On March 24, 2021,
Alexey Oblaukhov and
Aleksander Kutsenko defended their candidate dissertations, made under the scientific supervision of
Natalia Tokareva. Both papers are in the field of discrete mathematics and cryptography.
The work of Alexander Kutsenko is devoted to the description of characteristics of duality mapping in the class of Boolean bent functions. Due to their maximum nonlinearity, bent functions and their generalizations are actively used in various cryptographic systems such as ciphers AES, CAST, Grain, hash function HAVAL, etc. It is notable that the "key" to unsolved questions in the field of bent functions lies in the disclosure of mechanisms for displaying duality. In this case, bent functions that are equal to their own dual functions — self-dual bent functions — or to anti-self-dual bent functions are of particular interest. In Alexander's work, groups of automorphisms of self-dual and anti-self-dual bent functions are described, the full spectrum of distances between such functions is obtained, necessary and sufficient conditions for self-duality for functions from various classes are found, and a number of other results are obtained.
Alexey Oblaukhov studies metrically regular sets in a Boolean cube: their constructions, properties, classification issues. For any subset in the discrete metric space, its metric complement can be naturally defined — a set whose elements are as far from the initial set as possible. A set is called metrically regular if the metric complement to its metric complement coincides with itself. Such sets have various interpretations and applications in the theory of coding and cryptography; their study is associated with the solution of long-standing unsolved problems. In Alexey's thesis, a general form of the metric complement of an arbitrary linear subspace of the Boolean cube was obtained, original constructions of such sets and lower bounds for their number were proposed, and the metric regularity of almost all Reed-Muller codes of arbitrary length was proved using the covering toolkit.
Natalya Tokareva, head of the research project "Cryptography and Information Security" of the Mathematical Center in Akademgorodok, says:
The guys are very strong: they actively conduct scientific research, teach at NSU, participate in international seminars and conferences. Twice they underwent scientific internships in their scientific field at the University of Bergen (Norway).
The staff of MCA sincerely congratulate on joining the ranks of Candidates of physical and mathematical sciences!
