draft-irtf-cfrg-bls-signature-02.xml

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<front>
<title abbrev="BLS-signature">draft-irtf-cfrg-bls-signature-02</title><author initials="D." surname="Boneh" fullname="Dan Boneh"><organization>Stanford University</organization><address><postal><street></street>
<country>USA</country>
</postal><email>dabo@cs.stanford.edu</email>
</address></author>
<author initials="S." surname="Gorbunov" fullname="Sergey Gorbunov"><organization>Algorand and University of Waterloo</organization><address><postal><street></street>
<city>Boston, MA</city>
<country>USA</country>
</postal><email>sergey@algorand.com</email>
</address></author>
<author initials="R." surname="Wahby" fullname="Riad S. Wahby"><organization>Stanford University</organization><address><postal><street></street>
<country>USA</country>
</postal><email>rsw@cs.stanford.edu</email>
</address></author>
<author initials="H." surname="Wee" fullname="Hoeteck Wee"><organization>Algorand and ENS, Paris</organization><address><postal><street></street>
<city>Boston, MA</city>
<country>USA</country>
</postal><email>wee@di.ens.fr</email>
</address></author>
<author initials="Z." surname="Zhang" fullname="Zhenfei Zhang"><organization>Algorand</organization><address><postal><street></street>
<city>Boston, MA</city>
<country>USA</country>
</postal><email>zhenfei@algorand.com</email>
</address></author>
<date year="2020" month="March" day="9"></date>
<area>Internet</area><workgroup>CFRG</workgroup>
<abstract><t>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.</t>
</abstract>

</front>

<middle>

<section anchor="introduction" title="Introduction">
<t>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.</t>
<t>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.</t>
<t>This document describes the BLS signature scheme. The scheme enjoys a variety
of important efficiency properties:</t>
<t>
<list style="numbers">
<t>The public key and the signatures are encoded as single group elements.</t>
<t>Verification requires 2 pairing operations.</t>
<t>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).</t>
</list>
</t>
<t>Given the above properties,
the scheme enables many interesting applications.
The immediate applications include</t>
<t>
<list style="symbols">
<t>Authentication and integrity for Public Key Infrastructure (PKI) and blockchains.
<list style="symbols">
<t>The usage is similar to classical digital signatures, such as ECDSA.</t>
</list></t>
<t>Aggregating signature chains for PKI and Secure Border Gateway Protocol (SBGP).
<list style="symbols">
<t>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.</t>
</list></t>
<t>consensus protocols for blockchains.
<list style="symbols">
<t>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.</t>
</list></t>
</list>
</t>

<section anchor="comparison-with-ecdsa" title="Comparison with ECDSA">
<t>The following comparison assumes BLS signatures with curve BLS12-381, targeting
128 bits security.</t>
<t>For 128 bits security, ECDSA takes 37 and 79 micro-seconds to sign and verify
a signature on a typical laptop. In comparison, for the same level of security,
BLS takes 370 and 2700 micro-seconds to sign and verify
a signature.</t>
<t>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.</t>
</section>

<section anchor="organization-of-this-document" title="Organization of this document">
<t>This document is organized as follows:</t>
<t>
<list style="symbols">
<t>The remainder of this section defines terminology and the high-level API.</t>
<t><xref target="coreops"></xref> defines primitive operations used in the BLS signature scheme.
These operations MUST NOT be used alone.</t>
<t><xref target="schemes"></xref> defines three BLS Signature schemes giving slightly different
security and performance properties.</t>
<t><xref target="ciphersuites"></xref> defines the format for a ciphersuites and gives recommended ciphersuites.</t>
<t>The appendices give test vectors, etc.</t>
</list>
</t>
</section>

