Towards Optimal Toom-Cook Multiplication for Univariate and Multivariate Polynomials in Characteristic 2 and 0

Abstract

Abstract. Toom-Cook strategy is a well-known method for building algorithms to efficiently multiply dense univariate polynomials. Efficiency of the algorithm depends on the choice of interpolation points and on the exact sequence of operations for evaluation and interpolation. If carefully tuned, it gives the fastest algorithm for a wide range of inputs. This work smoothly extends the Toom strategy to polynomial rings, with a focus on GF2 [x]. Moreover a method is proposed to find the faster Toom multiplication algorithm for any given splitting order. New results found with it, for polynomials in characteristic 2, are presented. A new extension for multivariate polynomials is also introduced; through a new definition of density leading Toom strategy to be efficient.

Faster Multiplication in GF(2)[x]

“…In particular, the multiplication of polynomials over GF (2) has received much attention in the literature, both in hardware and software. It is indeed a key operation for cryptographic applications [22], for polynomial factorisation or irreducibility tests [20,3].…”

“…It is indeed a key operation for cryptographic applications [22], for polynomial factorisation or irreducibility tests [20,3]. Some applications are less known, for example in integer factorisation, where multiplication in GF (2)[x] can speed up Berlekamp-Massey's algorithm inside the (block) Wiedemann algorithm [17,1].…”

“…Apart from the classical O(n 2 ) algorithm, and Karatsuba's algorithm which readily extends to GF (2)[x], Schönhage in 1977 and Cantor in 1989 proposed algorithms of complexity O(n log n log log n) and O(n(log n) 1.5849... ) respectively [14,4]. In [12], Montgomery invented Karatsuba-like formulae splitting the inputs into more than two parts; the key feature of those formulae is that they involve no division, thus work over any field.…”

“…In [12], Montgomery invented Karatsuba-like formulae splitting the inputs into more than two parts; the key feature of those formulae is that they involve no division, thus work over any field. More recently, Bodrato [2] proposed good schemes for Toom-Cook 3, 4, and 5, which are useful cases of the Toom-Cook class of algorithms [7,18]. A detailed bibliography on multiplication and factorisation in GF (2)[x] can be found in [21].…”

“…More recently, Bodrato [2] proposed good schemes for Toom-Cook 3, 4, and 5, which are useful cases of the Toom-Cook class of algorithms [7,18]. A detailed bibliography on multiplication and factorisation in GF (2)[x] can be found in [21].…”