论文信息 - Reversed Dickson polynomials over finite fields

Reversed Dickson polynomials over finite fields

Reversed Dickson polynomials over finite fields are obtained from Dickson polynomials D"n(x,a) over finite fields by reversing the roles of the indeterminate x and the parameter a. We study reversed Dickson polynomials with emphasis on their permutational properties over finite fields. We show that reversed Dickson permutation polynomials (RDPPs) are closely related to almost perfect nonlinear (APN) functions. We find several families of nontrivial RDPPs over finite fields; some of them arise from known APN functions and others are new. Among RDPPs on F"q with q<200, with only one exception, all belong to the RDPP families established in this paper.

[1] Gary L. Mullen,et al. Value sets of Dickson polynomials over finite fields , 1988 .

[2] Shin-won Kang. REMARKS ON FINITE FIELDS III , 1985 .

[3] K. Conrad,et al. Finite Fields , 2018, Series and Products in the Development of Mathematics.

[4] Hans Dobbertin,et al. Almost Perfect Nonlinear Power Functions on GF(2n): The Welch Case , 1999, IEEE Trans. Inf. Theory.

[5] H. Dobbertin. Almost Perfect Nonlinear Power Functions on GF(2n): A New Case for n Divisible by 5 , 2001 .

[6] Claude Carlet,et al. Two Classes of Quadratic APN Binomials Inequivalent to Power Functions , 2008, IEEE Transactions on Information Theory.

[7] I. G. MacDonald,et al. Symmetric Functions and Orthogonal Polynomials , 1998 .

[8] Rudolf Lide,et al. Finite fields , 1983 .

[9] Gary L. Mullen,et al. Finite Fields and Applications , 2007, Student mathematical library.

[10] Stephen D. Cohen,et al. A class of exceptional polynomials , 1994 .

[11] L. Dickson. The Analytic Representation of Substitutions on a Power of a Prime Number of Letters with a Discussion of the Linear Group. , 1896 .

[12] Kaisa Nyberg,et al. Differentially Uniform Mappings for Cryptography , 1994, EUROCRYPT.

[13] Tor Helleseth,et al. New Families of Almost Perfect Nonlinear Power Mappings , 1999, IEEE Trans. Inf. Theory.

[14] Robert S. Coulter. Explicit evaluations of some Weil sums , 1998 .