Expansion of Sliding Window Method for Finding Shorter Addition/Subtraction-Chains

Abstract

Finding a shorter addition/subtraction-chain for an integer is an important problem for many cryptographic systems based on number theory. Especially, execution time of multiplication on an elliptic curve cryptosystem is directly proportional to the length of the addition/subtraction-chain. In this paper, we propose an algorithm to find an addition/subtraction-chain. The proposed algorithm is based on the small-window method, and reduces the number of windows by using subtractions. We show the proposed algorithm finds the shorter addition/subtraction-chain than what can be found by any other previous algorithm.