<section anchor="definitions" title="Terminology and definitions">
<t>The following terminology is used through this document:</t>
<t>
<list style="symbols">
<t>SK:  The secret key for the signature scheme.</t>
<t>PK:  The public key for the signature scheme.</t>
<t>message:  The input to be signed by the signature scheme.</t>
<t>signature:  The digital signature output.</t>
<t>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.</t>
<t>rogue key attack:
An attack in which a specially crafted public key (the &quot;rogue&quot; key) is used
to forge an aggregated signature.
<xref target="schemes"></xref> specifies methods for securing against rogue key attacks.</t>
</list>
</t>
<t>The following notation and primitives are used:</t>
<t>
<list style="symbols">
<t>a || b denotes the concatenation of octet strings a and b.</t>
<t>A pairing-friendly elliptic curve defines the following primitives
(see <xref target="I-D.irtf-cfrg-pairing-friendly-curves"></xref> for detailed discussion):
<list style="symbols">
<t>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.</t>
<t>G1, G2: subgroups of E1 and E2 (respectively) having prime order r.</t>
<t>P1, P2: distinguished points that generate G1 and G2, respectively.</t>
<t>GT: a subgroup, of prime order r, of the multiplicative group of a field extension.</t>
<t>e : G1 x G2 -&gt; GT: a non-degenerate bilinear map.</t>
</list></t>
<t>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.</t>
</list>
</t>
<t>
<list style="symbols">
<t>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.<vspace />
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.
<list style="symbols">
<t>point_to_octets(P) -&gt; ostr: returns the canonical representation of
the point P as an octet string.
This operation is also known as serialization.</t>
<t>octets_to_point(ostr) -&gt; 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.</t>
<t>subgroup_check(P) -&gt; 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., <xref target="Bowe19"></xref>.</t>
</list></t>
<t>I2OSP and OS2IP are the functions defined in <xref target="RFC8017"></xref>, Section 4.</t>
<t>hash_to_point(ostr) -&gt; 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 <xref target="I-D.irtf-cfrg-hash-to-curve"></xref>.
Each of the ciphersuites in <xref target="ciphersuites"></xref> specifies the hash_to_point
algorithm to be used.</t>
</list>
</t>
</section>

<section anchor="blsapi" title="API">
<t>The BLS signature scheme defines the following API:</t>
<t>
<list style="symbols">
<t>KeyGen(IKM) -&gt; SK: a key generation algorithm that
takes as input an octet string comprising secret keying material,
and outputs a secret key SK.</t>
<t>SkToPk(SK) -&gt; PK: an algorithm that takes as input a secret key
and outputs the corresponding public key.</t>
<t>Sign(SK, message) -&gt; signature: a signing algorithm that generates a
deterministic signature given a secret key SK and a message.</t>
<t>Verify(PK, message, signature) -&gt; VALID or INVALID:
a verification algorithm that outputs VALID if signature is a valid
signature of message under public key PK, and INVALID otherwise.</t>
<t>Aggregate((signature_1, ..., signature_n)) -&gt; signature:
an aggregation algorithm that aggregates a collection of signatures
into a single signature.</t>
<t>AggregateVerify((PK_1, ..., PK_n), (message_1, ..., message_n), signature) -&gt; 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.</t>
</list>
</t>
</section>

<section anchor="requirements" title="Requirements">
<t>The key words &quot;MUST&quot;, &quot;MUST NOT&quot;, &quot;REQUIRED&quot;, &quot;SHALL&quot;, &quot;SHALL NOT&quot;,
&quot;SHOULD&quot;, &quot;SHOULD NOT&quot;, &quot;RECOMMENDED&quot;, &quot;MAY&quot;, and &quot;OPTIONAL&quot; in this
document are to be interpreted as described in <xref target="RFC2119"></xref>.</t>
</section>
</section>

<section anchor="coreops" title="Core operations">
<t>This section defines core operations used by the schemes defined in <xref target="schemes"></xref>.
These operations MUST NOT be used except as described in that section.</t>

<section anchor="corevariants" title="Variants">
<t>Each core operation has two variants that trade off signature
and public key size:</t>
<t>
<list style="numbers">
<t>Minimal-signature-size: signatures are points in G1,
public keys are points in G2.
(Recall from <xref target="definitions"></xref> that E1 has a more compact representation than E2.)</t>
<t>Minimal-pubkey-size: public keys are points in G1,
signatures are points in G2.<vspace />
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.</t>
</list>
</t>
</section>

