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International Conference on Algebraic Informatics

CAI 2022: Algebraic Informatics pp 69–80Cite as

Finding Points on Elliptic Curves with Coppersmith’s Method

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Part of the Lecture Notes in Computer Science book series (LNCS,volume 13706)

Abstract

Several cryptosystems based on Elliptic Curve Cryptography such as ElGamal and KMOV process the message as a point \(M=(x_0,y_0)\) of an elliptic curve with an equation of the form \(y^2=x^3+ax+b\) over a finite field or a finite ring. In this paper, we present a method to find the small solutions of the former elliptic curve equation. Our method is based on Coppersmith’s technique and enables one to find the solutions \((x_0,y_0)\) when \(|x_0|^3|y_0|^2\) is smaller than the modulus.

Keywords

  • Elliptic curve cryptography
  • Coppersmith’s method
  • Lattice basis reduction
  • Cryptanalysis

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Author information

Authors and Affiliations

  1. Institut de Mathématiques et de Sciences Physiques, Dangbo, Bénin

    Virgile Dossou-Yovo

  2. Normandie Univ, UNICAEN, CNRS, LMNO, 14000, Caen, France

    Abderrahmane Nitaj

  3. Department of Mathematics and Statistics, Purdue University Northwest, Westville, USA

    Alain Togbé

Authors
  1. Virgile Dossou-Yovo
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  2. Abderrahmane Nitaj
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  3. Alain Togbé
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Corresponding author

Correspondence to Abderrahmane Nitaj .

Editor information

Editors and Affiliations

  1. Aristotle University of Thessaloniki, Thessaloniki, Greece

    Dimitrios Poulakis

  2. Aristotle University of Thessaloniki, Thessaloniki, Greece

    George Rahonis

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Dossou-Yovo, V., Nitaj, A., Togbé, A. (2022). Finding Points on Elliptic Curves with Coppersmith’s Method. 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_6

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  • DOI: https://doi.org/10.1007/978-3-031-19685-0_6

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-19684-3

  • Online ISBN: 978-3-031-19685-0

  • eBook Packages: Computer ScienceComputer Science (R0)

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