In Tomsk University of Control Systems and Radioelectronics (TUSUR) one of the main areas of research is information security. The work is carried out by a scientific group under the guidance of Professor Shelupanov. One of the directions is the development of a comprehensive approach to assessing the security of the information systems. This direction includes the construction of an information security threats model and a protection system model, which allow to compile a complete list of threats and methods of protection against them. The main directions of information security tools development are dynamic methods of biometrics, methods for generating prime numbers for data encryption, steganography, methods and means of data protection in Internet of Things (IoT) systems. The article presents the main results of research in the listed areas of information security. The resultant properties in symmetric cryptography are based on the properties of the power of the generating functions. The authors have obtained symmetric principles for the development of primality testing algorithms, as discussed in the Appendix.

In this paper, we study the problem of developing new combinatorial generation algorithms. The main purpose of our research is to derive and improve general methods for developing combinatorial generation algorithms. We present basic general methods for solving this task and consider one of these methods, which is based on AND/OR trees. This method is extended by using the mathematical apparatus of the theory of generating functions since it is one of the basic approaches in combinatorics (we propose to use the method of compositae for obtaining explicit expression of the coefficients of generating functions). As a result, we also apply this method and develop new ranking and unranking algorithms for the following combinatorial sets: permutations, permutations with ascents, combinations, Dyck paths with return steps, labeled Dyck paths with ascents on return steps. For each of them, we construct an AND/OR tree structure, find a bijection between the elements of the combinatorial set and the set of variants of the AND/OR tree, and develop algorithms for ranking and unranking the variants of the AND/OR tree.

This article deals with the enumeration of directed lattice walks on the integers with any finite set of steps, starting at a given altitude j and ending at a given altitude k, with additional constraints, for example, to never attain altitude 0 in-between. We first discuss the case of walks on the integers with steps \(-h, \dots , -1, +1, \dots , +h\). The case \(h=1\) is equivalent to the classical Dyck paths, for which many ways of getting explicit formulas involving Catalan-like numbers are known. The case \(h=2\) corresponds to “basketball” walks, which we treat in full detail. Then, we move on to the more general case of walks with any finite set of steps, also allowing some weights/probabilities associated with each step. We show how a method of wide applicability, the so-called kernel method, leads to explicit formulas for the number of walks of length n, for any h, in terms of nested sums of binomials. We finally relate some special cases to other combinatorial problems, or to problems arising in queuing theory.