In 2014, Harvey, Joris van der Hoeven and Lecerf gave a new algorithm that achieves a running time of , making explicit the implied constant in the exponent. They also proposed a variant of their algorithm which achieves but whose validity relies on standard conjectures about the distribution of Mersenne primes. In 2016, Covanov and Thomé proposed an integer multiplication algorithm based on a generalization of Fermat primes that conjecturally achieves a complexity bound of . This matches the 2015 conditional result of Harvey, van der Hoeven, and Lecerf but uses a different algorithm and relies on a different conjecture. In 2018, Harvey and van der Hoeven used an approach based on the existence of short lattice vectors guaranteed by Minkowski's theorem to prove an unconditional complexity bound of .
In 2014, Harvey, Joris van der Hoeven and Lecerf gave a new algorithm that achieves a running? time of , making explicit the implied constant in the. exponent. They also proposed a variant of their algorithm which achieves but whose validity relies on standard conjectures about the distribution of Mersenne primes. In 2016, Covanov and Thomé proposed an integer multiplicati.on algorithm based on a generalization of Fermat primes that conjecturally achieves a complexity bound of . This matches the 2015 conditional result of Harvey, van der Hoeven, and Lecerf but uses a different algorithm and relies on a different conjecture. In 2018, Harvey and van der Hoeven used an approach based on the existence of short lattice vectors guaranteed by Minkowski's theorem to !prove an unconditi,onal complexity bound of .