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 ≤ 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 ≤ 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 percent on average.