Cryptographic functions and their relations to other areas

Many classes of functions, like bent functions, generalized bent functions, almost perfect nonlinear (APN) functions, almost bent (AB) functions, pla- nar functions are characterized with their differential properties or with their behaviour with respect to some unitary transforms. These classes of functions have applications in cryptography (resistance against differential attacks and against linear attacks) and in coding the- ory, and they have also rich connections to other areas (and objects) in mathematics, like combinatorics and finite geometry (difference sets, rela- tive difference sets, projective planes, designs, strongly regular graphs). These classes of functions have been studied widely over the last 40 years, due to their rich connections to many areas, one can still observe increasing interest in these topics. Groups of researchers from various backgrounds and areas, such as combinatorics, (algebraic) number theory, finite geometry, coding theory, cryptography are working on these classes of functions. Some concrete research questions to be investigated in this project are - the construction and analysis of partitions of vector spaces over prime fields, which, similar to spreads, yield large classes of bent functions and corresponding difference sets; - the construction and analysis of generalized bent functions, of correspond- ing partitions and bent function spaces; - the introduction and analysis of concepts of equivalence for functions into the cyclic group; - the study of codes and designs connected with bent functions, APN func- tions and related functions; - a detailed study of some features for vectorial functions (extendability of Boolean bent functions to vectorial bent functions, constructions and anal- ysis of cryptographic properties of vectorial functions with maximal number of bent components, further analysis of relations between vectorial bent functions partial difference sets and strongly regular graphs); - the analysis of properties of (potential) Boolean and vectorial components of APN functions. As methods, character sum analysis and finite field arithmetic will play an important role. Due to the rich connections of the considered functions, also methods from combinatorics and finite geometry will have to be used.