groth-pairing

Testing "pairing function" in the context of Elliptic Curves constructed using a tower of field extensions.

Elliptic curves:

• BN128 (file src/BN128.hs)
• BLS12-381 (file src/BLS12381.hs)

Pairing function

Let G1 be the elliptic curve over Fp1 (the base field) and G2 the elliptic curve over Fp2 (the first extension). Then

pairing :: G1 -> G2 -> Fp12

where Fp12 is the extended field according to the curve's embedding degree 'k = 12' ("top floor" in the tower).

Points on G1 and G2

A point in G1 is constructed with ecExp g1Gen n:

ghci> :t ecExp g1Gen
ecExp g1Gen :: Integer -> G1

Likewise, a point in G2 is constructed with ecExp g2Gen n.

Here n is an integer between 0 and q, with

• For BN128:

q = 21888242871839275222246405745257275088548364400416034343698204186575808495617

• For BLS12-381:

q = 52435875175126190479447740508185965837690552500527637822603658699938581184513

Function ecExp is an efficient implementation of the exponential function on an elliptic curve, so that e.g. ecExp p 5 is effectively equivalent to p <> p <> p <> p <> p.

As a check, one can test that a constructed point p is on the elliptic curve with isOnCurve p,

ghci> :t isOnCurve 
isOnCurve :: Field a => EllipticCurve a -> Bool

Bilinearity

It is expected that the pairing satisfies the bilinearity property:

pairing (p1 <> p2) q == pairing p1 q * pairing p2 q

pairing p (q1 <> q2) == pairing p q1 * pairing p q2

The test of the property can be checked for the BN-128 curve with the following commands:

cabal test bilineal-property-128

And the BLS12-381 curve:

cabal test bilineal-property-381

ZK verification

To check the viability of using our pairing implementation in conjunction with an open source ZK tooling like snarkjs, we test the validation of a proof produced by such a tool; see files in src/Groth16 and corresponding log in directory logs.