<section anchor="coreparams" title="Parameters">
<t>The core operations in this section depend on several parameters:</t>
<t>
<list style="symbols">
<t>A signature variant, either minimal-signature-size or minimal-pubkey-size.
These are defined in <xref target="corevariants"></xref>.</t>
<t>A pairing-friendly elliptic curve, plus associated functionality
given in <xref target="definitions"></xref>.</t>
<t>H, a hash function that MUST be a secure cryptographic hash function,
e.g., SHA-256 <xref target="FIPS180-4"></xref>.
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.</t>
<t>hash_to_point, a function whose interface is described in <xref target="definitions"></xref>.
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 <xref target="I-D.irtf-cfrg-hash-to-curve"></xref>.</t>
</list>
</t>
<t>In addition, the following primitives are determined by the above parameters:</t>
<t>
<list style="symbols">
<t>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 <xref target="definitions"></xref>).
When the signature variant is minimal-pubkey-size, P is the
distinguished point P1 that generates the group G1.</t>
<t>r, the order of the subgroups G1 and G2 defined by the pairing-friendly curve.</t>
<t>pairing, a function that invokes the function e of <xref target="definitions"></xref>,
with argument order depending on signature variant:
<list style="symbols">
<t>For minimal-signature-size:<vspace />
pairing(U, V) := e(U, V)</t>
<t>For minimal-pubkey-size:<vspace />
pairing(U, V) := e(V, U)</t>
</list></t>
<t>point_to_pubkey and point_to_signature, functions that invoke the
appropriate serialization routine (<xref target="definitions"></xref>) depending on
signature variant:
<list style="symbols">
<t>For minimal-signature-size:<vspace />
point_to_pubkey(P) := point_to_octets_E2(P)<vspace />
point_to_signature(P) := point_to_octets_E1(P)</t>
<t>For minimal-pubkey-size:<vspace />
point_to_pubkey(P) := point_to_octets_E1(P)<vspace />
point_to_signature(P) := point_to_octets_E2(P)</t>
</list></t>
<t>pubkey_to_point and signature_to_point, functions that invoke the
appropriate deserialization routine (<xref target="definitions"></xref>) depending on
signature variant:
<list style="symbols">
<t>For minimal-signature-size:<vspace />
pubkey_to_point(ostr) := octets_to_point_E2(ostr)<vspace />
signature_to_point(ostr) := octets_to_point_E1(ostr)</t>
<t>For minimal-pubkey-size:<vspace />
pubkey_to_point(ostr) := octets_to_point_E1(ostr)<vspace />
signature_to_point(ostr) := octets_to_point_E2(ostr)</t>
</list></t>
<t>pubkey_subgroup_check and signature_subgroup_check, functions
that invoke the appropriate subgroup check routine (<xref target="definitions"></xref>)
depending on signature variant:
<list style="symbols">
<t>For minimal-signature-size:<vspace />
pubkey_subgroup_check(P) := subgroup_check_E2(P)<vspace />
signature_subgroup_check(P) := subgroup_check_E1(P)</t>
<t>For minimal-pubkey-size:<vspace />
pubkey_subgroup_check(P) := subgroup_check_E1(P)<vspace />
signature_subgroup_check(P) := subgroup_check_E2(P)</t>
</list></t>
</list>
</t>
</section>

<section anchor="keygen" title="KeyGen">
<t>The KeyGen algorithm generates a secret key SK deterministically from
a secret octet string IKM.</t>
<t>KeyGen uses HKDF <xref target="RFC5869"></xref> instantiated with the hash function H.</t>
<t>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.</t>
<t>Because KeyGen is deterministic, implementations MAY choose either to store
the resulting SK or to store IKM and call KeyGen to derive SK when necessary.</t>
<t>KeyGen takes an optional parameter, key_info.
This parameter MAY be used to derive multiple independent keys from the same IKM.
By default, key_info is the empty string.</t>

<figure><artwork>SK = KeyGen(IKM)

Inputs:
- IKM, a secret octet string. See requirements above.

Outputs:
- SK, a uniformly random integer such that 0 &lt;= SK &lt; r.

Parameters:
- 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).
- &quot;BLS-SIG-KEYGEN-SALT-&quot; is an ASCII string comprising 20 octets.

