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Trisymmetric Multiplication Formulae in Finite Fields

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Arithmetic of Finite Fields (WAIFI 2020)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 12542))

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Abstract

Multiplication is an expensive arithmetic operation, therefore there has been extensive research to find Karatsuba-like formulae reducing the number of multiplications involved when computing a bilinear map. The minimal number of multiplications in such formulae is called the bilinear complexity, and it is also of theoretical interest to asymptotically understand it. Moreover, when the bilinear maps admit some kind of invariance, it is also desirable to find formulae keeping the same invariance. In this work, we study trisymmetric, hypersymmetric, and Galois invariant multiplication formulae over finite fields, and we give an algorithm to find such formulae. We also generalize the result that the bilinear complexity and symmetric bilinear complexity of the two-variable multiplication in an extension field are linear in the degree of the extension, to trisymmetric bilinear complexity, and to the complexity of t-variable multiplication for any \(t\ge 3\).

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Notes

  1. 1.

    https://github.com/erou/TriSym.jl.

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Author information

Authors and Affiliations

  1. ANSSI, Paris, France

    Hugues Randriambololona

  2. Institut Polytechnique de Paris/Télécom Paris, Palaiseau, France

    Hugues Randriambololona & Édouard Rousseau

  3. Université Paris-Saclay/UVSQ, Versailles, France

    Édouard Rousseau

Authors
  1. Hugues Randriambololona
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  2. Édouard Rousseau
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Corresponding author

Correspondence to Édouard Rousseau .

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Editors and Affiliations

  1. Sorbonne Paris, Paris, France

    Jean Claude Bajard

  2. Sabancı University, Istanbul, Turkey

    Alev Topuzoğlu

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Randriambololona, H., Rousseau, É. (2021). Trisymmetric Multiplication Formulae in Finite Fields. In: Bajard, J.C., Topuzoğlu, A. (eds) Arithmetic of Finite Fields. WAIFI 2020. Lecture Notes in Computer Science(), vol 12542. Springer, Cham. https://doi.org/10.1007/978-3-030-68869-1_5

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  • DOI: https://doi.org/10.1007/978-3-030-68869-1_5

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  • Online ISBN: 978-3-030-68869-1

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