Co-factor Clearing and Subgroup Membership Testing on Pairing-Friendly Curves

An important cryptographic operation on elliptic curves is hashing to a point on the curve. When the curve is not of prime order, the point is multiplied by the cofactor so that the result has a prime order. This is important to avoid small subgroup attacks for example. A second important operation, in the composite-order case, is testing whether a point belongs to the subgroup of prime order. A pairing is a bilinear map e: G(1) x G(2) -> GT where G(1) and G(2) are distinct subgroups of prime order r of an elliptic curve, and G(T) is a multiplicative subgroup of the same prime order r of a finite field extension. Pairing-friendly curves are rarely of prime order. We investigate cofactor clearing and subgroup membership testing on these composite-order curves. First, we generalize a result on faster cofactor clearing for BLS curves to other pairingfriendly families of a polynomial form from the taxonomy of Freeman, Scott and Teske. Second, we investigate subgroup membership testing for G(1) and G(2). We fix a proof argument for the G(2) case that appeared in a preprint by Scott in late 2021 and has recently been implemented in different cryptographic libraries. We then generalize the result to both G(1) and G(2) and apply it to different pairing-friendly families of curves. This gives a simple and shared framework to prove membership tests for both cryptographic subgroups.