Complete permutation polynomials from exceptional polynomials

In this paper, we study polynomials of the form [Formula: see text], where [Formula: see text] and list all permutation polynomials (PPs) and complete permutation polynomials (CPPs) of this form. This type of polynomials were studied by Bassalygo and Zinoviev for the cases [Formula: see text] and [Formula: see text], Wu, Li, Helleseth and Zhang for the case [Formula: see text], [Formula: see text], Bassalygo and Zinoviev answered the question for the case [Formula: see text], [Formula: see text] and finally by Bartoli et al. for the case [Formula: see text]. Here, we determine all PPs and CPPs for the case [Formula: see text].

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Some permutations and complete permutation polynomials over finite fields

TURKISH JOURNAL OF MATHEMATICS ◽

10.3906/mat-1806-83 ◽

2019 ◽

Vol 43 (5) ◽

pp. 2154-2160

Author(s):

Pınar ONGAN ◽

Burcu GÜLMEZ TEMÜR

Keyword(s):

Finite Fields ◽

Permutation Polynomials ◽

Polynomials Over Finite Fields ◽

Complete Permutation Polynomials

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Some classes of complete permutation polynomials over $\mathbb{F}_q $

Science China Mathematics ◽

10.1007/s11425-014-4964-2 ◽

2015 ◽

Vol 58 (10) ◽

pp. 1-14 ◽

Cited By ~ 9

Author(s):

GaoFei Wu ◽

Nian Li ◽

Tor Helleseth ◽

YuQing Zhang

Keyword(s):

Permutation Polynomials ◽

Complete Permutation Polynomials

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New classes of complete permutation polynomials

Finite Fields and Their Applications ◽

10.1016/j.ffa.2018.10.001 ◽

2019 ◽

Vol 55 ◽

pp. 177-201 ◽

Cited By ~ 2

Author(s):

Lisha Li ◽

Chaoyun Li ◽

Chunlei Li ◽

Xiangyong Zeng

Keyword(s):

Permutation Polynomials ◽

Complete Permutation Polynomials

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On a Conjecture of Carlitz

Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics ◽

10.1017/s1446788700029657 ◽

1987 ◽

Vol 43 (3) ◽

pp. 375-384 ◽

Cited By ~ 7

Author(s):

Wan Daqing

Keyword(s):

Positive Integer ◽

Finite Field ◽

Permutation Polynomials ◽

Odd Order ◽

Equivalent Version ◽

Exceptional Polynomials

A conjecture of Carlitz on permutation polynomials is as follow: Given an even positive integer n, there is a constant Cn, such that if Fq is a finite field of odd order q with q > Cn, then there are no permutation polynomials of degree n over Fq. The conjecture is a well-known problem in this area. It is easily proved if n is a power of 2. The only other cases in which solutions have been published are n = 6 (Dickson [5]) and n = 10 (Hayes [7]); see Lidl [11], Lausch and Nobauer [9], and Lidl and Niederreiter [10] for remarks on this problem. In this paper, we prove that the Carlitz conjecture is true if n = 12 or n = 14, and give an equivalent version of the conjecture in terms of exceptional polynomials.

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