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International Conference on Information Security and Cryptology

ICISC 2017: Information Security and Cryptology – ICISC 2017 pp 245–263Cite as

On the Computational Complexity of ECDLP for Elliptic Curves in Various Forms Using Index Calculus

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Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 10779))

Abstract

The security of elliptic curve cryptography is closely related to the computational complexity of the elliptic curve discrete logarithm problem (ECDLP). Today, the best practical attacks against ECDLP are exponential-time, generic discrete logarithm algorithms such as Pollard’s rho method. Recently, there is a line of research on index calculus for ECDLP started by Semaev, Gaudry, and Diem. Under certain heuristic assumptions, such algorithms could lead to subexponential attacks to ECDLP in some cases. In this paper, we investigate the computational complexity of ECDLP for elliptic curves in various forms—including Hessian, Montgomery, (twisted) Edwards, and Weierstrass using index calculus. The research question we would like to answer is: Using index calculus, is there any significant difference in the computational complexity of ECDLP for elliptic curves in various forms? We will provide some empirical evidence and insights showing an affirmative answer in this paper.

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Acknowledgments

This work is partially supported by JSPS KAKENHI Grant (C)(JP15K00183) and (JP15K00189) and Japan Science and Technology Agency, CREST and Infrastructure Development for Promoting International S&T Cooperation and Project for Establishing a Nationwide Practical Education Network for IT Human Resources Development, Education Network for Practical Information Technologies.

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Authors and Affiliations

  1. Graduate School of Engineering, Osaka University, Suita, Japan

    Chen-Mou Cheng, Kenta Kodera & Atsuko Miyaji

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  1. Chen-Mou Cheng
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  2. Kenta Kodera
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  3. Atsuko Miyaji
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Correspondence to Chen-Mou Cheng .

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Editors and Affiliations

  1. Pusan National University, Busan, Korea (Republic of)

    Howon Kim

  2. Kookmin University, Seoul, Korea (Republic of)

    Dong-Chan Kim

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Cheng, CM., Kodera, K., Miyaji, A. (2018). On the Computational Complexity of ECDLP for Elliptic Curves in Various Forms Using Index Calculus. In: Kim, H., Kim, DC. (eds) Information Security and Cryptology – ICISC 2017. ICISC 2017. Lecture Notes in Computer Science(), vol 10779. Springer, Cham. https://doi.org/10.1007/978-3-319-78556-1_14

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  • DOI: https://doi.org/10.1007/978-3-319-78556-1_14

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  • Print ISBN: 978-3-319-78555-4

  • Online ISBN: 978-3-319-78556-1

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