The Existence And Enumeration Of A Class Of Composited Involutional Polynomials Over Finite Fields

Permutation polynomials（especially involutional polynomials）over finite fields are widely used in coding theory and cryptography.Since Charpin used the exponent for polynomials to study involutional Dickson polynomials in 2016,the involutional polynomial over finite fields is an interesting topic.So far,people have obtained some necessary and suffi-cient conditions for monomials or some special polynomials over finite fields to be involutional polynomials,and then studied the properties for their fixed points.However,there are few relevant results about the criterion for the composited polynomial of two polynomials to be involutional over finite fields.In this thesis,we obtain some necessary and sufficient conditions for that the composited polynomial of a permutation polynomial and an involutional polynomial over finite fields is an involutional polynomial,and then give a counting formula basing on the relationship between the fixed points set and the non-fixed points set of a polynomial.Furthermore,we obtain all the involutional polynomials and their truth values table on F₇,and a class of involutional polynomials in special forms and their truth values table on F₁₃.For a given involutional polynomial,another involutional polynomial satisfying the conditions of our theorem can be obtained by Lagrange interpolation formula,and then two involutional polynomials can be obtained by using this theorem.