Unicyclic strong permutations

  • Claude Gravel
  • Daniel Panario
  • David Thomson
Part of the following topical collections:
  1. Special Issue on Boolean Functions and Their Applications


In this paper, we study some properties of a certain kind of permutation σ over \(\mathbb {F}_{2}^{n}\), where n is a positive integer. The desired properties for σ are: (1) the algebraic degree of each component function is n − 1; (2) the permutation is unicyclic; (3) the number of terms of the algebraic normal form of each component is at least 2n− 1. We call permutations that satisfy these three properties simultaneously unicyclic strong permutations. We prove that our permutations σ always have high algebraic degree and that the average number of terms of each component function tends to 2n− 1. We also give a condition on the cycle structure of σ. We observe empirically that for n even, our construction does not provide unicylic permutations. For n odd, n ≤ 11, we conduct an exhaustive search of all σ given our construction for specific examples of unicylic strong permutations. We also present some empirical results on the difference tables and linear approximation tables of σ.


Boolean functions Finite fields Permutations Algebraic degree Differential uniformity Walsh spectra 

Mathematics Subject Classification (2010)

11T06 11T71 



The authors are grateful for the very careful reviews and the constructive suggestions received from the referees.


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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.MontrealCanada
  2. 2.School of Mathematics and StatisticsCarleton UniversityOttawaCanada

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