Dickson Polynomials That Are Permutations

A theorem of S.D. Cohen gives a characterization for Dickson polynomials of the second kind that permutes the elements of a finite field of cardinality the square of the characteristic. Here, a different proof is presented for this result. 1. Permutation polynomials of finite fields. In recent years cryptographers became interested in finding polynomials that induce a bijection of a finite field under substitution. This property has been used in several constructions of cryptographic systems for the secure transmission of data (see, for instance, [19], [22, Ch. IX], [23]). Permutations of this type have also notable applications in combinatorics (cf., e.g., [8], [28], [29]). However, the interest in permutation polynomials (shortly, PP) is not as recent as it might seem. A classical result of Hermite [17] provides a necessary and sufficient condition for 2000 Mathematics Subject Classification: 11T06, 13P10.