eth2.0-specs/specs/bls_verify.md

BLS signature verification

Warning: This document is pending academic review and should not yet be considered secure.

Table of contents

• BLS signature verification
  • Table of contents
  • Curve parameters
  • Point representations
    • G1 points
    • G2 points
  • Helpers
    • hash_to_G2
    • modular_squareroot
  • Signature verification
    • bls_verify
    • bls_verify_multiple

Curve parameters

The BLS12-381 curve parameters are defined here.

Point representations

We represent points in the groups G1 and G2 following zkcrypto/pairing. We denote by q the field modulus and by i the imaginary unit.

G1 points

A point in G1 is represented as a 384-bit integer z decomposed as a 381-bit integer x and three 1-bit flags in the top bits:

• x = z % 2**381
• a_flag = (z % 2**382) // 2**381
• b_flag = (z % 2**383) // 2**382
• c_flag = (z % 2**384) // 2**383

Respecting bit ordering, z is decomposed as (c_flag, b_flag, a_flag, x).

We require:

• x < q
• c_flag == 1
• if b_flag == 1 then a_flag == x == 0 and z represents the point at infinity
• if b_flag == 0 then z represents the point (x, y) where y is the valid coordinate such that (y * 2) // q == a_flag

G2 points

A point in G2 is represented as a pair of 384-bit integers (z1, z2). We decompose z1 as above into x1, a_flag1, b_flag1, c_flag1 and z2 into x2, a_flag2, b_flag2, c_flag2.

We require:

• x1 < q and x2 < q
• a_flag2 == b_flag2 == c_flag2 == 0
• c_flag1 == 1
• if b_flag1 == 1 then a_flag1 == x1 == x2 == 0 and (z1, z2) represents the point at infinity
• if b_flag1 == 0 then (z1, z2) represents the point (x1 * i + x2, y) where y is the valid coordinate such that the imaginary part y_im of y satisfies (y_im * 2) // q == a_flag1

Helpers

hash_to_G2

G2_cofactor = 305502333931268344200999753193121504214466019254188142667664032982267604182971884026507427359259977847832272839041616661285803823378372096355777062779109
q = 0x1a0111ea397fe69a4b1ba7b6434bacd764774b84f38512bf6730d2a0f6b0f6241eabfffeb153ffffb9feffffffffaaab

def hash_to_G2(message, domain):
    x1 = int.from_bytes(hash(bytes8(domain) + b'\x01' + message), 'big')
    x2 = int.from_bytes(hash(bytes8(domain) + b'\x02' + message), 'big')
    x_coordinate = FQ2([x1, x2]) # x1 + x2 * i
    while 1:
        x_cubed_plus_b2 = x_coordinate ** 3 + FQ2([4, 4])
        y_coordinate = modular_squareroot(x_cubed_plus_b2)
        if y_coordinate is not None:
            break
        x_coordinate += FQ2([1, 0]) # Add one until we get a quadratic residue
    assert is_on_G2((x_coordinate, y_coordinate))
    return multiply_in_G2((x_coordinate, y_coordinate), G2_cofactor)

modular_squareroot

modular_squareroot(x) returns a solution y to y**2 % q == x, and None if none exists. If there are two solutions the one with higher imaginary component is favored; if both solutions have equal imaginary component the one with higher real component is favored.

qmod = q ** 2 - 1
eighth_roots_of_unity = [FQ2([1,1]) ** ((qmod * k) // 8) for k in range(8)]

def modular_squareroot(value):
    candidate_squareroot = value ** ((qmod + 8) // 16)
    check = candidate_squareroot ** 2 / value
    if check in eighth_roots_of_unity[::2]:
        x1 = candidate_squareroot / eighth_roots_of_unity[eighth_roots_of_unity.index(check) // 2]
        x2 = -x1
        return x1 if (x1.coeffs[1].n, x1.coeffs[0].n) > (x2.coeffs[1].n, x2.coeffs[0].n) else x2
    return None

Signature verification

In the following e is the pairing function and g is the generator in G1.

bls_verify

Let bls_verify(pubkey: uint384, message: bytes32, signature: [uint384], domain: uint64) -> bool:

• Verify that pubkey is a valid G1 point.
• Verify that signature is a valid G2 point.
• Verify that e(pubkey, hash_to_G2(message, domain)) == e(g, signature).

bls_verify_multiple

Let bls_verify_multiple(pubkeys: [uint384], messages: [bytes32], signature: [uint384], domain: uint64) -> bool:

• Verify that each pubkey in pubkeys is a valid G1 point.
• Verify that signature is a valid G2 point.
• Verify that len(pubkeys) equals len(messages) and denote the length L.
• Verify that e(pubkeys[0], hash_to_G2(messages[0], domain)) * ... * e(pubkeys[L-1], hash_to_G2(messages[L-1], domain)) == e(g, signature).