Abstract
We design a class of Chudnovsky-type algorithms multiplying k elements of a finite extension \({\text {I}\!\text {F}}_{q^n}\) of a finite field \({\text {I}\!\text {F}}_q\), where \(k\ge 2\). 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 \({\text {I}\!\text {F}}_q\). 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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Ballet, S., Rolland, R. (2022). Chaining Multiplications in Finite Fields with Chudnovsky-Type Algorithms and Tensor Rank of the k-Multiplication. In: Poulakis, D., Rahonis, G. (eds) Algebraic Informatics. CAI 2022. Lecture Notes in Computer Science, vol 13706. Springer, Cham. https://doi.org/10.1007/978-3-031-19685-0_1
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DOI: https://doi.org/10.1007/978-3-031-19685-0_1
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