Low complexity bit-parallel finite field arithmetic using polynomial basis

Bit-parallel finite field multiplication in F-2m using polynomial basis can be realized in two steps: polynomial multiplication and reduction module 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(m) + x(k) + 1 over F-2 is used. Consequently, it is argued that to solve multiplicative inverse in F-2m using polynomial basis can be as good as using normal basis.