Classes of Permutation Polynomials Based on Cyclotomy and an Additive Analogue

I present a construction of permutation polynomials based on cyclotomy, an additive analogue of this construction, and a generalization of this additive analogue which appears to have no multiplicative analogue. These constructions generalize recent results of Jose Marcos.

[1]  Michael E. Zieve On some permutation polynomials over Fq of the form x^r h(x^{(q-1)/d}) , 2007, 0707.1110.

[2]  Michael E. Zieve Some families of permutation polynomials over finite fields , 2007, 0707.1111.

[3]  L. Dickson Linear Groups, with an Exposition of the Galois Field Theory , 1958 .

[4]  Luyan Wang,et al.  On Permutation Polynomials , 2002 .

[5]  Stephen D. Cohen,et al.  Exceptional Polynomials over Finite Fields , 1995 .

[6]  Harald Niederreiter,et al.  Cyclotomic R-orthomorphisms of finite fields , 2005, Discret. Math..

[7]  Ariane M. Masuda,et al.  Permutation binomials over finite fields , 2007, 0707.1108.

[8]  N. S. James,et al.  Permutation polynomials on matrices , 1987 .

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

[10]  José E. Marcos,et al.  Specific permutation polynomials over finite fields , 2008, Finite Fields Their Appl..

[11]  E. Mathieu Mémoire sur l'étude des fonctions de plusieurs quantités, sur la manière de les former et sur les substitutions qui les laissent invariables. , .

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

[13]  Anthony B. Evans Orthomorphism graphs of groups , 1992 .

[14]  Wang Daqing,et al.  Permutation polynomials over finite fields , 1987 .

[15]  Young Ho Park,et al.  Permutation polynomials with exponents in an arithmetic progression , 1998, Bulletin of the Australian Mathematical Society.

[16]  June-Bok Lee,et al.  Permutation polynomials and group permutation polynomials , 2001, Bulletin of the Australian Mathematical Society.

[17]  H. Niederreiter,et al.  Complete mappings of finite fields , 1982, Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics.

[18]  Qiang Wang,et al.  A generalized Lucas sequence and permutation binomials , 2005 .

[19]  L. Carlitz Permutations in a finite field , 1953 .

[20]  Charles Hermite,et al.  Œuvres de Charles Hermite: Sur les fonctions de sept lettres , 2009 .

[21]  Rudolf Lidl,et al.  Permutation polynomials of the formxrf(xq−1)/d) and their group structure , 1991 .

[22]  L. Dickson,et al.  Linear Groups: With an Exposition of the Galois Field Theory , 1902 .

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

[24]  Qiang Wang,et al.  On Polynomials of the Form xrf(x(q-1)/l) , 2007, Int. J. Math. Math. Sci..

[25]  Ding Decheng Non-p-generic and strongly nonbranching degree , 1994 .

[26]  C. Small,et al.  On permutation polynomials over finite fields , 1987 .

[27]  Franc Brioschi Des substitutions de la forme $$\theta (r) \equiv \varepsilon \left( {r^{n - 2} + ar^{\tfrac{{n - 3}}{2}} } \right)$$ pour un nombren premier de lettres , 1870 .

[28]  Leonard Carlitz,et al.  The number of solutions of a special system of equations in a finite field , 1966 .

[29]  Yann Laigle-Chapuy,et al.  Permutation polynomials and applications to coding theory , 2007, Finite Fields Their Appl..

[30]  Qiang Wang,et al.  Research Article On Polynomials of the Form x r f(x (q−1)/l ) , 2007 .

[31]  Qiang Wang,et al.  On some permutation polynomials over finite fields , 2005, Int. J. Math. Math. Sci..

[32]  L. Carlitz,et al.  Some theorems on permutation polynomials , 1962 .

[33]  A family of exceptional polynomials in characteristic three , 1996 .

[34]  Groups of permutation polynomials , 1967 .

[35]  Amir Akbary,et al.  ON SOME CLASSES OF PERMUTATION POLYNOMIALS , 2008 .

[36]  June-Bok Lee,et al.  SOME PERMUTING TRINOMIALS OVER FINITE FIELDS , 1997 .

[37]  Charles Wells A generalization of the regular representation of finite Abelian groups , 1968 .

[38]  H. Niederreiter,et al.  The structure of a group of permutation polynomials , 1985, Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics.