Fast computation of Tate pairing on general divisors for hyperelliptic curves of genus

For the Tate pairing implementation over hyperelliptic curves, there is a development by DuursmaLee and Barreto et al., and those computations are focused on degenerate divisors. As divisors are not degenerate form in general, it is necessary to find algorithms on general divisors for the Tate pairing computation. In this paper, we present two efficient methods for computing the Tate pairing over divisor class groups of the hyperelliptic curves y = x − x + d, d = ±1 of genus 3. First, we provide the point-wise approach, which is a generalization of the previous developments by Duursma-Lee and Barreto et al. In the second method, we use a Resultant for the Tate pairing computation. The approach by using the Resultant is approximately three times faster than the point-wise approach. These two methods are completely general in a sense that they work for general divisors, and they provide very explicit algorithms. keywords: Tate pairing; hyperelliptic curves; divisors; eta pairing; resultant; pairing-based cryptosystem