Parallel Montgomery multiplication in GF(2k) using trinomial residue arithmetic

We propose the first general multiplication algorithm in GF(2^k) with a subquadratic area complexity of O(k⁸5/) = O(k^1.6). Using the Chinese remainder theorem, we represent the elements of GF(2^k); i.e. the polynomials in GF(2) [X] of degree at most k-1, by their remainder modulo a set of n pairwise prime trinomials, T₁,...,T_n, of degree d and such that nd ≥ k. Our algorithm is based on Montgomery´s multiplication applied to the ring formed by the direct product of the trinomials.