Procedure:
1. PRK = HKDF-Extract(&quot;BLS-SIG-KEYGEN-SALT-&quot;, IKM || I2OSP(0, 1))
2. OKM = HKDF-Expand(PRK, key_info || I2OSP(L, 2), L)
3. SK = OS2IP(OKM) mod r
4. return SK
</artwork></figure>

</section>

<section anchor="sktopk" title="SkToPk">
<t>The SkToPk algorithm takes a secret key SK and outputs the corresponding
public key PK.</t>
<t>SK MUST be indistinguishable from uniformly random modulo r (<xref target="coreparams"></xref>)
and infeasible to guess, e.g., generated using a trusted source of randomness.
KeyGen (<xref target="keygen"></xref>) outputs SK meeting these requirements.
Other key generation approaches meeting these requirements MAY also be used;
details of such methods are beyond the scope of this document.</t>

<figure><artwork>PK = SkToPk(SK)

Inputs:
- SK, a secret integer such that 0 &lt;= SK &lt; 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
</artwork></figure>

</section>

<section anchor="keyvalidate" title="KeyValidate">
<t>The KeyValidate algorithm ensures that a public key is valid.</t>
<t>As an optimization, implementations MAY cache the result of KeyValidate
in order to avoid unnecessarily repeating validation for known keys.</t>

<figure><artwork>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 pubkey_subgroup_check(xP) is INVALID, return INVALID
4. return VALID
</artwork></figure>

</section>

<section anchor="coresign" title="CoreSign">
<t>The CoreSign algorithm computes a signature from SK, a secret key,
and message, an octet string.</t>

<figure><artwork>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
</artwork></figure>

</section>

<section anchor="coreverify" title="CoreVerify">
<t>The CoreVerify algorithm checks that a signature is valid for
the octet string message under the public key PK.</t>

<figure><artwork>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
</artwork></figure>

</section>

<section anchor="aggregate" title="Aggregate">
<t>The Aggregate algorithm aggregates multiple signatures into one.</t>

<figure><artwork>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 &gt;= 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
</artwork></figure>

</section>

<section anchor="coreaggregateverify" title="CoreAggregateVerify">
<t>The CoreAggregateVerify algorithm checks an aggregated signature
over several (PK, message) pairs.</t>

<figure><artwork>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 &gt;= 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
</artwork></figure>

</section>
</section>

<section anchor="schemes" title="BLS Signatures">
<t>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 (<xref target="definitions"></xref>).</t>
<t>All of the schemes in this section are built on a set of core operations
defined in <xref target="coreops"></xref>.
Thus, defining a scheme requires fixing a set of parameters as
defined in <xref target="coreparams"></xref>.</t>
<t>All three schemes expose the KeyGen, SkToPk, and Aggregate operations
that are defined in <xref target="coreops"></xref>.
The sections below define the other API functions (<xref target="blsapi"></xref>)
for each scheme.</t>

<section anchor="schemenul" title="Basic scheme">
<t>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.</t>
<t>The Sign and Verify functions are identical to CoreSign and
CoreVerify (<xref target="coreops"></xref>), respectively.
AggregateVerify is defined below.</t>

<section anchor="aggregateverify" title="AggregateVerify">
<t>This function first ensures that all messages are distinct, and then
invokes CoreAggregateVerify.</t>

<figure><artwork>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 &gt;= 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)
</artwork></figure>

</section>
</section>

<section anchor="schemeaug" title="Message augmentation">
<t>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.</t>

<section anchor="sign" title="Sign">
<t>To match the API for Sign defined in <xref target="blsapi"></xref>, 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.</t>
<t>Note that the point P and the point_to_pubkey function are
defined in <xref target="coreparams"></xref>.</t>

<figure><artwork>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)
</artwork></figure>

</section>

<section anchor="verify" title="Verify">

<figure><artwork>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)
</artwork></figure>

</section>

<section anchor="aggregateverify-1" title="AggregateVerify">

<figure><artwork>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 &gt;= 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)
</artwork></figure>

</section>
</section>

<section anchor="schemepop" title="Proof of possession">
<t>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.</t>
<t>The Sign, Verify, and AggregateVerify functions
are identical to CoreSign, CoreVerify, and CoreAggregateVerify
(<xref target="coreops"></xref>), respectively.
In addition, a proof of possession scheme defines three functions beyond
the standard API (<xref target="blsapi"></xref>):</t>
<t>
<list style="symbols">
<t>PopProve(SK) -&gt; proof: an algorithm that generates a proof of possession
for the public key corresponding to secret key SK.</t>
<t>PopVerify(PK, proof) -&gt; VALID or INVALID:
an algorithm that outputs VALID if proof is valid for PK, and INVALID otherwise.</t>
<t>FastAggregateVerify((PK_1, ..., PK_n), message, signature) -&gt; VALID or INVALID:
a verification algorithm for the aggregate of multiple signatures on
the same message.
This function is faster than AggregateVerify.</t>
</list>
</t>
<t>All public keys used by Verify, AggregateVerify, and FastAggregateVerify
MUST be accompanied by a proof of possession, and the result of evaluating
PopVerify on the public key and proof MUST be VALID.</t>

