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Efficient Algorithms for Isogeny Sequences and Their Cryptographic Applications

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Mathematical Modelling for Next-Generation Cryptography

Part of the book series: Mathematics for Industry ((MFI,volume 29))

Abstract

We summarize efficient isogeny sequence computations on elliptic and genus 2 Jacobians. For cryptographic purposes, sequences of low-degree isogenies are important. Then we focus on sequences of 2- and 3-isogenies on elliptic curves and (2, 2)- and (3, 3)-isogenies on genus 2 Jacobians. Our aim is to explicitly describe the low-degree isogeny sequence computations and improve them for cryptographic applications such as post-quantum cryptosystems and random self-reducibility of discrete logarithm problem (DLP).

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Acknowledgements

The author would like to thank Kazuto Matsuo for his valuable comments on genus 2 division polynomials given in Sect. 4.3.

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Authors and Affiliations

  1. Mitsubishi Electric, 5-1-1 Ofuna, Kamakura, Kanagawa 247-8501, Japan

    Katsuyuki Takashima

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  1. Katsuyuki Takashima
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Correspondence to Katsuyuki Takashima .

Editor information

Editors and Affiliations

  1. Kyushu University, Fukuoka, Japan

    Tsuyoshi Takagi

  2. Kyushu University, Fukuoka, Japan

    Masato Wakayama

  3. Tokyo Institute of Technology, Tokyo, Japan

    Keisuke Tanaka

  4. The University of Tokyo, Kashiwa, Japan

    Noboru Kunihiro

  5. University of the Ryukyus, Nakagami-gun, Japan

    Kazufumi Kimoto

  6. Institute of Mathematics for Industry, Kyushu University, Fukuoka, Japan

    Dung Hoang Duong

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Takashima, K. (2018). Efficient Algorithms for Isogeny Sequences and Their Cryptographic Applications. In: Takagi, T., Wakayama, M., Tanaka, K., Kunihiro, N., Kimoto, K., Duong, D. (eds) Mathematical Modelling for Next-Generation Cryptography. Mathematics for Industry, vol 29. Springer, Singapore. https://doi.org/10.1007/978-981-10-5065-7_6

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  • DOI: https://doi.org/10.1007/978-981-10-5065-7_6

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  • Online ISBN: 978-981-10-5065-7

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