Tate Pairing Implementation for Hyperelliptic Curves y 2 = x p – x + d

The Weil and Tate pairings have been used recently to build new schemes in cryptography. It is known that the Weil pairing takes longer than twice the running time of the Tate pairing. Hence it is necessary to develop more efficient implementations of the Tate pairing for the practical application of pairing based cryptosystems. In 2002, Barreto et al. and Galbraith et al. provided new algorithms for the fast computation of the Tate pairing in characteristic three. In this paper, we give a closed formula for the Tate pairing on the hyperelliptic curve y2 = xp – x + d in characteristic p. This result improves the implementations in [BKLS02], [GHS02] for the special case p=3.