Parallel Montgomery Multiplication in GF (2^k) Using Trinomial Residue Arithmetic
Jean-Claude Bajard, Laurent Imbert, GRAHAM A. JULLIEN · 2005
We propose the first general multiplication algorithm in GF(2/sup k/) with a subquadratic area complexity of O(k/sup 8/5/) = O(k/sup 1.6/). Using the Chinese remainder theorem, we represent the elements of GF(2/sup 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, T/sub 1/,...,T/sub n/, of degree d and such that nd /spl ges/ k. Our algorithm is based on Montgomery's multiplication applied to the ring formed by the direct product of the trinomials.