Constructions of complete permutation polynomials

  • Xiaofang Xu
  • Chunlei Li
  • Xiangyong Zeng
  • Tor Helleseth


Based on the Feistel and MISTY structures, this paper presents several new constructions of complete permutation polynomials (CPPs) of the finite field \({\mathbb {F}}_{2^{n}}^2\) for a positive integer n and three constructions of CPPs over \({\mathbb {F}}_{p^{n}}^m\) for any prime p and positive integer \(m\ge 2\). In addition, we investigate the upper bound on the algebraic degree of these CPPs and show that some of them can have nearly optimal algebraic degree.


Permutation polynomial Complete permutation polynomial Algebraic degree 

Mathematics Subject Classification

05A05 11T06 11T55 



The authors would like to thank Prof. Xiwang Cao and Dr. Nian Li for their valuable discussions and suggestions. The authors also thank the anonymous referees whose comments significantly improved the quality of the paper. X. Zeng was supported by National Natural Science Foundation of China under Grant 61472120 and National Natural Science Foundation of Hubei Province of China under Grant 2017CFB143. The work of T. Helleseth and C. Li is supported by the Research Council of Norway; C. Li’s work is also partly supported by the research project (No. 720025) from UH-nett Vest in Norway and the National Natural Science Foundation of China under Grant 61771021.


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Authors and Affiliations

  1. 1.Faculty of Mathematics and Statistics, Hubei Key Laboratory of Applied MathematicsHubei UniversityWuhanChina
  2. 2.Department of InformaticsUniversity of BergenBergenNorway

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