On metric complements and metric regularity in finite metric spaces

This review deals with the metric complements and metric regularity in the Boolean cube and in arbitrary finite metric spaces. Let A be an arbitrary subset of a finite metric space M, and A be the metric complement of A — the set of all points of M at the maximal possible distance from A. If the metric complement of the set A coincides with A, then the set A is called a metrically regular set. The problem of investigating metrically regular sets was posed by N. Tokareva in 2012 when studying metric properties of bent functions, which have important applications in cryptography and coding theory and are also one of the earliest examples of a metrically regular set. In this paper, main known problems and results concerning the metric regularity are overviewed, such as the problem of finding the largest and the smallest metrically regular sets, both in the general case and in the case of fixed covering radius, and the problem of obtaining metric complements and establishing metric regularity of linear codes. Results concerning metric regularity of partition sets of functions and Reed — Muller codes are presented

Mathematical methods in solutions of the problems from the Third International Students' Olympiad in Cryptography

The mathematical problems and their solutions of the Third International Students' Olympiad in Cryptography NSUCRYPTO'2016 are presented. We consider mathematical problems related to the construction of algebraic immune vectorial Boolean functions and big Fermat numbers, problems about secrete sharing schemes and pseudorandom binary sequences, biometric cryptosystems and the blockchain technology, etc. Two open problems in mathematical cryptography are also discussed and a solution for one of them proposed by a participant during the Olympiad is described. It was the first time in the Olympiad history

Mathematical methods in solutions of the problems presented at the Third International Students' Olympiad in Cryptography

The mathematical problems, presented at the Third International Students’ Olympiad in Cryptography NSUCRYPTO’2016, and their solutions are considered. They are related to the construction of algebraic immune vectorial Boolean functions and big Fermat numbers, the secrete sharing schemes and pseudorandom binary sequences, biometric cryptosystems and the blockchain technology, etc. Two open problems in mathematical cryptography are also discussed and a solution for one of them proposed by a participant during the Olympiad is described. It was the first time in the Olympiad history. The problem is the following: construct F : ^ with maximum possible component algebraic immunity 3 or prove that it does not exist. Alexey Udovenko from University of Luxembourg has found such a function