Fast integer multiplication using generalized Fermat primes

Svyatoslav Covanov (1) , Emmanuel Thomé (1)

1 CARAMBA - Cryptology, arithmetic : algebraic methods for better algorithms

For almost 35 years, Schönhage-Strassen's algorithm has been the fastest algorithm known for multiplying integers, with a time complexity O(n · log n · log log n) for multiplying n-bit inputs. In 2007, Fürer proved that there exists K > 1 and an algorithm performing this operation in O(n · log n · K log n). Recent work by Harvey, van der Hoeven, and Lecerf showed that this complexity estimate can be improved in order to get K = 8, and conjecturally K = 4. Using an alternative algorithm, which relies on arithmetic modulo generalized Fermat primes, we obtain conjecturally the same result K = 4 via a careful complexity analysis in the deterministic multitape Turing model.

Mots clés: Integer multiplication; complexity; FFT; arithmetic

Domaines: Calcul formel [cs.SC]; Complexité [cs.CC]; Algorithme et structure de données [cs.DS]; Mathématique discrète [cs.DM]

Svyatoslav Covanov, Emmanuel Thomé. Fast integer multiplication using generalized Fermat primes. 2016. ⟨hal-01108166v3⟩