On Permutation Polynomial And Orthogonal Systems Over Finite Field

Permutation polynomials have been widely used in number theory, combinatorial theory,group theory, and the combination of algebra. Since the1970s, permutation polynomials haveattracted more attention from the field of mathematic, engineer, and technology, owing to thenecessary of study of cryptography and information security. In1971, orthogonal system, as ageneralization of permutation polynomial, was introduced by Niederreiter. Permutationpolynomial and orthogonal system over finite fields have applications in a variety of areas,such as cryptography for the secure transmission of information, and combinatorics for theconstruction of various kinds of combinatorial designs used in statistical or experimentaldesign theory. This paper mainly studies a kind of special orthogonality of polynomial groupover the finite field.The thesis is arranged as follows：Chapter1Make an introduction about the definition and properties of permutationpolynomials, orthogonal polynomial group, trace function, additive character and so on.Chapter2In the finite field, based on the properties of the additive character, thisthesis presents how to use the additive character to justify a special kind of polynomial grouporthogonality. This kind of polynomial group is defined as follows:LetF qbe the finite fields of q elements with q2and be a primitive elementofF q. Let q,ddenote the system defined by the two polynomials Where d is a positive integer coprime to q1. We provide some sufficient conditions forthe system q,dto be orthogonal or not.Chapter3Make an introduction on orthogonal system of polynomials and permutationpolynomial in several indeterminates over Z mZ.