Research On Permutation And Differential Properties Of Cryptographic Functions

Designing safe and highly efficient symmetric and asymmetric cryptography is the key problem in cryptography.Many symmetric ciphers use S-boxes to generate nonlin-ear functions in encryption algorithm.S-boxes use functions over the finite fields with both low differential uniformity and.high nonlinearity to resist the differential attacks as well as the linear attacks.American AES program greatly improve the development in theory and method of block ciphers.AES encryption standard uses Inverse function,a differentially 4-uniform permutation over F28,in S-box.Recently,the domestic and in-ternational experts and scholars are devoted to finding new permutation functions with low differential uniformity to replace the Inverse function.This problem is involved with two important properties of cryptographic functions—permutation properties and differential properties.The content of this thesis is mainly about these two properties of cryptographic functions.Investigating the permutation properties of cryptographic functions,we mainly solve the problems about finding new permutation polynomials over the finite fields.Permutation polynomials are widely applied in coding theory,cryptography and com-binatorial design.In 2015,Tu Ziran and Zeng Xiangyong constructed permutation polynomials of the form f(x)= {x2m + x + ?)s + x over F22m,where s(2m + 1)?2m + 1(mod 22m-1)or s(2m-1)? 2m-1(mod22m-1).Inspired by their re-search,we consider permutation polynomials of the form f(x)=(Trmn(x)+?)s + L(x)with the special exponent s satisfying s(2m + 1)= 2m + 1(mod 22m-1)or s(2m-1)? 2m-1(mod 22m-1)in the thesis and this form is completely new.We combine trace function over finite fields with the special exponent and obtain some regulation by the computer experiments and through derivation and proofs we construct new permutation polynomials over finite fields.For the differential properties of cryptography,calculating the differential spec-trum of Boolean functions is one of the main research topics.Differential spectrum is the reflection of the differential properties of cryptography functions in detail and it's a CCZ-equivalent invariant.In 2013,Sung-Tai Choi and others determined the differen-tial spectrum of x(?)over Fpn with odd prime p,as well as the differential spectrum of x(?)over Fpn with odd prime p satisfying p ? 3(mod 4)and odd n with m | n.Based on their work,we consider the differential spectrum of power functions with the exponent d satisfying d(pk+1)? 2(mod pn-1),where e = gcd(n,k)and n/e is odd.This work greatly generalizes the result of Sung-Tai Choi,since their exponents account for only a small proportion in this thesis.We take a similar method as that of Sung-Tai Choi to divide the finite fields into sets and classify the exponent d,also we determine the relations between some image sets and finally obtain the differential spectrum of such power functions.