Bent Functions and Generalizations, and APN-Functions

Bent functions and related functions like perfect nonlinear functions, planar functions, negabent functions, almost bent functions and APN functions have many applications in cryptography and coding theory, and rich connections to other areas (and objects) in mathematics, like combinatorics (difference sets) or finite geometry (projective planes). Boolean bent functions have highest nonlinearity, APN-functions optimal differential properties, both important cryptographic features. For these reasons one observes increasing interest in this field and extensive research activities by many groups of researchers all over the world, resulting in many presentations on these classes of functions in international conferences and workshops and in many research papers in a large number of journals. The high quality journal Cryptography and Communications (CCDS) is entirely dedicated to this topic in a broader sense. With this project this research area shall be established further at an Austrian institution. Besides form the applicant, the post-doc researcher N. Anbar shall be employed within the project with the goal to qualify her for a professorship. Due to the rich connections to several areas and objects in mathematics, besides from character sum analysis, also methods from combinatorics, finite geometry, algebraic number theory will play a role. Of particular importance in the analysis of APN and related functions will be methods from curves over finite fields, which already proved to be successful in this area. With the applications in APN functions, also basic research on curves over finite fields shall be pushed further. Some concrete research questions to be addressed in the project are the - the analysis of the normality of p-ary bent functions, and the analysis of bent functions quadratic on the elements of a spread; - the analysis of equivalence of (vectorial) negabent functions and their relative difference sets; - the analysis of generalizations of bent functions into cyclic groups, and their differential properties; - the construction of strongly regular graphs from vectorial bent functions; - the construction of APN functions, functions with low differential uniformity, and the analysis of their components via curves over finite fields. One of the most interesting problems in the area of curves over finite fields are - to construct new families of curves having many rational points; - to investigate the asymptotic behaviour of their p-torsion points; - to obtain lattices based on cryptography arising from the curves with many extremal properties.

I.2. Summary Bent functions, vectorial bent functions, APN functions and related functions have applications in cryptography and in coding theory, and are also widely studied for connections to objects in combinatorics and finite geometry, like difference sets, designs or strongly regular graphs. In the framework of this project it is investigated which bent functions are components of vectorial bent functions. There are many constructions of bent functions, a few of vectorial bent functions. It is though an open problem if there are bent functions that are lonely, i.e., not a component of a vectorial bent function. We present the first construction of vectorial bent functions with non-weakly regular bent component functions (but dual-bent) and the first one, of which the components are non-dual-bent. Coding-theoretic criteria were obtained for a Boolean bent function (not) to be lonely. Initiating a detailed study of vanishing flats, a quadruple system attached to vectorial Boolean functions, a further design-theoretic criterion was obtained. As another application of vanishing flats, totally skew covers of n-dimensional vector spaces were constructed. As a further highlight, an upper bound was shown for the nonlinearity of plateaued vectorial Boolean functions in dimension n=2m with the maximal possible number 2^n-2^m of bent components, and a class of functions of which some attain this bound was analysed. As another result, further connections between vectorial bent functions and strongly regular graphs have been established. Generalized bent functions, a superclass of bent functions which map into the cyclic group with 2^k elements, can be seen as a bent function with a corresponding partition of F_2^n with certain properties. In the framework of the project it is shown that Maiorana-McFarland (MMF) bent functions allow the largest possible such partitions. The existence of bent functions into the cyclic group with 2^k, k > 2, elements, which do not come from spreads, has been an open problem. We could give a positive answer by constructing some classes which provable do not come from the spread construction. As a further achievement, a new method was introduced for determining the nonlinearity of some classes of quadratic functions. The nonlinearity of a new APN function presented by Taniguchi in 2019 was determined, and some shorter proofs for the nonlinearity of some other functions were found. Further, in the framework of the project, nonlinearity and differential uniformity of (n,n)-functions that contain vectorial MMF bent functions, were analysed in detail. Within this project, in total 12 articles have been submitted to recognized scientific journals or peer-reviewed conference proceedings (8 appeared, 4 are still in the reviewing process). Several of the results were presented at international conferences and in seminar talks.