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

Finite field GF(2^m) arithmetic is becoming increasingly important for a variety of different applications including cryptography, coding theory and computer algebra. Among finite field arithmetic operations, GF(2^m) 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(2^m). For irreducible GF(2^m) generating polynomials f(x)=xm+xkt+xk^t-1+⋯+xk¹+1 with m ≥ 2k_t-1, the proposed multiplier has a theoretical latency of 2kt+1 cycles . This latency is the lowest one found in the literature for GF(2^m) 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.