Polynomial Basis Multipliers for Irreducible Trinomials

We show that the step “modulo the degree-n field generating irreducible polynomial” in the classical definition of the GF (2) multiplication operation can be avoided. This leads to an alternative representation of the finite field multiplication operation. Combining this representation and the Chinese Remainder Theorem, we design bit-parallel GF (2) multipliers for irreducible trinomials u + u + 1 on GF (2) where 1 < k ≤ n/2. For some values of n, our architectures have the same time complexity as the fastest bit-parallel multipliers – the quadratic multipliers, but their space complexities are reduced. Take the special irreducible trinomial u + u + 1 for example, the space complexity of the proposed design is reduced by about 1/8, while the time complexity matches the best result. Our experimental results show that among the 539 values of n such that 4 < n < 1000 and x+x+1 is irreducible over GF (2) for some k in the range 1 < k ≤ n/2, the proposed multipliers beat the current fastest parallel multipliers for 290 values of n when (n − 1)/3 ≤ k ≤ n/2: they have the same time complexity, but the space complexities are reduced by 8.4% on average.