Low Latency $GF(2^{m})$ Polynomial Basis Multiplier

Abstract

Finite field GF(2m) arithmetic is becoming increasingly important for a variety of different applications including cryptography, coding theory and computer algebra. Among finite field arithmetic operations, GF(2m) multiplication is of special interest because it is considered the most important building block. This contribution describes a new low latency parallel-in/parallel-out sequential polynomial basis multiplier over GF(2m). For irreducible GF(2m) generating polynomials f(x)=xm+xkt+xkt-1+⋯+xk1+1 with m ≥ 2kt-1, the proposed multiplier has a theoretical latency of 2kt+1 cycles . This latency is the lowest one found in the literature for GF(2m) multipliers. Furthermore, the condition m ≥ 2kt-1 is specially important because the five binary irreducible polynomials recommended by NIST for elliptic curve cryptography (ECC) implementation verify this condition.