Construction And Equivalence Of Functions With Good Cryptographic Properties

In the design of block ciphers,the choice of substitutions boxes?S-boxes?is central.Their cryptographic properties are crucial to the security of the underlying ciphers.This dissertation focuses on the design and analysis of functions with many good cryptographic properties with respect to the design of S-boxes in block ciphers.In more details,we study the construction and CCZ-equivalence of differentially 4-uniform permutations with high nonlinearity and high algebraic degree,the construction of?n,m?-functions?m>n/2?with low differential uniformity,high nonlinearity and high algebraic degree and the con-struction of low-weight Boolean functions with high order correlation immunity.The obtained results aim to provide more choices for the design of S-boxes in SPN ciphers,the design of S-boxes in Feistel ciphers and the design of S-boxes masking to prevent the side channel attacks.Further,we study the problem of calculating the number of equiv-alent classes of vectorial Boolean functions,which is helpful to construct cryptographic functions.The main results are obtained as follows:1.We firstly give an equivalent condition on the switching construction of differ-entially 4-uniform permutations over F_[2^2K] from the inverse function.Then we construct a new infinite family of differentially 4-uniform permutations.These differentially 4-uniform permutations have high nonlinearity and achieve the highest algebraic degree.2.We creatively propose a new notion named“Projected Differential Spectrum”.By considering this notion,we present a new method to investigate the CCZ-equivalence between cryptographic functions.Based on these results,we mathematically prove some results on the CCZ-inequivalence between the only three infinite family of differentially4-uniform permutations overF_[2^2K],which number grows double exponentially when k in-creases.A method to judge whether an isolated differentially 4-uniform permutation is CCZ-inequivalent to a given function in the aforementioned three classes on small fields is given at last.3.We characterize the differential uniformity of those?n,m?-functions of the form F?x,z?=??z?I?x?.Using this characterization,we construct an infinite family of differ-entially 2^m-1+2-uniform?2m-1,m?-functions?achieving Nyberg's bound?,which also have high nonlinearity and high algebraic degree.We then discuss an infinite family of differentially 4-uniform?m+1,m?-functions in this form,which leads to many differen-tially 4-uniform permutations.We also present a method to construct infinite families of?m+k,m?-functions with low differential uniformity for 1?k?m-2.4.We propose methods to construct low-weight Boolean functions with high order correlation immunity based on the Fourier transform and the Kronecker product of vec-tors.According to the aforementioned methods,we obtain some low-weight d-CI Boolean functions with n variables,which achieving the known lowest Hamming weight.5.According to the cycle index and Burnside's lemma,we give a compact estimation of the number of vectorial Boolean functions up to extended affine equivalence and affine equivalence.We also obtain an explicit formula and calculate the exact number of affine equivalence of?n,m?-functions with 1?m,n?11.