Notes on Low Degree Toom-Cook Multiplication with Small Characteristic

The use of Toom-Cook sub-quadratic polynomial multiplication was recently shown to be possible also when the coefficient field does not have elements enough, particularly for F2[x]. This paper focus on how Toom’s strategies can be adapted to polynomials on non-binary small fields. In particular we describe the Toom-3 algorithm for Fp[x] with p ∈ {3, 5, 7}, and some other unbalanced algorithms. Algorithms are described with full details, not only the asymptotic complexity is given, but the exact sequence of operations for possibly optimal implementations. Algorithms given here for Fp[x] perfectly works, as well, for Fpn [x], and may be useful for computations inside Fpn for large enough n. Moreover some connections with FFT-based multiplication are found, slightly improving the worst cases of FFT.