A Chinese Remainder Theorem Approach to Bit-Parallel GF(2^n) 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 $GF(2^{n})$ 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 $GF(2^{n})$ multipliers for irreducible trinomials $u^n+u^k+1$ on $GF(2)$ where $1 < k \\le n/2$ . 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 $u^{2k}+u^k+1$ 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 $4< n < 1{,}000$ and $x^n+x^k+1$ is irreducible over $GF(2)$ for some $k$ in the range $1 < k \\le n/2$ , the proposed multipliers beat the current fastest parallel multipliers for 290 values of $n$ when $(n-1)/3 \\le k \\le n/2$ : they have the same time complexity, but the space complexities are reduced by $8.4$ percent on average.