Applications of the Hasse-Weil bound to permutation polynomials

Abstract Riemann's hypothesis on function fields over a finite field implies the Hasse–Weil bound for the number of zeros of an absolutely irreducible bi-variate polynomial over a finite field. The Hasse–Weil bound has extensive applications in the arithmetic of finite fields. In this paper, we use the Hasse–Weil bound to prove two results on permutation polynomials over F q where q is sufficiently large. To facilitate these applications, the absolute irreducibility of certain polynomials in F q [ X , Y ] is established.

[1]  Xiang-dong Hou.  Lectures on Finite Fields , 2018, Graduate Studies in Mathematics.

[2]  Xiang-dong Hou,et al.  Determination of a type of permutation trinomials over finite fields, II , 2013, Finite Fields Their Appl..

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

[4]  Elodie Leducq,et al.  Functions which are PN on infinitely many extensions of $$\mathbb {F}_p,\,p$$Fp,p odd , 2010, Des. Codes Cryptogr..

[5]  Michael E. Zieve.  Planar functions and perfect nonlinear monomials over finite fields , 2013, Des. Codes Cryptogr..

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

[7]  Stephen D. Lappano.  A note regarding permutation binomials over Fq2 , 2015, Finite Fields Their Appl..

[8]  A. Weil.  Sur les courbes algébriques et les variétés qui s'en déduisent , 1948 .

[9]  Henning Stichtenoth,et al.  Algebraic function fields and codes , 1993, Universitext.

[10]  W. Schmidt.  Equations over Finite Fields: An Elementary Approach , 1976 .

[12]  A. Blokhuis,et al.  On the roots of certain Dickson polynomials , 2018, Journal of Number Theory.

[13]  Xiang-dong Hou,et al.  Solution to a problem of S. Payne , 2003 .

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

[15]  S. Lang,et al.  NUMBER OF POINTS OF VARIETIES IN FINITE FIELDS. , 1954 .

[16]  David R. Hayes.  A geometric approach to permutation polynomials over a finite field , 1967 .

[17]  Xiang-Dong Hou,et al.  A new approach to permutation polynomials over finite fields , 2012, Finite Fields Their Appl..

[18]  Xiang-dong Hou,et al.  Determination of a type of permutation binomials over finite fields , 2015 .

[19]  Tor Helleseth,et al.  Permutation polynomials of the form (xpm-x+δ)s+L(x) over the finite field Fp2m of odd characteristic , 2015, Finite Fields Their Appl..

[20]  D. J. Lewis,et al.  NOTES ON CONGRUENCES (I) , 1963 .

[21]  Francisco Monserrat,et al.  On the classification of exceptional planar functions over $$\mathbb {F}_{p}$$Fp , 2013, ArXiv.

[22]  Qiang Wang,et al.  Cyclotomic Mapping Permutation Polynomials over Finite Fields , 2007, SSC.

[23]  Nian Li,et al.  On two conjectures about permutation trinomials over 𝔽32k , 2017, Finite Fields Their Appl..

[24]  Cunsheng Ding,et al.  Permutation polynomials of the form cx+Trql/q(xa)$cx+\text {Tr}_{q^{l}/ q}(x^{a})$ and permutation trinomials over finite fields with even characteristic , 2014, Cryptography and Communications.