Fast integer multiplication using \goodbreak generalized Fermat primes

Svyatoslav Covanov, Emmanuel Thomé
2018 Mathematics of Computation  
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
more » ... algorithm, which relies on arithmetic modulo generalized Fermat primes (of the form r 2 λ + 1), we obtain conjecturally the same result K = 4 via a careful complexity analysis in the deterministic multitape Turing model.
doi:10.1090/mcom/3367 fatcat:fykwtfvmtfc5fcchyjzfvqcfui