Low space complexity CRT-based bit-parallel GF(2n) polynomial basis multipliers for irreducible trinomials

By selecting the largest possible value of k ∈ (n/2, 2n/3], we further reduce the AND and XOR gate complexities of the CRT-based hybrid parallel GF (2) polynomial basis multipliers for the irreducible trinomial f = u + u + 1 over GF (2): they are always less than those of the current fastest parallel multipliers – quadratic multipliers, i.e., n AND gates and n− 1 XOR gates. Our experimental results show that among the 539 values of n ∈ [5, 999] such that f is irreducible for some k ∈ [2, n− 2], there are 317 values of n such that k ∈ (n/2, 2n/3]. For these irreducible trinomials, the AND and XOR gate complexities of the CRT-based hybrid multipliers are reduced by 15.3% on average. Especially, for the 124 values of such n, the two kinds of multipliers have the same time complexity, but the space complexities are reduced by 15.5% on average. As a comparison, the previous CRT-based multipliers consider the case k ∈ [2, n/2], and the improvement rate is only 8.4% on average.