The generalized Weil pairing and the discrete logarithm problem on elliptic curves

We review the construction of a generalization of the Weil pairing, which is non-degenerate and bilinear, and use it to construct a reduction from the discrete logarithm problem on elliptic curves to the discrete logarithm problem in finite fields. We show that the new pairing can be computed efficiently for curves with trace of Frobenius congruent to 2 modulo the order of the base point. This leads to an efficient reduction for this class of curves. The reduction is as simple to construct as that of Menezes et al. (IEEE Trans. Inform. Theory, 39, 1993), and is provably equivalent to that of Frey and Rück (Math. Comput. 62 (206) (1994) 865).