Warning: This is toy POC and an academic review is required.

Validate multiple signatures on Ethereum chain using BLS.

Proofs-of-possession(POP) at registration to address the rogue public key attack see here

• This code does not check for POP.
• We can test POP at validator/user registration time. We can get any data signed and verify using pubkey. However, we will add this later.

• We are currently using py_ecc py_ecc/fork for testing/poc purpose
• In the future we will use rust code for faster results

BN256G2 is used for solidity G2 point operations and pre-compiles ECMUL, ECADD and ECPARING

• Test data/points generated using BLSSmall.py

Generator for curve over FQ

G1 = (FQ(1), FQ(2))

Generator for twisted curve over FQ2

G2 = (
    FQ2([11559732032986387107991004021392285783925812861821192530917403151452391805634,
        10857046999023057135944570762232829481370756359578518086990519993285655852781]),
    FQ2([4082367875863433681332203403145435568316851327593401208105741076214120093531,
        8495653923123431417604973247489272438418190587263600148770280649306958101930]))
)

Securely generated number

Public keys are G1 points on the curve. pk = mul(G1, sk)

• Message m = data to be signed
• h = Hash of Message m or some numeric deterministic representation
• H is Point on G2
• Currently, for toy version we are using hashToG2 => mul(G2, h)
• Some better methods for hashToG1/G2 [try-and-increment, hashingToBNCurves] can be used for real use cases.

• sig is a valid G2 point
• sig = mul(H, sk)

aggPubkey
for each pubkey in pubkeys:
  aggPubkey = add(aggPubkey, pubkey)

where add is the elliptic curve addition operation over the G1 curve and the empty aggSigs is the G1 point at infinity.

aggSigs
for each sig in sigs:
  aggSigs = add(aggSig, sig)

where add is the elliptic curve addition operation over the G2 curve and the empty aggSigs is the G2 point at infinity.

In the following, e is the pairing function with the following coordinates (see here):

aggPubkey = aggregated pubkeys aggSig is aggregation of n sigs from n privkeys/validators

• Each pubkey in pubkeys is a valid G1 point.
• signature is a valid G2 point.
• Verify pairing e(add(pubkeys[0]...[n]), hashtoG2(message)) == e(G1, aggSig).

where add is the elliptic curve addition operation over the G1 curve and the empty aggSigs is the G1 point at infinity.

def hash_to_G2(message_hash: Bytes32, domain: uint64) -> [uint384]:
    # Initial candidate x coordinate
    x_re = int.from_bytes(hash(message_hash + bytes8(domain) + b'\x01'), 'big')
    x_im = int.from_bytes(hash(message_hash + bytes8(domain) + b'\x02'), 'big')
    x_coordinate = Fq2([x_re, x_im])  # x = x_re + i * x_im

    # Test candidate y coordinates until a one is found
    while 1:
        y_coordinate_squared = x_coordinate ** 3 + Fq2([4, 4])  # The curve is y^2 = x^3 + 4(i + 1)
        y_coordinate = modular_squareroot(y_coordinate_squared)
        if y_coordinate is not None:  # Check if quadratic residue found
            return multiply_in_G2((x_coordinate, y_coordinate), G2_cofactor)
        x_coordinate += Fq2([1, 0])  # Add 1 and try again

Above algorithm from ETH2.0 can be used for hashToG2.

1. https://www.di.ens.fr/~fouque/pub/latincrypt12.pdf
2. https://crypto.stanford.edu/~dabo/pubs/papers/BLSmultisig.html
3. https://crypto.stanford.edu/pbc/thesis.pdf
4. http://www.craigcostello.com.au/pairings/PairingsForBeginners.pdf
5. https://eprint.iacr.org/2007/264.pdf
6. https://eprint.iacr.org/2008/530.pdf