Permutation Polynomials Over Finite Fields And Their Applications In Cryptography

The permutation polynomials have extensive applications in the area of number theory, combinatorial theory, group theory and non-associative algebra and so on. From 70 ' s of the last century for the necessary of the study of cryptography, permutation polynomials have been extensively concerned by mathematicians, engineering and technical personnel. Recently, nice applications of specific types of permutation polynomials in Turbo codes and LDPC codes are found. The paper is dedicated to the constructions of permutation polynomials over finite fields. The main results are as follows:Firstly, we give an introduction about the development and applications of permutation polynomials, and provide some open problems and conjectures. We also give a survey of the judgement methods for permutation polynomials, including Hermite's criterion and through the investigation of the number of their value sets and so on.Secondly, in this paper, we construct some permutation polynomials, and discuss the properties of some specific types of polynomials over finite fields, and prove that they are permutation polynomials under certain conditions. We give some linearised permutation polynomials and orthomorphic permutation polynomials, including the expression and enumeration of the orthomorphic permutations of degree 4 over F_2⁴ and F_2⁵ , and linearised orthomophic permutations of degree 8 overF_2⁴, and the constructions of general orthomorphic permutation polynomials over finite fields.Finally, we introduce some applications of permutation polynomials. We also give simple introduction for cryptographic systems, and provide some examples on the applications of APN permutation and balanced word. We also calculate the inverse of a class of permutation polynomials, so we can construct a security key exchange protocol.