The Design And Analysis Of The Almost Perfect Nonlinear Functions

Almost perfect nonlinear (APN) functions have good difference properties, and they are important candidate functions for designing S-box with desired cryptographic properties. These functions should also have high nonlinearity to resist linear attack. In this paper, we study and analyze a family of almost perfect nonlinear functions, The main contributions are as follows.In this paper, the definition of the linear cryptanalysis and differential cryptanalysis are reviewed, and the link between linear attack and differential attack are introduced too. When used as an S-box in a block cipher, it contributes optimally to the resis-tance to differential cryptanalysis. So the almost perfect nonlinear mappings which have the optimally differential uniformity attracted much attention. Then two notion of equivalence has been introduced which are extended affine (EA) equivalent and Carlet-Charpin-Zinoviev (CCZ) equivalence.In chapter3, the all known almost perfect nonlinear functions are listed and we classify the almost perfect nonlinear functions up to the algebraic degree and the dif-ference of the finite fields. In particular, an infinite class of quadratic almost perfect nonlinear functions studied, and then their nonlinearity is determined. Further, the distribution is also completely determined. At last, a few more date were given by computer. In section3, we construct a family of quadratic almost perfect functions, and for a odd prime p, the family of quadratic almost perfect functions is a family of perfect functions in GF(p2k), besides we generalize the family of perfect functions.