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Journal Articles Mathematics of Computation Year : 2019

Fast integer multiplication using generalized Fermat primes

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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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Dates and versions

hal-01108166 , version 1 (22-01-2015)
hal-01108166 , version 2 (28-01-2016)
hal-01108166 , version 3 (10-08-2017)
hal-01108166 , version 4 (13-04-2018)

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Svyatoslav Covanov, Emmanuel Thomé. Fast integer multiplication using generalized Fermat primes. Mathematics of Computation, 2019, 88 (317), pp.1449-1477. ⟨10.1090/mcom/3367⟩. ⟨hal-01108166v4⟩
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