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On the Scalar Complexity of Chudnovsky\(^2\) Multiplication Algorithm in Finite Fields

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Algebraic Informatics (CAI 2019)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11545))

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Abstract

We propose a new construction for the multiplication algorithm of D.V. and G.V. Chudnovsky in order to improve scalar algebraic complexity. In particular, we improve the Baum-Shokrollahi construction for multiplication in \(\mathbb F_{256}/\mathbb F_4\) based on the elliptic Fermat curve \(x^3+y^3=1\).

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References

  1. Atighehchi, K., Ballet, S., Bonnecaze, A., Rolland, R.: Arithmetic in finite fields based on Chudnovsky’s multiplication algorithm. Math. Comput. 86(308), 297–3000 (2017)

    Article  MathSciNet  Google Scholar 

  2. Ballet, S.: Curves with many points and multiplication complexity in any extension of \(\mathbb{F}_q\). Finite Fields Their Appl. 5, 364–377 (1999)

    Article  MathSciNet  Google Scholar 

  3. Ballet, S.: Quasi-optimal algorithms for multiplication in the extensions of \(\mathbb{F}_{16}\) of degree \(13\), \(14\), and \(15\). J. Pure Appl. Algebra 171, 149–164 (2002)

    Article  MathSciNet  Google Scholar 

  4. Baum, U., Shokrollahi, A.: An optimal algorithm for multiplication in \(\mathbb{F}_{256}/\mathbb{F}_4\). Appl. Algebra Eng. Commun. Comput. 2(1), 15–20 (1991)

    Article  MathSciNet  Google Scholar 

  5. Chaumine, J.: On the bilinear complexity of multiplication in small finite fields. Comptes Rendus de l’Académie des Sciences, Série I(343), 265–266 (2006)

    MathSciNet  MATH  Google Scholar 

  6. Chudnovsky, D., Chudnovsky, G.: Algebraic complexities and algebraic curves over finite fields. J. Complex. 4, 285–316 (1988)

    Article  MathSciNet  Google Scholar 

  7. De Groote, H.: Characterization of division algebras of minimal rank and the structure of their algorithm varieties. SIAM J. Comput. 12(1), 101–117 (1983)

    Article  MathSciNet  Google Scholar 

  8. Pieltant, J.: Tours de corps de fonctions algébriques et rang de tenseur de la multiplication dans les corps finis. Ph.D. thesis, Université d’Aix-Marseille, Institut de Mathématiques de Luminy (2012)

    Google Scholar 

  9. Shokhrollahi, A.: Optimal algorithms for multiplication in certain finite fields using algebraic curves. SIAM J. Comput. 21(6), 1193–1198 (1992)

    Article  MathSciNet  Google Scholar 

  10. Stichtenoth, H.: Algebraic Function Fields and Codes. Lectures Notes in Mathematics, vol. 314. Springer, Heidelberg (1993)

    MATH  Google Scholar 

  11. Winograd, S.: On multiplication in algebraic extension fields. Theor. Comput. Sci. 8, 359–377 (1979)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Aix Marseille Univ, CNRS, Centrale Marseille, I2M, Marseille, France

    Stéphane Ballet, Alexis Bonnecaze & Thanh-Hung Dang

  2. Institut de Mathématiques de Marseille, UMR 7373, CNRS, Aix-Marseille Université, case 930, 13288, Marseille cedex 9, France

    Stéphane Ballet, Alexis Bonnecaze & Thanh-Hung Dang

Authors
  1. Stéphane Ballet
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  2. Alexis Bonnecaze
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  3. Thanh-Hung Dang
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Correspondence to Thanh-Hung Dang .

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Editors and Affiliations

  1. University of Niš, Niš, Serbia

    Miroslav Ćirić

  2. University of Leipzig , Leipzig, Germany

    Manfred Droste

  3. Université Paris Denis Diderot and CNRS, Paris, France

    Jean-Éric Pin

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Ballet, S., Bonnecaze, A., Dang, TH. (2019). On the Scalar Complexity of Chudnovsky\(^2\) Multiplication Algorithm in Finite Fields. In: Ćirić, M., Droste, M., Pin, JÉ. (eds) Algebraic Informatics. CAI 2019. Lecture Notes in Computer Science(), vol 11545. Springer, Cham. https://doi.org/10.1007/978-3-030-21363-3_6

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  • DOI: https://doi.org/10.1007/978-3-030-21363-3_6

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-21362-6

  • Online ISBN: 978-3-030-21363-3

  • eBook Packages: Computer ScienceComputer Science (R0)

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