Fast arithmetic for faster integer multiplication
Résumé
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 multi-plying n-bit inputs. In 2007, Fürer proved that there ex-ists K > 1 and an algorithm performing this operation in O(n · log n · K log * n). Recent work showed that this com-plexity estimate can be made more precise with 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 rely on a conjecture about the existence of sufficiently many primes of a certain form.
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