A New Type of Fast Endomorphisms on Jacobians of Hyperelliptic Curves and Their Cryptographic Application

Abstract

The Gallant-Lambert-Vanstone method [14] (GLV method for short) is a scalar multiplication method for elliptic curve cryptography (ECC). In WAP WTLS [49], SEC 2 [44], ANSI X9.62 [1] and X9.63 [2], several domain parameters for applications of the GLV method are described. Curves with those parameters have efficiently-computable endomorphisms. Recently the GLV method for Jacobians of hyperelliptic curve (HEC) has also been studied. In this paper, we discuss applications of the GLV method to curves with real multiplication (RM). It is the first time to use RM for efficient scalar multiplication as far as we know. We describe the general algorithm for using such RM, and we show that some genus 2 curves with RM have enough effciency to be used in the GLV method as in the previous CM case. Moreover, we will see that such RM curves can be obtained abundantly unlike the previously proposed CM curves of genus 2.