Necessary conditions for reversed Dickson polynomials of the second kind to be permutational

Hong, Shaofang; Qin, Xiaoer; Zhao, Wei

doi:10.1016/j.ffa.2015.09.001

Cited by 14 publications

(19 citation statements)

References 5 publications

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“…In [2], Hong, Qin, and Zhao explored the reversed Dickson polynomials of the second kind and found many of their properties and necessary conditions for them to be a permutation of F q .…”

Section: Introductionmentioning

confidence: 99%

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Reversed Dickson polynomials of the (k+ 1)-th kind over finite fields

Fernando

2017

Journal of Number Theory

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Abstract. We discuss the properties and the permutation behaviour of the reversed Dickson polynomials of the (k + 1)-th kind D n,k (1, x) over finite fields. The results in this paper unify and generalize several recently discovered results on reversed Dickson polynomials over finite fields. IntroductionLet p be a prime and q a power of p. Let F q be the finite field with q elements. A polynomial f ∈ F q [x] is called a permutation polynomial (PP) of F q if the associated mapping x → f (x) from F q to F q is a permutation of F q . Permutation polynomials over finite fields have important applications in Coding Theory, Cryptography, Finite Geometry, Combinatorics and Computer Science, among other fields.Recently, reversed Dickson polynomials over finite fields have been studied extensively by many for their general properties and permutation behaviour. The concept of the reversed Dickson polynomial D n (a, x) was first introduced by Hou, Mullen, Sellers and Yucas in [5] by reversing the roles of the variable and the parameter in the Dickson polynomial D n (x, a).The n-th reversed Dickson polynomial of the first kind D n (a, x) is defined bywhere a ∈ F q is a parameter. In [5], it was shown that the reversed Dickson polynomials of the first kind are closely related to Almost Perfect Nonlinear (APN) functions which have applications in cryptography. Hou and Ly found more properties of the reversed Dickson polynomials of the first kind and necessary conditions for them to be a permutation of F q ; see [4].By reversing the roles of the variable and the parameter in the Dickson polynomial of the second kind E n (x, a), the n-th reversed Dickson polynomial of the second kind E n (a, x) can be defined byn − i i (−x) i a n−2i , 2010 Mathematics Subject Classification. 11T06, 11T55.

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“…In [2], Hong, Qin, and Zhao explored the reversed Dickson polynomials of the second kind and found many of their properties and necessary conditions for them to be a permutation of F q .…”

Section: Introductionmentioning

confidence: 99%

“…Motivated by [1], [2], [3], [4], and [5], we fix k and study the general properties and permutation property of the reversed Dickson polynomials of the (k + 1)-th kind over finite fields. The results obtained in this paper unify and generalize many existing results on reversed Dickson polynomials.…”

Section: Introductionmentioning

confidence: 99%

Reversed Dickson polynomials of the (k+ 1)-th kind over finite fields

Fernando

2017

Journal of Number Theory

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“…Note that (1.2) is a genelarization of (1.3), (1.4), and (1.5) for any k. The selfreciprocal property of (1.3), (1.4), and (1.5) was used in [10], [9], and [4], respectively, by the aforementioned authors to find necessary conditions for the corresponding reversed Dickson polynomials to be a permutation of F q . These observations led to the inquisitive question "when is f n,k a self-reciprocal?".…”

Section: Introductionmentioning

confidence: 99%

Self-reciprocal polynomials and coterm polynomials

Fernando

2017

Des. Codes Cryptogr.

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“…Permutation polynomials are the subject of many research efforts in mathematics [12,13,21]. The well-studied Dickson polynomials [8,15] provide examples of permutation polynomials. Most of the results on these polynomials assume that their coefficients are in a finite field.…”

Section: Introductionmentioning

confidence: 99%