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+⋯+
xk
¹
+1 with
m
≥ 2
k
_t
-1, the proposed multiplier has a theoretical latency of 2
kt
+1 cycles . This latency is the lowest one found in the literature for
GF
(2
^m
) multipliers. Furthermore, the condition
m
≥ 2
kt
-1 is specially important because the five binary irreducible polynomials recommended by NIST for elliptic curve cryptography (ECC) implementation verify this condition.