Constructions And Properties Analysis On Low Differential Uniformity Functions

Functions with better cryptographic properties are one of focuses of the security research of block ciphers all the time. Low differential uniformity functions usually have lower uniformity and higher nonlinearity, and so are those functions with better cryptographic properties. Alone with the development of cryptanalysis especially the maturity of differential cryptanalysis and linear cryptanalysis, the studies of low differential uniformity functions are more important. Moreover, low differential uniformity functions are widely used in algebra combinatorics.In this paper, with the interrelated knowledge of finite field, our research is focused on some new constructions of almost perfect nonlinear functions, perfect nonlinear functions and low differential uniformity functions. The main results are as follows:Some APN functions are constructed. We investigate Dillon's idea of switching construction, deduce a construction of APN functions from the research on APN functions with the form of F(x)+f(x) where F(x) are APN functions, and construct some APN functions which are EA-inequivalent to known APN functions in binary field.Some constructions of PN functions are investigated. By using two known perfect nonlinear functions, we construct a quadratic binomial perfect nonlinear function over finite fields of odd characteristic based on determining the root of a linearized function, prove that this quadratic function is inequivalent to x 2, and analyze on its inequivalence to x ps+1 by an example. Moreover, according to Dillon's idea of switching construction and the construction of APN functions with the form of F(x)+f(x) above, we deduce some conclusions on odd characteristic finite fields from even characteristic finite fields, and finally acquire a construction of PN functions with the form of F(x)+f(x) where F(x) are PN functions.Two types of low differential uniformity functions with higher nonlinearity are constructed. Firstly, we analyze on quadratic functions in binary field, give a binomial differential 4 uniformity function, and prove that it has higher nonlinearity. Secondly, we analyze on cubic functions on even characteristic finite fields, give two binomial differential 6 uniformity functions, and give a lower bound of the nonlinearity of the two functions by using the properties of second order nonlinearity. Finally, we investigate an open conjecture, and prove that the functions introduced have lower differential uniformity.