<section anchor="popparams" title="Parameters">
<t>In addition to the parameters required to instantiate the core operations
(<xref target="coreparams"></xref>), a proof of possession scheme requires one further parameter:</t>
<t>
<list style="symbols">
<t>hash_pubkey_to_point(PK) -&gt; 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.<vspace />
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 <xref target="I-D.irtf-cfrg-hash-to-curve"></xref>.<vspace />
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 <xref target="I-D.irtf-cfrg-hash-to-curve"></xref>, Section 3.1).
<xref target="ciphersuites-format"></xref> discusses the RECOMMENDED way to construct the
domain separation tag.</t>
</list>
</t>
</section>

<section anchor="popprove" title="PopProve">
<t>This function recomputes the public key coresponding to the input SK.
Implementations MAY instead implement an interface that takes the
public key as input.</t>
<t>Note that the point P and the point_to_pubkey and point_to_signature
functions are defined in <xref target="coreparams"></xref>.
The hash_pubkey_to_point function is defined in <xref target="popparams"></xref>.</t>

<figure><artwork>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
</artwork></figure>

</section>

<section anchor="popverify" title="PopVerify">
<t>PopVerify uses several functions defined in <xref target="coreops"></xref>.
The hash_pubkey_to_point function is defined in <xref target="popparams"></xref>.</t>
<t>As an optimization, implementations MAY cache the result of PopVerify
in order to avoid unnecessarily repeating validation for known keys.</t>

<figure><artwork>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
</artwork></figure>

</section>

<section anchor="fastaggregateverify" title="FastAggregateVerify">
<t>FastAggregateVerify uses several functions defined in <xref target="coreops"></xref>.</t>

<figure><artwork>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.

Precondition: n &gt;= 1, otherwise return INVALID.

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)
</artwork></figure>

</section>
</section>
</section>

<section anchor="ciphersuites" title="Ciphersuites">
<t>This section defines the format for a BLS ciphersuite.
It also gives concrete ciphersuites based on the BLS12-381 pairing-friendly
elliptic curve <xref target="I-D.irtf-cfrg-pairing-friendly-curves"></xref>.</t>

<section anchor="ciphersuites-format" title="Ciphersuite format">
<t>A ciphersuite specifies all parameters from <xref target="coreparams"></xref>,
a scheme from <xref target="schemes"></xref>, and any parameters the scheme requires.
In particular, a ciphersuite comprises:</t>
<t>
<list style="symbols">
<t>ID: the ciphersuite ID, an ASCII string. The REQUIRED format for
this string is<vspace />
&quot;BLS_SIG_&quot; || H2C_SUITE_ID || SC_TAG || &quot;_&quot;
<list style="symbols">
<t>Strings in double quotes are ASCII-encoded literals.</t>
<t>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.</t>
<t>SC_TAG is a string indicating the scheme and, optionally, additional information.
The first three characters of this string MUST chosen as follows:
<list style="symbols">
<t>&quot;NUL&quot; if SC is basic,</t>
<t>&quot;AUG&quot; if SC is message-augmentation, or</t>
<t>&quot;POP&quot; if SC is proof-of-possession.</t>
<t>Other values MUST NOT be used.</t>
</list>
<vspace />
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).<vspace />
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, &quot;NUL:version=2&quot; is an appropriate SC_TAG value.</t>
</list>
<vspace />
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.</t>
<t>SC: the scheme, one of basic, message-augmentation, or proof-of-possession.</t>
<t>SV: the signature variant, either minimal-signature-size or
minimal-pubkey-size.</t>
<t>EC: a pairing-friendly elliptic curve, plus all associated functionality
(<xref target="definitions"></xref>).</t>
<t>H: a cryptographic hash function.</t>
<t>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 <xref target="I-D.irtf-cfrg-hash-to-curve"></xref>.<vspace />
The RECOMMENDED hash-to-curve domain separation tag is the ciphersuite ID string defined above.</t>
<t>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 <xref target="I-D.irtf-cfrg-hash-to-curve"></xref>.<vspace />
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:<vspace />
&quot;BLS_POP_&quot; || H2C_SUITE_ID || SC_TAG || &quot;_&quot;</t>
</list>
</t>
</section>

