Research Of Generalized Self-conjugate-reciprocal Polynomials And Password Functions Over Finite Fields

Let Fq be a finite field with q elements,where q is a prime power.In this paper,firstly,we generalize self-conjugate-reciprocal polynomials over finite fields.Furthermore,we construct several classes of mappings with low differential uniformity and complete PN functions.Detailed works and results are as follows:In Chapter two:We present a factorization of xqn+1-? over Fq2 and the enumeration of self-conjugate-reciprocal monic polynomials over Fq2,where ??Fq*.Moreover,we propose the explicit number of monic self-conjugate-reciprocal irreducible factors of xn+1.As an application to negacyclic codes,the number of linear Hermitian complementary dual negacyclic codes is evaluated.In Chapter three,we firstly present a new approach to prove that x3n+3/2 is a perfect nonlinear function in by employing some properties of Dickson polynomials.Secondly,with the piecewise functions,some power mappings with low differential uniformity in IFpn is proposed.We show that xd is differentially 3 uniform function over Fpn under the condition that d=±pn-1/2+pk+1,where n/gcd(n,k)is odd,p is a prime and p?1(mod4).We proposed firstly the conception of complete perfect nonlinear functions,and some of such functions are provided.