Almost perfect nonlinear functions and related combinatorial structures

A map f(x) from the finite field Fpn to itself is said to be differentially k-uniform if k is the maximum number of solutions of the equation f(x + a) - f(x) = b where a, b [is in] Fpn , a ≠ 0. In particular, 2-uniform maps over F2n are called almost perfect nonlinear (APN) maps. These maps are of interest in cryptography because they offer optimum resistance to linear and differential attacks on certain cryptosystems. They can also be used to construct several combinatorial structures of interest.;In this dissertation, we characterize and classify all known power maps f(x) = xd over F2n , which are APN or of low uniformity. We discuss some basic properties of APN maps, collect all known APN power maps, and give a classification of APN power maps up to equivalence. We also give some insight regarding efforts to find other APN functions or prove that others do not exist and classify all power maps according to their degree of uniformity for n up to 13.;In the latter part of this dissertation, through the introduction of an incidence structure, we study how these functions can be used to construct semi-biplanes utilizing the method of Robert S. Coulter and Marie Henderson. We then consider a particular class of APN functions, from which we construct symmetric association schemes of class two and three. Using the result of E. R. van Dam and D. Fon-Der-Flaass, we can see that the relation graphs of some of these association schemes are distance-regular graphs. We discuss the local structure of these distance-regular graphs and characterize them

Classification of small class association schemes coming from certain combinatorial objects

We explore two- or three-class association schemes. We study aspects of the structure of the relation graphs in association schemes which are not easily revealed by their parameters and spectra. The purpose is to develop some combinatorial methods to characterize the graphs and classify the association schemes, and also to delve deeply into several specific classification problems. We work with several combinatorial objects, including strongly regular graphs, distance-regular graphs, the desarguesian complete set of mutually orthogonal Latin squares, orthogonal arrays, and symmetric Bush-type Hadamard matrices, all of which give rise to many small-class association schemes. We work within the framework of the theory of association schemes.;Our focus is placed on the search for all isomorphism classes of association schemes and characterization of small-class association schemes of specific order. In particular, we examine two-class association schemes (strongly regular graphs) of order 64 and their three-class fission schemes. After we collect \u27feasible\u27 parameter sets for the putative association schemes, we make an attempt to check the realization (existence) of the parameter sets and describe the structure of the schemes chiefly by investigating the structure of their relation graphs. In the course of this thesis, we find a new way to construct orthogonal arrays and investigate their implications for strongly regular graphs, symmetric Bush-type Hadamard matrices, and three-class association schemes. We obtain several results regarding the characterization and classification of two- or three-class association schemes of order 64