<section anchor="ciphersuites-for-bls12-381" title="Ciphersuites for BLS12-381">
<t>The following ciphersuites are all built on the BLS12-381 elliptic curve.
The required primitives for this curve are given in <xref target="bls12381def"></xref>.</t>
<t>These ciphersuites use the hash-to-curve suites
BLS12381G1_XMD:SHA-256_SSWU_RO_ and
BLS12381G2_XMD:SHA-256_SSWU_RO_
defined in <xref target="I-D.irtf-cfrg-hash-to-curve"></xref>, 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).</t>

<section anchor="basic" title="Basic">
<t>BLS_SIG_BLS12381G1_XMD:SHA-256_SSWU_RO_NUL_ is defined as follows:</t>
<t>
<list style="symbols">
<t>SC: basic</t>
<t>SV: minimal-signature-size</t>
<t>EC: BLS12-381, as defined in <xref target="bls12381def"></xref>.</t>
<t>H: SHA-256</t>
<t>hash_to_point: BLS12381G1_XMD:SHA-256_SSWU_RO_ with the ASCII-encoded domain separation tag<vspace />
BLS_SIG_BLS12381G1_XMD:SHA-256_SSWU_RO_NUL_</t>
</list>
</t>
<t>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:</t>
<t>
<list style="symbols">
<t>SV: minimal-pubkey-size</t>
<t>hash_to_point: BLS12381G2_XMD:SHA-256_SSWU_RO_ with the ASCII-encoded domain separation tag<vspace />
BLS_SIG_BLS12381G2_XMD:SHA-256_SSWU_RO_NUL_</t>
</list>
</t>
</section>

<section anchor="message-augmentation" title="Message augmentation">
<t>BLS_SIG_BLS12381G1_XMD:SHA-256_SSWU_RO_AUG_ is defined as follows:</t>
<t>
<list style="symbols">
<t>SC: message-augmentation</t>
<t>SV: minimal-signature-size</t>
<t>EC: BLS12-381, as defined in <xref target="bls12381def"></xref>.</t>
<t>H: SHA-256</t>
<t>hash_to_point: BLS12381G1_XMD:SHA-256_SSWU_RO_ with the ASCII-encoded domain separation tag<vspace />
BLS_SIG_BLS12381G1_XMD:SHA-256_SSWU_RO_AUG_</t>
</list>
</t>
<t>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:</t>
<t>
<list style="symbols">
<t>SV: minimal-pubkey-size</t>
<t>hash_to_point: BLS12381G2_XMD:SHA-256_SSWU_RO_ with the ASCII-encoded domain separation tag<vspace />
BLS_SIG_BLS12381G2_XMD:SHA-256_SSWU_RO_AUG_</t>
</list>
</t>
</section>

<section anchor="proof-of-possession" title="Proof of possession">
<t>BLS_SIG_BLS12381G1_XMD:SHA-256_SSWU_RO_POP_ is defined as follows:</t>
<t>
<list style="symbols">
<t>SC: proof-of-possession</t>
<t>SV: minimal-signature-size</t>
<t>EC: BLS12-381, as defined in <xref target="bls12381def"></xref>.</t>
<t>H: SHA-256</t>
<t>hash_to_point: BLS12381G1_XMD:SHA-256_SSWU_RO_ with the ASCII-encoded domain separation tag<vspace />
BLS_SIG_BLS12381G1_XMD:SHA-256_SSWU_RO_POP_</t>
<t>hash_pubkey_to_point: BLS12381G1_XMD:SHA-256_SSWU_RO_ with the ASCII-encoded domain separation tag<vspace />
BLS_POP_BLS12381G1_XMD:SHA-256_SSWU_RO_POP_</t>
</list>
</t>
<t>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:</t>
<t>
<list style="symbols">
<t>SV: minimal-pubkey-size</t>
<t>hash_to_point: BLS12381G2_XMD:SHA-256_SSWU_RO_ with the ASCII-encoded domain separation tag<vspace />
BLS_SIG_BLS12381G2_XMD:SHA-256_SSWU_RO_POP_</t>
<t>hash_pubkey_to_point: BLS12381G2_XMD:SHA-256_SSWU_RO_ with the ASCII-encoded domain separation tag<vspace />
BLS_POP_BLS12381G2_XMD:SHA-256_SSWU_RO_POP_</t>
</list>
</t>
</section>
</section>
</section>

