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 ...

The origin of this work was the search for a “Descartes’ rule” for finite fields a nontrivial upper bound for the number of roots of sparse polynomials. In [2], Bi, Cheng, and Rojas establish such an upper bound. Then, in [3], Cheng, Gao, Rojas, and Wan show that the bound is essentially optimal in an infinite number of cases by constructing t-nomials with many roots in Fpt . However, the bound...