Side Channel Attack Resistance of the Elliptic Curve Point Multiplication using Gaussian Integers

Abstract

Elliptic curve cryptography is a cornerstone of embedded security. However, hardware implementations of the elliptic curve point multiplication are prone to side channel attacks. In this work, we present a new key expansion algorithm which improves the resistance against timing and simple power analysis attacks. Furthermore, we consider a new concept for calculating the point multiplication, where the points of the curve are represented as Gaussian integers. Gaussian integers are subset of the complex numbers, such that the real and imaginary parts are integers. Since Gaussian integer fields are isomorphic to prime fields, this concept is suitable for many elliptic curves. Representing the key by a Gaussian integer expansion is beneficial to reduce the computational complexity and the memory requirements of a secure hardware implementation.