Some Classes Of Almost Perfect Nonlinear Functions

Almost Perfect Nonlinear (APN) functions have been widely applied in cryptography and many other fields for their low differential uniformity (Difference=2) and good resistance of differential attacks. They are the optimal functions to construct S boxes. It has been a hotspot of designing and constructing Almost Perfect Nonlinear (APN) functions. In this dissertation, we construct and analyze some classes of APN functions. The paper's organization and the main contributions are as follows.In Chapter 2, the APN functions on even characteristic fields are discussed. We firstly introduce the six classes of known APN power functions, some results searched by computers and several types of APN polynomial functions. We prove the EA equivalence between two types of APN polynomial functions by applying linear displacement. In Section 3, we construct a new type of APN polynomial functions deducing from a known type of that functions by adding some items. We calculate the Walsh spectrum of a type of APN polynomial functions by trace functions and middle variables in Section 4, and the result indicates that they have the same Walsh spectrum with the GOLD functions.In Chapter 3, the APN power functions on odd character fields have been introduced, listing some results searched by computers. The general method of building APN functions is searching APN functions by computers firstly, then constructing some proper conditions which can extend the results to infinite APN functions, and proving it finally. In Section 2, a families of APN polynomial functions on odd character fields are deduced from the known results of APN polynomial functions over p=3. Their APN property can be proved by applying quadratic character.