<section anchor="security-considerations" title="Security Considerations">

<section anchor="pubkeyvalid" title="Validating public keys">
<t>All algorithms in <xref target="coreops"></xref> and <xref target="schemes"></xref> 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.</t>
</section>

<section anchor="skipping-membership-check" title="Skipping membership check">
<t>Some existing implementations skip the signature_subgroup_check invocation
in CoreVerify (<xref target="coreverify"></xref>), 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.</t>
<t>
<list style="numbers">
<t>For most pairing-friendly elliptic curves used in practice, the pairing
operation e (<xref target="definitions"></xref>) 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.</t>
<t>Even if the pairing operation behaves properly on inputs that are outside
the correct subgroups, skipping the subgroup check breaks the strong
unforgeability property <xref target="ADR02"></xref>.</t>
</list>
</t>
</section>

<section anchor="side-channel-attacks" title="Side channel attacks">
<t>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.</t>
</section>

<section anchor="randomness-considerations" title="Randomness considerations">
<t>BLS signatures are deterministic. This protects against attacks
arising from signing with bad randomness, for example, the nonce reuse
attack on ECDSA <xref target="HDWH12"></xref>.</t>
<t>As discussed in <xref target="keygen"></xref>, 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.</t>
<t>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.</t>
</section>

<section anchor="implementing-hash-to-point-and-hash-pubkey-to-point" title="Implementing hash_to_point and hash_pubkey_to_point">
<t>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
<xref target="I-D.irtf-cfrg-hash-to-curve"></xref>.</t>
<t>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 <xref target="ciphersuites"></xref> is the RECOMMENDED one.</t>
</section>
</section>

<section anchor="implementation-status" title="Implementation Status">
<t>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.</t>
<t>
<list style="symbols">
<t>Algorand: <eref target="https://github.com/kwantam/bls_sigs_ref">bls_sigs_ref</eref>.</t>
<t>Chia: <eref target="https://github.com/Chia-Network/bls-signatures/blob/master/SPEC.md">spec</eref>
<eref target="https://github.com/Chia-Network/bls-signatures">python/C++</eref>.  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.</t>
<t>Dfinity: <eref target="https://github.com/dfinity/go-dfinity-crypto">go</eref> <eref target="https://github.com/dfinity/bls">BLS</eref>.  The current implementations do not seem to implement the membership check.</t>
<t>Ethereum 2.0: <eref target="https://github.com/ethereum/eth2.0-specs/blob/master/specs/bls_signature.md">spec</eref>.</t>
</list>
</t>
</section>

<section anchor="related-standards" title="Related Standards">
<t>
<list style="symbols">
<t>Pairing-friendly curves, <xref target="I-D.irtf-cfrg-pairing-friendly-curves"></xref></t>
<t>Pairing-based Identity-Based Encryption <eref target="https://ieeexplore.ieee.org/document/6662370">IEEE 1363.3</eref>.</t>
<t>Identity-Based Cryptography Standard <eref target="https://tools.ietf.org/html/rfc5091">rfc5901</eref>.</t>
<t>Hashing to Elliptic Curves <xref target="I-D.irtf-cfrg-hash-to-curve"></xref>, in order to implement the hash function hash_to_point.</t>
<t>EdDSA <eref target="https://tools.ietf.org/html/rfc8032">rfc8032</eref>.</t>
</list>
</t>
</section>

<section anchor="iana-considerations" title="IANA Considerations">
<t>TBD (consider to register ciphersuite identifiers for BLS signature and underlying
  pairing curves)</t>
</section>

