Fast integer multiplication using generalized Fermat primes

Svyatoslav Covanov 1 Emmanuel Thomé 1
1 CARAMBA - Cryptology, arithmetic : algebraic methods for better algorithms
Inria Nancy - Grand Est, LORIA - ALGO - Department of Algorithms, Computation, Image and Geometry
Abstract : For almost 35 years, Schönhage-Strassen's algorithm has been the fastest algorithm known for multiplying integers, with a time complexity $O(n \cdot \log n \cdot \log \relax \log n)$ for multiplying $n$-bit inputs. In 2007, Fürer proved that there exists $K>1$ and an algorithm performing this operation in $O(n \cdot \log n \cdot K^{\log^* n})$. 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 $K=4$. We obtain here the same result $K=4$ using simple modular arithmetic as a building block, and a careful complexity analysis. We obtain a similar result $K=4$ using an alternative somewhat simpler algorithm, which relies on arithmetic modulo generalized Fermat primes.
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https://hal.inria.fr/hal-01108166
Contributeur : Svyatoslav Covanov <>
Soumis le : jeudi 28 janvier 2016 - 09:05:27
Dernière modification le : vendredi 25 novembre 2016 - 01:04:09
Document(s) archivé(s) le : vendredi 29 avril 2016 - 10:15:31

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  • HAL Id : hal-01108166, version 2
  • ARXIV : 1502.02800

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Svyatoslav Covanov, Emmanuel Thomé. Fast integer multiplication using generalized Fermat primes. 2016. <hal-01108166v2>

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