Elliptic Curve Cryptography over Gaussian Integers

Abstract

Thus far, we have reviewed the point multiplication \(k\cdot P\) according to ( 2.9) with the binary expansion of the key. In this chapter, we alternatively consider the \(\tau \)-adic expansion of the integer k with a non-binary basis \(\tau \). We call the scalar multiplication with a point on the elliptic curve a complex point multiplication if the basis \(\tau \) is a complex number such as a Gaussian, Eisenstein, or Kleinian integer [33, 88]. Non-binary expansions were proposed to speed up the point multiplication [30–32, 34–36]. Moreover, it was demonstrated in [22, 27, 29, 32, 34, 36, 75] that non-binary expansions are beneficial to harden ECC implementations against SCA [14, 25–28]. Implementations of the elliptic curve point multiplication are prone to SCA. There exist different attacks on the point multiplication in the literature, such as TA, SPA, DPA, RPA, and ZPA [14, 25–28].