Paper 2005/166

Tate pairing computation on the divisors of hyperelliptic curves for cryptosystems

Eunjeong Lee and Yoonjin Lee

Abstract

In recent papers \cite{Bar05} and \cite{CKL}, Barreto et al and Choie et al worked on hyperelliptic curves $H_b$ defined by $y^2+y = x^5 + x^3 + b$ over a finite field $\Ftn$ with $b=0$ or $1$ for a secure and efficient pairing-based cryptosystems. We find a completely general method for computing the Tate-pairing over divisor class groups of the curves $H_b$ in a very explicit way. In fact, the Tate-pairing is defined over the entire divisor class group of a curve, not only over the points on a curve. So far only pointwise approach has been made in ~\cite{Bar05} and ~\cite{CKL} for the Tate-pairing computation on the hyperelliptic curves $H_b$ over $\Ftn$. Furthermore, we obtain a very efficient algorithm for the Tate pairing computation over divisors by reducing the cost of computing. We also find a crucial condition for divisor class group of hyperelliptic curve to have a significant reduction of the loop cost in the Tate pairing computation.

Note: Minor corrections have been made.