point
functions.
- SC_TAG is a string indicating the scheme and, optionally, additional information.
The first three characters of this string MUST chosen as follows:
- "NUL" if SC is basic,
- "AUG" if SC is message-augmentation, or
- "POP" if SC is proof-of-possession.
- Other values MUST NOT be used.
SC_TAG MAY be used to encode other information about the
ciphersuite, for example, a version number.
When used in this way, SC_TAG MUST contain only ASCII characters
between 0x21 and 0x7e (inclusive), except that it MUST NOT contain
underscore (0x5f).
The RECOMMENDED way to add user-defined information to SC_TAG is to
append a colon (':', ASCII 0x3a) and then the informational string.
For example, "NUL:version=2" is an appropriate SC_TAG value.
Note that hash-to-curve suite IDs always include a trailing underscore,
so no field separator is needed between H2C_SUITE_ID and SC_TAG.
- SC: the scheme, one of basic, message-augmentation, or proof-of-possession.
- SV: the signature variant, either minimal-signature-size or
minimal-pubkey-size.
- EC: a pairing-friendly elliptic curve, plus all associated functionality
().
- H: a cryptographic hash function.
- hash_to_point: a hash from arbitrary strings to elliptic curve points.
hash_to_point MUST be defined in terms of a hash-to-curve suite .
The RECOMMENDED hash-to-curve domain separation tag is the ciphersuite ID string defined above.
- hash_pubkey_to_point (only specified when SC is proof-of-possession):
a hash from serialized public keys to elliptic curve points.
hash_pubkey_to_point MUST be defined in terms of a
hash-to-curve suite .
The hash-to-curve domain separation tag MUST be distinct from the domain
separation tag used for hash_to_point.
It is RECOMMENDED that the domain separation tag be constructed similarly to
the ciphersuite ID string, namely:
"BLS_POP_" || H2C_SUITE_ID || SC_TAG || "_"
Ciphersuites for BLS12-381
The following ciphersuites are all built on the BLS12-381 elliptic curve.
The required primitives for this curve are given in .
These ciphersuites use the hash-to-curve suites
BLS12381G1_XMD:SHA-256_SSWU_RO_ and
BLS12381G2_XMD:SHA-256_SSWU_RO_
defined in , Section 8.7.
Each ciphersuite defines a unique hash_to_point function by specifying
a domain separation tag (see [@I-D.irtf-cfrg-hash-to-curve, Section 3.1).
Basic
BLS_SIG_BLS12381G1_XMD:SHA-256_SSWU_RO_NUL_ is defined as follows:
- SC: basic
- SV: minimal-signature-size
- EC: BLS12-381, as defined in .
- H: SHA-256
- hash_to_point: BLS12381G1_XMD:SHA-256_SSWU_RO_ with the ASCII-encoded domain separation tag
BLS_SIG_BLS12381G1_XMD:SHA-256_SSWU_RO_NUL_
BLS_SIG_BLS12381G2_XMD:SHA-256_SSWU_RO_NUL_ is identical to
BLS_SIG_BLS12381G1_XMD:SHA-256_SSWU_RO_NUL_, except for the
following parameters:
- SV: minimal-pubkey-size
- hash_to_point: BLS12381G2_XMD:SHA-256_SSWU_RO_ with the ASCII-encoded domain separation tag
BLS_SIG_BLS12381G2_XMD:SHA-256_SSWU_RO_NUL_
Message augmentation
BLS_SIG_BLS12381G1_XMD:SHA-256_SSWU_RO_AUG_ is defined as follows:
- SC: message-augmentation
- SV: minimal-signature-size
- EC: BLS12-381, as defined in .
