On a Relation between the Ate Pairing and the Weil Pairing for Supersingular Elliptic Curves
The hyperelliptic curve Ate pairing provides an efficient way to compute a bilinear pairing on the Jacobian variety of a hyperelliptic curve. We prove that, for supersingular elliptic curves with embedding degree two, square of the Ate pairing is nothing but the Weil pairing. Using the formula, we develop an X -coordinate only pairing inversion method. However, the algorithm is still infeasible for cryptographic size problems.