Permutation polynomials of degree 6 or 7 over finite fields of characteristic 2

Li, Jiyou; Chandler, David B.; Xiang, Qing

doi:10.1016/j.ffa.2010.07.001

Cited by 29 publications

(17 citation statements)

References 9 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…(See also Shallue and Wanless [10] for a reconfirmation of the completeness of Dickson's classification.) Recently, the classification of all PPs of degree 6 and 7 over F 2 t was obtained by Li, Chandler and Xiang [7]. The present paper contributes to this line by classifying all PPs of degree 7 over F q with any odd q, up to linear transformations.…”

Section: Introductionmentioning

confidence: 74%

See 1 more Smart Citation

A classification of permutation polynomials of degree 7 over finite fields

Fan

2019

Finite Fields and Their Applications

View full text Add to dashboard Cite

show abstract

Section: Introductionmentioning

confidence: 74%

“…Section 4 will realize this by SageMath [11], a free open-source mathematics software system based on Python and many open-source packages. We omit the cases of q = 2 r , which are covered the following lemma quoted from [7].…”

Section: Introductionmentioning

confidence: 99%

A classification of permutation polynomials of degree 7 over finite fields

Fan

2019

Finite Fields and Their Applications

View full text Add to dashboard Cite

show abstract

“…Information about properties, constructions, and applications of permutation polynomials can be found in Cohen [7], Lidl and Niederreiter [24], and Mullen [26]. Some recent progress on permutation polynomials can be found in [1,4,18,20,22,23,27,32,33,34,35,36].Permutation polynomials with fewer terms over finite fields with even characteristics are in particular interesting. For example, in the study of Almost perfect nonlinear (APN) mappings which are of interest for their applications in cryptography, Dobbertin first proved a well-known conjecture of Welch stating that for odd n = 2m + 1, the power function x 2m+3 is even maximally nonlinear [15] or, in other terms, that the crosscorrelation function between a binary maximum-length linear shift register sequence of degree n and a decimation of that sequence by 2m + 3 takes on precisely the three values −1, −1 ± 2 m+1 .…”

mentioning

confidence: 99%

Permutation Trinomials Over Finite Fields with Even Characteristic

Ding¹,

Qu²,

Wang³

et al. 2015

SIAM J. Discrete Math.

View full text Add to dashboard Cite

Abstract. Permutation polynomials have been a subject of study for a long time and have applications in many areas of science and engineering. However, only a small number of specific classes of permutation polynomials are described in the literature so far. In this paper we present a number of permutation trinomials over finite fields, which are of different forms.Key words. Cryptography, difference set, linear code, permutation polynomial, trinomial AMS subject classifications. 11C08, 05A051. Introduction. A permutation polynomial f (x) over a finite field is a polynomial that acts as a permutation of the elements of the field, i.e. the map x → f (x) is one-to-one. Permutation polynomials are closely related to highly nonlinear functions [13,14] and other areas of mathematics. They have applications in combinatorial designs [13,14], coding theory [8,21,31] and cryptography [28,29,30]. For instance, Dickson permutation polynomials of order five, i.e., D 5 (x, a) = x 5 + ax 3 − a 2 x over F 3 m , led to a 70-year research breakthrough in combinatorics [13], gave a family of perfect nonlinear functions for cryptography [13], generated good linear codes [5] for data communication and storage, and produced optimal signal sets for CDMA communications [12], to mention only a few applications of these Dickson permutation polynomials. Information about properties, constructions, and applications of permutation polynomials can be found in Cohen [7], Lidl and Niederreiter [24], and Mullen [26]. Some recent progress on permutation polynomials can be found in [1,4,18,20,22,23,27,32,33,34,35,36].Permutation polynomials with fewer terms over finite fields with even characteristics are in particular interesting. For example, in the study of Almost perfect nonlinear (APN) mappings which are of interest for their applications in cryptography, Dobbertin first proved a well-known conjecture of Welch stating that for odd n = 2m + 1, the power function x 2m+3 is even maximally nonlinear [15] or, in other terms, that the crosscorrelation function between a binary maximum-length linear shift register sequence of degree n and a decimation of that sequence by 2m + 3 takes on precisely the three values −1, −1 ± 2 m+1 . The method in fact relies on the discovery of a class of permutation trinomials. Around the same time period, Dobbertin proved Niho's conjecture similarly using a class of permutation pentanomials [16]. Another example of demonstrating the application of permutation polynomials with fewer terms in constructing cyclic codes can be found in [10]. We note that these interesting

show abstract

“…For a recent verification of the classification of the PPs of degree 6 of F q for odd q, see [69] by Shallue and Wanless. Li, Chandler, Xiang [55] obtained the classification of all PPs of degree 6 and 7 of F 2 t . Theorem 6.16.…”

Section: Classification Of Pps Of Small Degreementioning

confidence: 99%