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Acceleration of Index Calculus for Solving ECDLP over Prime Fields and Its Limitation

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Cryptology and Network Security (CANS 2018)

Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 11124))

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Abstract

In 2018, Amadori et al. proposed a new variant of index calculus to solve the elliptic curve discrete logarithm problem (ECDLP), using Semaev’s summation polynomials. The variant drastically decreases the number of required Gröbner basis computations, and it outperforms other index calculus algorithms for the ECDLP over prime fields. In this paper, we provide several improvements to accelerate to solve systems of multivariate equations arising in the variant. A main improvement is to apply the hybrid method, which mixes exhaustive search and Gröbner bases techniques to solve multivariate systems over finite fields. We also make use of symmetries of summation polynomials. We show experimental results of our improvements, and give their complexity analysis to discuss a limitation of our acceleration in both theory and practice.

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Acknowledgments

This work was supported by JST CREST Grant Number JPMJCR14D6, Japan.

Author information

Authors and Affiliations

  1. Kobe City College of Technology, 8-3, Gakuen-Higashimachi, Nishi-ku, Kobe, 651-2194, Japan

    Momonari Kudo

  2. Graduate School of Mathematics, Kyushu University, 744 Motooka, Nishi-ku, Fukuoka, 819-0395, Japan

    Yuki Yokota & Yasushi Takahashi

  3. Institute of Mathematics for Industry, Kyushu University, 744 Motooka, Nishi-ku, Fukuoka, 819-0395, Japan

    Masaya Yasuda

Authors
  1. Momonari Kudo
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  2. Yuki Yokota
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  3. Yasushi Takahashi
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  4. Masaya Yasuda
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Corresponding author

Correspondence to Masaya Yasuda .

Editor information

Editors and Affiliations

  1. IBM Research - Zurich, Rüschlikon, Switzerland

    Jan Camenisch

  2. KTH Royal Institute of Technology, Stockholm, Sweden

    Panos Papadimitratos

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Kudo, M., Yokota, Y., Takahashi, Y., Yasuda, M. (2018). Acceleration of Index Calculus for Solving ECDLP over Prime Fields and Its Limitation. In: Camenisch, J., Papadimitratos, P. (eds) Cryptology and Network Security. CANS 2018. Lecture Notes in Computer Science(), vol 11124. Springer, Cham. https://doi.org/10.1007/978-3-030-00434-7_19

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  • DOI: https://doi.org/10.1007/978-3-030-00434-7_19

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-00433-0

  • Online ISBN: 978-3-030-00434-7

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