Low Complexity Bit-Parallel Finite Field Arithmetic Using Polynomial Basis

Bit-parallel finite field multiplication in F2m using polynomial basis can be realized in two steps: polynomial multiplication and reduction modulo the irreducible polynomial. In this article, we prove that the modular polynomial reduction can be done with (r − 1)(m − 1) bit additions, where r is the Hamming weight of the irreducible polynomial. We also show that a bit-parallel squaring operation using polynomial basis costs not more than ⌊ m+ k − 1 2 ⌋ bit operations if an irreducible trinomial of form x +x +1 over F2 is used. Consequently, it is argued that to solve multiplicative inverse in F2m using polynomial basis can be as good as using normal basis.