Cryptographic functions and their relat...

Many classes of functions, like bent functions, generalized bent functions (which map into the cyclic group), almost perfect nonlinear (APN) functions, almost bent (AB) functions, planar 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 theory, and they have also rich connections to other areas (and objects) in mathematics, like combinatorics and finite geometry (difference sets, relative difference sets, projective planes, designs, strongly regular graphs, association schemes).

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 (e.g. semifield spreads) yield large classes of bent functions and corresponding difference sets;

- the analysis of these "bent partitions" respectively "generalized semifield spreads" in connections with objects from combinatorics, like strongy regular graphs, association schemes, patial difference set packings.

- 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 functions and related functions;

- a detailed study of some features for vectorial functions (extendability of Boolean bent functions to vectorial bent functions, constructions and analysis 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.