Pseudo-random scalar multiplication based on group isomorphism

Abstract

Elliptic curve cryptography is an essential cryptography which is widely used in data encryption, key agreement, digital signature, and other applications. Scalar multiplication is the fundamental operation in all ECC schemes, such as ECDSA, ECIES, and ECMQV. Even a slight improvement of scalar multiplication is precious. Various methods have been proposed for improving efficiency of the scalar multiplication, including the windows method, the NAF method, and the w-NAF method. However, the endeavour in this direction almost exhausted in the past several years since it is hard to find a method substantially better than the w-NAF method. This paper focuses on the scalar multiplication algorithm for the case when the scalar is a pseudo-random number. A faster pseudo-random scalar multiplication method is proposed based on a group isomorphism between the pseudo-random number group and elliptic curve point group. Experimental results show that the proposed method can considerably reduce the computation time compared with those traditional methods. The pseudo-random scalar multiplication accounts for a significant proportion of total scalar multiplication operations in almost all ECC schemes. Therefore, the proposed method is promising and applicable for various ECC applications in the fields such as Internet of things, edge computing, and swarm robotics.