Structure And Property Analysis Of Some Cryptographic Functions

With the development of modern cryptography, especially the maturity of differential attack and linear attack technology, structure and properties of cryptographic functions attracts more and more people’s attention, especially for the studies of Bent function and low differential function has become one of hot issues in current research in the field of cryptography. In this dissertation, our research is mainly focused on two cryptographic properties of cryptographic functions include the nonlinearity and the differential uniformity, we construct and analyze a class of Bent functions and several classes of low differential uniformity functions. The main results are listed as follows:1. A class of Bent functions in odd prime field are constructed. Using the theory of quadratic form and exponential sum, through the common factor of polynomials over the finite field the Bent properties of a class of quadratic functions are described. Specifically, the enumeration problem of such a class of Bent functions is completely solved when n satisfies some certain conditions. The class of Bent function constructed is more general than the known quadratic Bent function.2. A class of new APN polynomial functions on binary field and a family of new PN functions in odd prime field are constructed. By adding to a switching postfix function from a known APN function on binary field, we obtain a new APN polynomial function on binary field and the APN functions are not EA equivalent to the currently known APN power functions are proved. Moreover, we further study the APN polynomial functions and then get a family of new PN polynomial functions in odd prime field. Then the proof that the PN polynomial functions constructed are not CCZ equivalent to some currently known PN functions is given.3. A class of new differential4-uniformity functions is constructed. Based on the idea that exchanging two values of a cryptographic function, its differential uniformity change only a little, by exchanging any two values of the APN Kasami function on binary field, we obtain a class of differential4-uniformity cryptographic functions and the equivalent condition of the existence of such a class of differential4-uniformity is given. Through the analysis of the property of nonlinearity and algebraic degree about such differential4-uniformity functions, we prove that they are permutation functions of high nonlinearity and high algebraic degree, which can provide a more reference for the design of the symmetric cryptography.