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arXiv · 1502.02800

Fast integer multiplication using generalized Fermat primes

Svyatoslav Covanov·Emmanuel Thomé
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Abstract

For almost 35 years, Sch{\"o}nhage-Strassen's algorithm has been the fastest algorithm known for multiplying integers, with a time complexity O(n $\times$ log n $\times$ log log n) for multiplying n-bit inputs. In 2007, F{\"u}rer proved that there exists K > 1 and an algorithm performing this operation in O(n $\times$ log n $\times$ 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.

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Svyatoslav Covanov, Emmanuel Thomé. 2015-02-10. Fast integer multiplication using generalized Fermat primes. https://arxiv.org/abs/1502.02800

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