Implementation of Tate Pairing on Hyperelliptic Curves of Genus 2 YoungJu Choie and Eunjeong Lee ⋆ Dept. of Mathematics,POSTECH, Pohang, Korea {yjc,ejlee}@postech.ac.kr Abstract. Since Tate pairing was suggested to construct a cryptosys- tem, fast computation of Tate pairing has been researched recently. Bar- reto et. al[3] and Galbraith[8] provided efficient algorithms for Tate pair- ing on y 2 = x 3 - x + b in characteristic 3 and Duursma and Lee[6] gave a closed formula for Tate pairing on y 2 = x p - 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. Keywords- elliptic curve cryptosystem, Tate pairing implementation, hy- perelliptic curve cryptosystem 1 Introduction Since Weil pairing was proposed to cryptanalysis[16], Weil pairing and Tate pair- ing have contributed to two different aspects in cryptography community; one is attacking a cryptosystem[7] and the other side is building a cryptosystem[1]. Recently, the cryptosystem based on pairings becomes one of the most active research fields ([3],[2],[6],[9],[18],[21]). Tate pairing can be computed using an al- gorithm first suggested by Miller [14] which is described in [2], [3] and [9] for the case of elliptic curves. Miller algorithm on elliptic curves is basically the usual scalar point multiplication with an evaluation of certain intermediate rational functions which are straight lines used in the addition process. ⋆ This work was partially supported by University ITRC fund