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Fixed points of rational functions satisfying the Carlitz property

  • Kaitlyn Chubb
  • Daniel Panario
  • Qiang WangEmail author
Original Paper
  • 11 Downloads

Abstract

Recent research within the field of cryptography has suggested that S-boxes should be chosen to contain few fixed points, motivating analysis of the fixed points of permutations. This paper presents a novel mean of obtaining fixed points for all functions satisfying a property put forth by Carlitz. We determine particular results concerning the fixed points of rational functions. Such concepts allow the derivation of an algorithm which cyclically generates fixed points for all three classes of functions satisfying the Carlitz property, the most renowned of which are Rédei rational functions. Specifically, we present all fixed points for any given Rédei function in a single cycle, generated by a particular non-constant rational transformation. For the other two classes of functions, we present their fixed points in cycles consisting of smaller cycles of fixed points. Finally, we provide an explicit expression for the fixed points of all Rédei functions over \({\mathbb {F}}_q\).

Keywords

Permutations Fixed points Redei functions Carltiz property Cycles 

Notes

Acknowledgements

This work served as partial fulfillment of the requirements for a Master of Science in Mathematics by the first author [7], and was conducted under the supervision of the latter two authors. The second and third authors are partially funded by NSERC of Canada. We also thank the anonymous referees for their helpful suggestions.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsCarleton UniversityOttawaCanada

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