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Translating the Discrete Logarithm Problem on Jacobians of Genus 3 Hyperelliptic Curves with \((\ell ,\ell ,\ell )\)-Isogenies

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Abstract

We give an algorithm to compute \((\ell ,\ell ,\ell )\)-isogenies from the Jacobians of genus three hyperelliptic curves to the Jacobians of non-hyperelliptic curves over a finite field of characteristic different from 2 in time \(\tilde{O}(\ell ^3)\), where \(\ell \) is an odd prime which is coprime to the characteristic. An important application is to reduce the discrete logarithm problem in the Jacobian of a hyperelliptic curve to the corresponding problem in the Jacobian of a non-hyperelliptic curve.

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Acknowledgements

Part of this work has been done while the author worked as a postdoc with Claus Diem at the University of Leipzig. I thank him for suggesting me to make use of Jean-Marc Couveignes and Tony Ezome’s approach. I thank Christophe Ritzenthaler for telling me about Milio’s paper. I thank Jean-Marc Couveignes for giving me the idea about how to construct the eight good functions on the Kummer variety. I also thank Nicholas Shepherd-Barron for answering many questions about his paper. Finally, I thank the anonymous reviewers for their very helpful comments and suggestions.

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Authors and Affiliations

  1. State Key Laboratory of Information Security, Institute of Information Engineering, Chinese Academy of Sciences, Beijing, 100093, China

    Song Tian

  2. State Key Laboratory of Cryptology, P.O. Box 5159, Beijing, 100878, China

    Song Tian

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  1. Song Tian
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Correspondence to Song Tian.

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Communicated by Nigel Smart

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Song Tian is supported by the China Scholarship Council, the National Natural Science Foundation of China (Grant No. 61802401), the National Cryptography Development Fund (No. MMJJ20180207), and Beijing Municipal Science and Technology Commission (Project No. Z191100007119006)

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Tian, S. Translating the Discrete Logarithm Problem on Jacobians of Genus 3 Hyperelliptic Curves with \((\ell ,\ell ,\ell )\)-Isogenies. J Cryptol 34, 32 (2021). https://doi.org/10.1007/s00145-021-09401-3

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  • DOI: https://doi.org/10.1007/s00145-021-09401-3

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