(with Joris van der Hoeven)
arXiv preprint (December 2017).
We present an algorithm that computes the product of two n-bit integers in O(n log n (4√2)log* n) bit operations. Previously, the best known bound was O(n log n 6log* n). We also prove that for a fixed prime p, polynomials in Fp[X] of degree n may be multiplied in O(n log n 4log* n) bit operations; the previous best bound was O(n log n 8log* n).