AEI-DFG: Boolean Functions meet Combinatorics

• Applicant: Professor Dr. Alexander Pott
• Subject Area: Mathematics
• Term: since 2024
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 541511634

In cryptography, one uses quite often functions (defined on finite fields) which are highly nonlinear. To construct such highly nonlinear functions, one uses in many situations algebraic properties of finite fields. So called "almost perfect nonlinear" (APN) functions as well as bent functions are quite popular. One can define nonlinearity in these cases as follows: consider the two-dimensional affine subspaces in a vector space defined over a field with just two elements. One tries to rearrange the points in this vector space such that no subspace is maintained. This is possible if and only if there is a bijective APN function. Note that the 2-dimensional subspaces form a combinatorial design, so that one may ask the same question for combinatorial designs in general. We expect that this new viewpoint sheds new light on some of the main open problems about APN functions, notably the problem about the existence of bijective APN functions. Is it perhaps easier to obtain such derangements for combinatorial designs different from the point-subspace design? If yes, what is the reason that it is difficult to find such derangements in the classical case? In this project, we will also consider new construction methods for APN functions using other algebraic structures different from finite fields. If we restrict ourselves to quadratic functions, it might be possible to consider semifields, since in the quadratic case associativity is not needed (and associativity is a property that does not necessarily hold in semifields). In this way, we also expect to construct new sequences. Moreover, we want to investigate cryptographic schemes which rely on masking quadratic functions. In one research direction, we plan to replace quadratic functions over finite fields by other functions (see above). We also want to replace the classical approach of masking (which is related to affine equivalence of functions) by the more general conecpt of CCZ equivalence.

• DFG Programme: Research Grants
• International Connection: Spain
• Partner Organisation: Agencia Estatal de Investigación
• Cooperation Partner: Professor Dr. Domingo Gomez-Perez