C-differential Uniformity Of Some Monomials

Let p be a prime and m be a positive integer.Let Fpm be a finite field and f(x)?Fpm[x].A polynomial f(x)is called a differential ? uniformity function if the equation f(x+a)-f(x)=b has at most ? solutions for any a,b?Fpm with(a?0).In particular,f(x)is said to be a perfect nonlinear(Perfect Nonlinear)or al-most perfect nonlinear(Almost Perfect Nonlinear)function if ?=1 or 2,respectively.In recent years,PN functions and APN functions have been extensively studied due to their widely applications in cryptography,coding theory,combinational design and other fields.Recently,based on the multiplicative differential[8]analysis of cryptographic al-gorithms,Ellingsen,Felke,et.al.proposed the concept of c-difference uniformity of functions[24].A polynomial f(x)is said to have ? c-differential uniformity if the e-quation f(x+a)-cf(x)=b(c? 1)has at most ? solutions for any a,b?Fom.In this thesis,we first construct a new class of PcN functions and show that f(x)=x 3m+1/4 is a PcN over Fpm,where c=-1,m is an odd integer.Second,we obtain the c-differential spectrum of three class of monomials:x2,xpk+1,xpm-2 over Fpm.