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International Conference on Information Security and Cryptology

Inscrypt 2015: Information Security and Cryptology pp 167–177Cite as

On Generating Coset Representatives of \(PGL_{2}(\mathbb {F}_{q})\) in \(PGL_{2}(\mathbb {F}_{q^{2}})\)

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Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 9589))

Abstract

There are \( q^3 + q \) right \( PGL_{2}({\mathbb F}_{q})-\)cosets in the group \( PGL_{2}({\mathbb F}_{q^2}) \). In this paper, we present a method of generating all the coset representatives, which runs in time \( \tilde{O}(q^3) \), thus achieves the optimal time complexity up to a constant factor. Our algorithm has applications in solving discrete logarithms and finding primitive elements in finite fields of small characteristic.

J. Zhuang—This work was partially supported by the National Natural Science Foundation of China under Grant 61502481, the Strategic Priority Research Program of the Chinese Academy of Sciences under Grant XDA06010701, and the Open Project Program of the State Key Laboratory of Mathematical Engineering and Advanced Computing for Jincheng Zhuang.

Q. Cheng—This work was partially supported by China 973 Program under Grant 2013CB834201 and by US NSF under Grant CCF-1409294 for Qi Cheng.

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Acknowledgements

The authors would like to thank anonymous reviewers, Eleazar Leal, Robert Granger and Frederik Vercauteren for helpful comments and discussions.

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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

    Jincheng Zhuang

  2. State Key Laboratory of Mathematical Engineering and Advanced Computing, Wuxi, 214125, China

    Jincheng Zhuang

  3. School of Computer Science, The University of Oklahoma, Norman, OK, 73019, USA

    Qi Cheng

Authors
  1. Jincheng Zhuang
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  2. Qi Cheng
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Corresponding author

Correspondence to Jincheng Zhuang .

Editor information

Editors and Affiliations

  1. Chinese Academy of Sciences, SKLOIS Institute of Engineering, Beijing, China

    Dongdai Lin

  2. Center for Security Informatics, Indiana University at Bloomington, Indiana, USA

    XiaoFeng Wang

  3. Computer Science, Columbia University, New York, New York, USA

    Moti Yung

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Zhuang, J., Cheng, Q. (2016). On Generating Coset Representatives of \(PGL_{2}(\mathbb {F}_{q})\) in \(PGL_{2}(\mathbb {F}_{q^{2}})\) . In: Lin, D., Wang, X., Yung, M. (eds) Information Security and Cryptology. Inscrypt 2015. Lecture Notes in Computer Science(), vol 9589. Springer, Cham. https://doi.org/10.1007/978-3-319-38898-4_10

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  • DOI: https://doi.org/10.1007/978-3-319-38898-4_10

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-38897-7

  • Online ISBN: 978-3-319-38898-4

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