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

An important cryptographic operation on elliptic curves is hashing to a point the curve. When curve not of prime order, multiplied by cofactor so that result has order. This avoid small subgroup attacks for example. A second operation, in composite-order case, testing whether belongs pairing bilinear map $$e :\mathbb G_1 \times \mathbb G_2 \rightarrow G_T$$ where $$\mathbb G_1$$ and G_2$$ are distinct subgroups order r an curve, multiplicative same finite field extension. Pairing-friendly rarely We investigate clearing membership these curves. First, we generalize faster BLS other pairing-friendly families polynomial form from taxonomy Freeman, Scott Teske. Second, . fix proof argument case appeared preprint late 2021 recently been implemented different libraries. then both apply it gives simple shared framework prove tests subgroups.