Elliptic Curve Point Multiplication in GF(2<sup>n</sup>) using Polynomial Residue Arithmetic

Abstract

Elliptic Curve Point Multiplication is the main operation employed in all elliptic curve cryptosystems, as it forms the basis of the Elliptic Curve Discrete Logarithm Problem. Therefore, the efficient realization of an Elliptic Curve Point Multiplier is of fundamental importance, as its performance is decisive for the performance of the overall cryptosystem. This work presents the first practical implementation of an Elliptic Curve Point Multiplier in GF(2n) using Polynomial Residue Arithmetic. Unlike the typical representation of GF(2n) elements as polynomials in GF(2)[x] of degree at most n − 1, data are represented as their remainder modulo a set of L pairwise prime polynomials m 1 ,m 2 , … ,m L of degree w and such that Lw ≥ 2n. The methodology for incorporating Polynomial Residue Arithmetic in the elliptic curve point addition and doubling algorithms, as well as the VLSI architecture of the proposed point multiplier are analyzed, thus forming an interesting alternative to Elliptic Curve Cryptography realization.