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A general construction of regular complete permutation polynomials

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Abstract

Let \(r\ge 3\) be a positive integer and \({\mathbb {F}}_q\) the finite field with q elements. In this paper, we consider the r-regular complete permutation property of maps with the form \(f=\tau \circ \sigma _M\circ \tau ^{-1}\) where \(\tau \) is a PP over an extension field \({\mathbb {F}}_{q^d}\) and \(\sigma _M\) is an invertible linear map over \({\mathbb {F}}_{q^d}\). When \(\tau \) is additive, we give a general construction of r-regular CPPs for any positive integer r. When \(\tau \) is not additive, we give many examples of regular CPPs over the extension fields for \(r=3,4,5,6,7\) and for arbitrary odd positive integer r. These examples are the generalization of the first class of r-regular CPPs constructed by Xu et al. (Des Codes Cryptogr 90:545–575, 2022).

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Authors and Affiliations

  1. School of Mathematics, Southeast University, Nanjing, 210096, China

    Wei Lu, Xia Wu & Yufei Wang

  2. Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, 211100, China

    Xiwang Cao

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  1. Wei Lu
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  2. Xia Wu
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  3. Yufei Wang
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  4. Xiwang Cao
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Correspondence to Xia Wu.

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Communicated by P. Charpin.

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Supported by NSFC (Nos. 11971102, 11801070, 11771007), the Fundamental Research Funds for the Central Universities.

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Lu, W., Wu, X., Wang, Y. et al. A general construction of regular complete permutation polynomials. Des. Codes Cryptogr. 91, 2627–2647 (2023). https://doi.org/10.1007/s10623-023-01224-6

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