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On the Menezes-Teske-Weng conjecture

Abstract

In 2003, Alfred Menezes, Edlyn Teske and Annegret Weng presented a conjecture on properties of the solutions of a type of quadratic equations over the binary extension fields, which had been confirmed by extensive experiments but the proof was unknown until now. We prove that this conjecture is correct. Furthermore, using this proved conjecture, we have completely determined the null space of a class of linearized polynomials.

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Acknowledgments

The authors deeply thank Alfred Menezes for checking our proof of the Menezes-Teske-Weng conjecture. They also thank the Assoc. Edit. and the anonymous reviewers for their valuable comments which have highly improved the manuscript.

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Correspondence to Sihem Mesnager.

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Mesnager, S., Kim, K.H., Choe, J. et al. On the Menezes-Teske-Weng conjecture. Cryptogr. Commun. 12, 19–27 (2020). https://doi.org/10.1007/s12095-019-00359-5

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Keywords

  • Binary finite fields
  • Elliptic curve
  • Discrete Logarithm Problem (DLP)
  • Quadratic equation
  • Trace function

Mathematics Subject Classification (2010)

  • 68R01
  • 11G05
  • 12E12
  • 12E20