Chaining Multiplications in Finite Fields with Chudnovsky-Type Algorithms and Tensor Rank of the k-Multiplication

Abstract

We design a class of Chudnovsky-type algorithms multiplying k elements of a finite extension F q n of a finite field Fq, where k ≥ 2. We prove that these algorithms give a tensor decomposition of the k-multiplication for which the rank is in O(n) uniformly in q. We give uniform upper bounds of the rank of k-multiplication in finite fields. They use interpolation on algebraic curves which transforms the problem in computing the Hadamard product of k vectors with components in Fq. This generalization of the widely studied case of k = 2 is based on a modification of the Riemann-Roch spaces involved and the use of towers of function fields having a lot of places of high degree.Finite Field and Tensor Rank of the Multiplication and Function Field