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 · 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.
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Submitted on : Friday, April 13, 2018 - 3:18:20 PM
Last modification on : Tuesday, December 18, 2018 - 4:18:26 PM

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Svyatoslav Covanov, Emmanuel Thomé. Fast integer multiplication using generalized Fermat primes. Mathematics of Computation, American Mathematical Society, In press, ⟨10.1090/mcom/3367 ⟩. ⟨hal-01108166v4⟩

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