- H: SHA-256
- hash_to_point: BLS12381G1_XMD:SHA-256_SSWU_RO_ with the ASCII-encoded domain separation tag
BLS_SIG_BLS12381G1_XMD:SHA-256_SSWU_RO_AUG_
BLS_SIG_BLS12381G2_XMD:SHA-256_SSWU_RO_AUG_ is identical to
BLS_SIG_BLS12381G1_XMD:SHA-256_SSWU_RO_AUG_, except for the
following parameters:
- SV: minimal-pubkey-size
- hash_to_point: BLS12381G2_XMD:SHA-256_SSWU_RO_ with the ASCII-encoded domain separation tag
BLS_SIG_BLS12381G2_XMD:SHA-256_SSWU_RO_AUG_
Proof of possession
BLS_SIG_BLS12381G1_XMD:SHA-256_SSWU_RO_POP_ is defined as follows:
- SC: proof-of-possession
- SV: minimal-signature-size
- EC: BLS12-381, as defined in .
- H: SHA-256
- hash_to_point: BLS12381G1_XMD:SHA-256_SSWU_RO_ with the ASCII-encoded domain separation tag
BLS_SIG_BLS12381G1_XMD:SHA-256_SSWU_RO_POP_
- hash_pubkey_to_point: BLS12381G1_XMD:SHA-256_SSWU_RO_ with the ASCII-encoded domain separation tag
BLS_POP_BLS12381G1_XMD:SHA-256_SSWU_RO_POP_
BLS_SIG_BLS12381G2_XMD:SHA-256_SSWU_RO_POP_ is identical to
BLS_SIG_BLS12381G1_XMD:SHA-256_SSWU_RO_POP_, except for the
following parameters:
- SV: minimal-pubkey-size
- hash_to_point: BLS12381G2_XMD:SHA-256_SSWU_RO_ with the ASCII-encoded domain separation tag
BLS_SIG_BLS12381G2_XMD:SHA-256_SSWU_RO_POP_
- hash_pubkey_to_point: BLS12381G2_XMD:SHA-256_SSWU_RO_ with the ASCII-encoded domain separation tag
BLS_POP_BLS12381G2_XMD:SHA-256_SSWU_RO_POP_
Security Considerations
Choosing a salt value for KeyGen
KeyGen uses HKDF to generate keys.
The security analysis of HKDF assumes that the salt value is either
empty (in which case it is replaced by the all-zeros string) or an
unstructured string.
Versions of this document prior to number 4 set the salt parameter
to "BLS-SIG-KEYGEN-SALT-", which does not meet this requirement.
If, as is common, the hash function H is modeled as a random oracle,
this requirement is obviated.
When choosing a salt value other than "BLS-SIG-KEYGEN-SALT-" or
H("BLS-SIG-KEYGEN-SALT-"), the RECOMMENDED method is to fix a
uniformly random octet string whose length equals the output
length of H.
Validating public keys
All algorithms in and that operate on public keys
require first validating those keys.
For the basic and message augmentation schemes, the use of KeyValidate
is REQUIRED.
For the proof of possession scheme, each public key MUST be accompanied by
a proof of possession, and use of PopVerify is REQUIRED.
KeyValidate requires all public keys to represent valid, non-identity points
in the correct subgroup.
A valid point and subgroup membership are required to ensure that the pairing
operation is defined ().
A non-identity point is required because the identity public key has the
property that the corresponding secret key is equal to zero, which means
that the identity point is the unique valid signature for every message
under this key.
A malicious signer could take advantage of this fact to equivocate about
which message he signed.
While non-equivocation is not a required property for a signature scheme,
equivocation is infeasible for BLS signatures under any nonzero secret key
because it would require finding colliding inputs to the hash_to_point
function, which is assumed to be collision resistant.
Prohibiting SK == 0 eliminates the exceptional case, which may help to prevent
equivocation-related security issues in protocols that use BLS signatures.
Skipping membership check
Some existing implementations skip the signature_subgroup_check invocation
in CoreVerify (), 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 () 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 .
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.
Randomness considerations
BLS signatures are deterministic. This protects against attacks
arising from signing with bad randomness, for example, the nonce reuse
attack on ECDSA .
As discussed in , 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.
Implementing hash_to_point and hash_pubkey_to_point
The security analysis models hash_to_point and hash_pubkey_to_point
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
.
