International Journal of Mathematics and Mathematical Sciences
Volume 10 (1987), Issue 3, Pages 535-543

On permutation polynomials over finite fields

R. A. Mollin1,2 and C. Small1,2

1Department of Mathematics and Statistics, University of Calgary, Calgary T2N 1N4, Alberta, Canada
2Department of Mathematics and Statistics, Queen's University, Kingston K7L 3N6, Ontario, Canada

Received 31 July 1986; Revised 3 October 1986

Copyright © 1987 R. A. Mollin and C. Small. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


A polynomial f over a finite field F is called a permutation polynomial if the mapping FF defined by f is one-to-one. In this paper we consider the problem of characterizing permutation polynomials; that is, we seek conditions on the coefficients of a polynomial which are necessary and sufficient for it to represent a permutation. We also give some results bearing on a conjecture of Carlitz which says essentially that for any even integer m, the cardinality of finite fields admitting permutation polynomials of degree m is bounded.