A METHOD OF CONSTRUCTING PERMUTATION POLYNOMIALS OVER FINITE FIELDS
Melsik K. Kyureghyan, Sergey Abrahamyan · 2010
In this paper we consider the problem of characterizing permutation polynomials of the shape (ݔ) = ݔ + ߛ(ݔ) + ߜ�(ݔ) + (ݔ) over the field ܨ ; that is, we seek conditions on the coefficients of a polynomial finite field into itself is given by polynomial. A polynomial ܨ(ݔ) is called a permutation polynomial of ܨ �if it induces a permutation on ܨ. These polynomials were first explored in the research of Betti (Betti,1851), Mathieu and Hermite (Hermite 1863) as a way of representing permutations. A general theory was developed by Hermite (Hermite 1863) and Dickson (Dickson 1896), with many subsequent developments by Carlitz et.al. The construction of permutation polynomials over any finite fields is a challenging mathematical problem. Interest in permutation polynomials stems from both mathematical theory as well as practical applications such as cryptography. Recent papers (Betti,1851)- (Markos 2011) highlight a method of construction of permutation polynomials. The given article considers permutations of the form ݔ+ ߛ(ݔ) +ߜ �(ݔ) + (ݔ) over ܨ . Preliminaries Let's start with recalling some definitions and basic results that will be helpful to derive our main result. 1. Definition 1. Let :ܨ →ܨ and ∈ܨ . We say that ߙ∈ܨ ∗ is a c linear structure of the function f if ( ݔ+ ߙ) − (ݔ) = for all ݔ∈ܨ . Note that if ߙ is a -linear structure of , then necessarily = (ߙ) −( 0) Definition 2. Define ܨ(ݔ) =ܩ (ݔ) °ܪ(ݔ) composition of the mapping ܩ with ܪ.