Characterizations And Constructions Of Triple-cycle Permutations Over Finite Fields

Permutation polynomials over finite fields are widely used in many fields,such as RSA encryption system,block cipher and coding theory.In this thesis,we mainly investigate two special permutation polynomials.They are triple-cycle permutation polynomials and complete permutation polynomials.Let Fq be the finite field with q elements and let f(x)be a permutation polynomial over Fq.In this thesis,we mainly investigate the map that equal to an identity map after two compound operations,i.e.,f(?)f(?)f=I,where I is the identity map,then f(x)is called a triple-cycle permutation.We mainly study the general characterizations of triple-cycle permutation polynomials of the form f(x)=xrh(xs)when sd=q-1 over the finite field of Fq.Some triple-cycle permutation polynomials can be given over Fq from a known triple-cycle permutation polynomial on ?d and by constructing a special h(x).Meanwhile,this thesis gives some complete permutation polynomials as the form of f(x)=xh(xs)and f(x)=g(xqi-x+?)+bx,and sums up some special polynomials both are triple-cycle permutation polynomials and complete permutation polynomials.Fristly,we construct a general condition such that f(?)f(?)f=I after two compound operations on the polynomials as the form of f(x)=xrh(xs)over Fq.We discuss the elements of ?d that is the subset of Fq to characterize the general condition of a triple-cycle permutation polynomial.We give the necessary and sufficient conditions such that f(x)is a triple-cycle permutation polynomial when d equals to two,three and the others.Then,we investigate some triple-cycle permutation polynomials over Fq by constructing an appropriate h(x).Besides,as g(x)=xrh(x)s is a known triple-cycle permutation polynomialson on ?d,we characterize the condition that f(x)is atriplecycle permutation polynomial over Fq.Finally,some special complete permutation polynomials as the form of f(x)=xh(xs)and f(x)=g(xqi-x+?)+bx are given,and some examples that both are complete permutation polynomials and triple-cycle permutation polynomials are summarized.