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Improving the Gaudry–Schost algorithm for multidimensional discrete logarithms

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Abstract

The discrete logarithm problem arises from various areas, including counting the number of points of certain curves and diverse cryptographic schemes. The Gaudry–Schost algorithm and its variants are state-of-the-art low-memory methods solving the multi-dimensional discrete logarithm problem through finding collisions between pseudorandom tame walks and wild walks. In this work, we explore the impact on the choice of tame and wild sets of the Gaudry–Schost algorithm, and give two variants with improved average case time complexity for the multidimensional case under certain heuristic assumptions. We explain why the second method is asymptotically optimal.

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Acknowledgements

This work was partially supported by the National Key Research and Development Project (Grant No. 2018YFA0704702) and the Major Basic Research Project of Natural Science Foundation of Shandong Province, China (Grant No. ZR202010220025). The authors thank the editor and anonymous reviewers for very helpful suggestions which improved the paper.

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Authors and Affiliations

  1. Key Laboratory of Cryptologic Technology and Information Security, Ministry of Education, Shandong University, Qingdao, 266237, China

    Haoxuan Wu & Jincheng Zhuang

  2. School of Cyber Science and Technology, Shandong University, Qingdao, 266237, China

    Haoxuan Wu & Jincheng Zhuang

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  1. Haoxuan Wu
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  2. Jincheng Zhuang
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Correspondence to Jincheng Zhuang.

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Communicated by S. D. Galbraith.

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Wu, H., Zhuang, J. Improving the Gaudry–Schost algorithm for multidimensional discrete logarithms. Des. Codes Cryptogr. 90, 107–119 (2022). https://doi.org/10.1007/s10623-021-00966-5

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