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Fixed points of rational functions satisfying the Carlitz property

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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 \({\mathbb {F}}_q\).

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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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Authors and Affiliations

  1. School of Mathematics and Statistics, Carleton University, Ottawa, Canada

    Kaitlyn Chubb, Daniel Panario & Qiang Wang

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  1. Kaitlyn Chubb
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  2. Daniel Panario
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  3. Qiang Wang
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Correspondence to Qiang Wang.

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Appendices

Appendices

Let \(q=p^t\), \(a \in {\mathbb {F}}_q\), and \(\Phi \in {\mathbb {F}}_{q^2}\), with \(a = \Phi ^2\). Algorithms for producing all of the fixed points in \({\mathbb {F}}_q\) for functions f satisfying the Carlitz property are given in [7]. In the following, we give sample outputs from these algorithms.

1.1 A: Sample fixed points for Type-1 functions

See Table 1.

Table 1 List of the N fixed points for f a Type-1 function with \(a=0\) and \(q=p^t\)
Full size table

1.2 B: Sample fixed points for Rédei functions

See Tables 2 and 3.

Table 2 List of the N fixed points for sample Rédei functions with a square
Full size table
Table 3 List of the N fixed points for sample Rédei functions with a non-square. The fixed points are represented in \({\mathbb {F}}_{q^2}\) but exist in \({\mathbb {F}}_q\)
Full size table

1.3 C: Sample fixed points for Type-3 functions

See Table 4.

Table 4 List of the N fixed points for f a Type-3 function with \(a \ne 0\) and \(q=2^t\)
Full size table

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Chubb, K., Panario, D. & Wang, Q. Fixed points of rational functions satisfying the Carlitz property. AAECC 30, 417–439 (2019). https://doi.org/10.1007/s00200-019-00382-2

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