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arXiv · 2410.12383

Chaining Multiplications in Finite Fields with Chudnovsky-type Algorithms and Tensor Rank of the k-multiplication

Stéphane Ballet·Robert Rolland
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Abstract

We design a class of Chudnovsky-type algorithms multiplying k elements of a finite extension of order n a finite field K. We prove that these algorithms give a tensor decomposition of the k-multiplication for which the rank is linear in 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 K. 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.

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Stéphane Ballet, Robert Rolland. 2024-10-16. Chaining Multiplications in Finite Fields with Chudnovsky-type Algorithms and Tensor Rank of the k-multiplication. https://doi.org/10.1007/978-3-031-19685-0_1

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