| In block cipher algorithm,an S-box is the only nonlinear component.Cryptographic functions used as S-boxes should have low differential uniformity and high nonlinearity,and those functions usually should be permutations.In addition,the permutations with low differential uniformity can be used to construct linear codes,and are also widely used in combinatorial designs.Compared with the differential uniformity,the differential spectra more accurately describe the differential properties of the cryptographic functions.In this thesis,the cryptographic properties of four classes of permutation polynomials over finite field F2n are studied,and their differential spectra and nonlinearity are calculated.The applications of these functions in coding theory and combinatorial design are further studied.The main contributions are as follows:1)The differential spectra and Walsh spectra of three kinds of permutations over finite field F2n are studied.The first one is F1(x)=x2k+1+cx2n-k+1,where k is any integer and d=(k,n)and n/d is odd,and c≠αt(2d-1)for any integer t.The second one is F2(x)=x22k+1+c2k+1x2k+1+cx2,where n=3k,and c#αt(2k-1)for any integer t.The third one is F3(x)=x5+x3+x,where n is odd.Utilizing the theory of linearized polynomials,the differential spectra are characterized.The results show that the first and second permutations are differentially two-valued functions.Based on the theory of quadratic form and the power sum method,the Walsh spectra of these functions are further obtained.2)The differential spectrum of the cryptographic function F(x)=x2t+2+γx with BCT 4 over funite field F2n is determined,where n≡2(mod4),t=n/2 and ord(γ2t-1)=3,and the result shows that F(x)is differentially two-valued function.The Walsh spectrum of F(x)is also calculated,and thus its nonlinearity is obtained.As applications,a class of binary linear codes is constructed from F(x),and the weight distributions of the linear codes are determined.Finally,based on the relationship between differentially two-valued functions and 2-designs,a 2-design is constructed. |
