Gaps between prime numbers and tensor rank of multiplication in finite fields

Abstract

We present effective upper bounds on the symmetric bilinear complexity of multiplication in extensions of a base finite field $$\mathbb {F}_{p^2}$$ of prime square order, obtained by combining estimates on gaps between prime numbers together with an optimal construction of auxiliary divisors for multiplication algorithms by evaluation-interpolation on curves. Most of this material dates back to a 2011 unpublished work of the author, but it still provides the best results on this topic at the present time. Then a few updates are given in order to take recent developments into account, including comparison with a similar work of Ballet and Zykin, generalization to classical bilinear complexity over $$\mathbb {F}_p$$ , and to short multiplication of polynomials, as well as a discussion of open questions on gaps between prime numbers or more generally values of certain arithmetic functions.