Low Space Complexity $GF(2^m)$ Multiplier for Trinomials Using $n$ -Term Karatsuba Algorithm

We propose bit-parallel GF(2
^m
) multipliers for irreducible trinomials using an n-term Karatsuba algorithm and Mastrovito approach, which are generalizations of the newly proposed multiplication scheme for a specific trinomial. The complexities of the proposed multipliers for GF(2
^m
) depend on the choice of an irreducible trinomial x
^m
+ x
^k
+ 1 defining GF(2
^m
) and values n, m
₀
such that m = nm
₀
or m = nm
₀
+ 1. It is possible to achieve a space-time tradeoff by choosing proper values for k, n, and m
₀
. For the purpose of a specific comparison, we compare the proposed multipliers with the best-known multipliers for an odd m ∈ [399, 450] for which there exists an irreducible trinomial of degree m. As a result, the proposed multipliers achieve the lowest space complexities among similar bit-parallel multipliers (they have roughly 40% reduced space complexities compared with the fastest multiplier). On the other hand, their time complexities match or are at most 2T
_X
higher than the fastest multipliers, where T
_X
is the delay of one 2-input XOR gate.