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

TL;DR

This work extends Chudnovsky-type interpolation methods to the k-multiplication problem in finite field extensions ${\mathbb{F}}_{q^n}$ by expressing the operation as a tensor decomposition facilitated by Hadamard products of evaluations on algebraic curves. By generalizing to places of arbitrary degree and leveraging towers of function fields with many high-degree places, the authors obtain uniform, linear-in-$n$ upper bounds on the k-multiplication rank, valid for all $q$ and large $n$. The construction uses a Garcia–Stichtenoth tower and a descent tower to satisfy interpolation conditions, with a strategy anchored in the Drinfeld–Vladut bound of order $r$ (where $r$ is the smallest even integer exceeding $2\log_q(k+1)$). The resulting bounds provide practical, asymptotically tight estimates for the multilinear complexity and introduce a framework for uniform control of tensor rank in finite-field multiplication across extensions. This has potential implications for cryptographic primitives, coding theory, and fast arithmetic in finite fields.

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.

Theorems & Definitions (24)

• Definition 1.1
• Definition 1.2
• Definition 1.3
• Definition 1.4
• Lemma 1.5
• proof
• Theorem 2.1: Algorithm
• proof
• Theorem 2.2
• proof