In addition, ciphersuites MUST specify unique domain separation tags
for hash_to_point and hash_pubkey_to_point.
The domain separation tag format used in is the RECOMMENDED one.
Implementation Status
This section will be removed in the final version of the draft.
There are currently several implementations of BLS signatures using the BLS12-381 curve.
- Algorand: bls_sigs_ref.
- Chia: spec
python/C++. Here, they are
swapping G1 and G2 so that the public keys are small, and the benefits
of avoiding a membership check during signature verification would even be more
substantial. The current implementation does not seem to implement the membership check.
Chia uses the Fouque-Tibouchi hashing to the curve, which can be done in constant time.
- Dfinity: go BLS. The current implementations do not seem to implement the membership check.
- Ethereum 2.0: spec.
Related Standards
- Pairing-friendly curves,
- Pairing-based Identity-Based Encryption IEEE 1363.3.
- Identity-Based Cryptography Standard rfc5901.
- Hashing to Elliptic Curves , in order to implement the hash function hash_to_point.
- EdDSA rfc8032.
IANA Considerations
TBD (consider to register ciphersuite identifiers for BLS signature and underlying
pairing curves)
Normative References
BLS12-381
Electric Coin Company
Informative References
Threshold Signatures, Multisignatures and Blind Signatures Based on the Gap-Diffie-Hellman-Group Signature Scheme
University of California, San Diego
Faster subgroup checks for BLS12-381
Electric Coin Company
On the Security of Joint Signature and Encryption
SoftMax Inc.
New York University
IBM T.J. Watson Research Center
Short signatures from the Weil pairing
Stanford University
Stanford University
Stanford University
A note on group membership tests for G1, G2 and GT on BLS pairing-friendly curves
Technical Innovation Institute
Sequential Aggregate Signatures and Multisignatures Without Random Oracles
University of California, Los Angeles
University of California, Los Angeles
University of California, Los Angeles
Weizmann Institute
SRI International
The Power of Proofs-of-Possession: Securing Multiparty Signatures against Rogue-Key Attacks
University of California, San Diego
University of California, San Diego
Cocks–Pinch curves of embedding degrees five to eight and optimal ate pairing computation
Aggregate and verifiably encrypted signatures from bilinear maps
Stanford University
Stanford University
Stanford University
Stanford University
Unrestricted aggregate signatures
University of California, San Diego
Thammasat University
Katholieke Universiteit Leuven
Compact multi-signatures for shorter blockchains
Stanford University
DFINITY
ETH Zurich
FIPS Publication 180-4: Secure Hash Standard
National Institute of Standards and Technology (NIST)
Mining your Ps and Qs: Detection of widespread weak keys in network devices
University of California, San Diego
The University of Michigan
The University of Michigan
The University of Michigan
BLS12-381
The ciphersuites in are based upon the BLS12-381
pairing-friendly elliptic curve.
The following defines the correspondence between the primitives
in and the parameters given in Section 4.2.1 of
.
- E1, G1: the curve E and its order-r subgroup.
- E2, G2: the curve E' and its order-r subgroup.
- GT: the subgroup G_T.
- P1: the point BP.
- P2: the point BP'.
- e: the optimal Ate pairing defined in Appendix A of
.
- point_to_octets and octets_to_point use the compressed
serialization formats for E1 and E2 defined by .
- subgroup_check MAY use either the naive check described
in or the optimized checks given by or .
Test Vectors
TBA: (i) test vectors for both variants of the signature scheme
(signatures in G2 instead of G1) , (ii) test vectors ensuring
membership checks, (iii) intermediate computations ctr, hm.
Security analyses
The security properties of the BLS signature scheme are proved in .
prove the security of aggregate signatures over distinct messages,
as in the basic scheme of .
prove security of the message augmentation scheme of .
prove security of constructions related to the proof
of possession scheme of .
prove the security of another rogue key defense; this
defense is not standardized in this document.