</middle>

<back>
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      <organization>The University of Michigan</organization>
    </author>
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      <organization>The University of Michigan</organization>
    </author>
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      <organization>The University of Michigan</organization>
    </author>
    <date year="2012" month="August"></date>
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<reference anchor="RY07" target="https://link.springer.com/chapter/10.1007%2F978-3-540-72540-4_13">
  <front>
    <title>The Power of Proofs-of-Possession: Securing Multiparty Signatures against Rogue-Key Attacks</title>
    <author fullname="Thomas Ristenpart" initials="T." surname="Ristenpart">
      <organization>University of California, San Diego</organization>
    </author>
    <author fullname="Scott Yilek" initials="S." surname="Yilek">
      <organization>University of California, San Diego</organization>
    </author>
    <date year="2007" month="May"></date>
  </front>
</reference>
<?rfc include="https://xml2rfc.ietf.org/public/rfc/bibxml/reference.RFC.8017.xml"?>
<?rfc include="https://xml2rfc.ietf.org/public/rfc/bibxml/reference.RFC.5869.xml"?>
<reference anchor="ADR02" target="https://doi.org/10.1007/3-540-46035-7_6">
  <front>
    <title>On the Security of Joint Signature and Encryption</title>
    <author fullname="Jee Hea An" initials="J. H." surname="An">
      <organization>SoftMax Inc.</organization>
    </author>
    <author fullname="Yevgeniy Dodis" initials="Y." surname="Dodis">
      <organization>New York University</organization>
    </author>
    <author fullname="Tal Rabin" initials="T." surname="Rabin">
      <organization>IBM T.J. Watson Research Center</organization>
    </author>
    <date year="2002" month="April"></date>
  </front>
</reference>
<reference anchor="BNN07" target="https://link.springer.com/chapter/10.1007%2F978-3-540-73420-8_37">
  <front>
    <title>Unrestricted aggregate signatures</title>
    <author fullname="Mihir Bellare" initials="M." surname="Bellare">
      <organization>University of California, San Diego</organization>
    </author>
    <author fullname="Chanathip Namprepre" initials="C." surname="Namprempre">
      <organization>Thammasat University</organization>
    </author>
    <author fullname="Gregory Neven" initials="G." surname="Neven">
      <organization>Katholieke Universiteit Leuven</organization>
    </author>
    <date year="2007" month="July"></date>
  </front>
</reference>
<reference anchor="Bol03" target="https://link.springer.com/chapter/10.1007%2F3-540-36288-6_3">
  <front>
    <title>Threshold Signatures, Multisignatures and Blind Signatures Based on the Gap-Diffie-Hellman-Group Signature Scheme</title>
    <author fullname="Alexandra Boldyreva" initials="A." surname="Boldyreva">
      <organization>University of California, San Diego</organization>
    </author>
    <date year="2003" month="January"></date>
  </front>
</reference>
</references>

<section anchor="bls12381def" title="BLS12-381">
<t>The ciphersuites in <xref target="ciphersuites"></xref> are based upon the BLS12-381
pairing-friendly elliptic curve.
The following defines the correspondence between the primitives
in <xref target="definitions"></xref> and the parameters given in Section 4.2.2 of
<xref target="I-D.irtf-cfrg-pairing-friendly-curves"></xref>.</t>
<t>
<list style="symbols">
<t>E1, G1: the curve E and its order-r subgroup.</t>
<t>E2, G2: the curve E' and its order-r subgroup.</t>
<t>GT: the subgroup G_T.</t>
<t>P1: the point BP.</t>
<t>P2: the point BP'.</t>
<t>e: the optimal Ate pairing defined in Appendix A of
<xref target="I-D.irtf-cfrg-pairing-friendly-curves"></xref>.</t>
<t>point_to_octets and octets_to_point use the compressed
serialization formats for E1 and E2 defined by <xref target="ZCash"></xref>.</t>
<t>subgroup_check MAY use either the naive check described
in <xref target="definitions"></xref> or the optimized check given by <xref target="Bowe19"></xref>.</t>
</list>
</t>
</section>

<section anchor="test-vectors" title="Test Vectors">
<t>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.</t>
</section>

<section anchor="security-analyses" title="Security analyses">
<t>The security properties of the BLS signature scheme are proved in <xref target="BLS01"></xref>.</t>
<t><xref target="BGLS03"></xref> prove the security of aggregate signatures over distinct messages,
as in the basic scheme of <xref target="schemenul"></xref>.</t>
<t><xref target="BNN07"></xref> prove security of the message augmentation scheme of <xref target="schemeaug"></xref>.</t>
<t><xref target="Bol03"></xref><xref target="LOSSW06"></xref><xref target="RY07"></xref> prove security of constructions related to the proof
of possession scheme of <xref target="schemepop"></xref>.</t>
<t><xref target="BDN18"></xref> prove the security of another rogue key defense; this
defense is not standardized in this document.</t>
</section>

</back>

</rfc>