Implementation of Tate Pairing on Hyperelliptic Curves of Genus 2

Since Tate pairing was suggested to construct a cryptosystem, fast computation of Tate pairing has been researched recently. Barreto et. al[3] and Galbraith[8] provided efficient algorithms for Tate pairing on y = x − x + b in characteristic 3 and Duursma and Lee[6] gave a closed formula for Tate pairing on y = x − x + d in characteristic p. In this paper, we present completely general and explicit formulae for computing of Tate pairing on hyperelliptic curves of genus 2. We have computed Tate parings on a supersingular hyperelliptic curve over prime fields and the detailed algorithms are explained. This is the first attempt to present the implementation results for Tate pairing on a hyperelliptic curve of genus bigger than 1. Keywordselliptic curve cryptosystem, Tate pairing implementation, hyperelliptic curve cryptosystem