The mathematical apparatus of the multidimensional discrete Fourier transform over finite fields is considered. Methods for the description of linear block codes in the frequency domain are investigated. It is shown that, in contrast to iterative codes (code-products), cascade codes in the general case cannot be described in the frequency domain in terms of multidimensional spectra. Analytic expressions are obtained that establish a one-to-one functional correspondence between the spectrum of a sequence over a finite field and the spectra of the corresponding words obtained by limiting this word to a subfield. A general solution of the problem of representation of cascade codes in the frequency domain is obtained, which allows constructing in the frequency domain using computationally efficient algorithms of encoding and decoding, and the derived analytic dependences of components of multidimensional spectra.

The code-based schemes, which were submitted to the contest of post-quantum crypto algorithms NIST PQC, are studied in this work. The general characteristics of the algorithms are explored and basic properties and parameters are estimated. A comparative analysis of the electronic digital signature schemes, public-key cryptosystems and key encapsulation schemes are carried out according to the criteria of speed and length of the main cryptographic parameters.

"Provable" strength generators of pseudo-random sequences have been considered in this paper, whose cryptanalysis problem reduces to solving a well-known and extremely complex mathematical problem related to the NP-complex class. In particular, the generators Blum-Blum-Shub, Rivest-Shamir-Adleman, Dual Elliptic Curve Deterministic Random Bit Generator and Pseudo-Random Generator Provably as Secure as Syndrome Decoding are considered. The periodic properties of generated pseudorandom sequences are investigated. It is shown that the considered generators do not allow forming sequences of the maximum period. In addition, for each generator there are initial states (weak keys), leading to catastrophically small lengths of the periods of generated sequences.

The methods for computing of Algebraic Immunity of cryptographic Boolean functions and non-linear substitutions (substitutions) blocks of symmetric ciphers are explored. The results of a comparative analysis of the Algebraic Immunity of nonlinear blocks of symmetric ciphers are denoted.

Linear block noise-proof codes constructed according to algebraic curves (algebraic geometric codes) are considered, their design properties are evaluated, algorithms of construction and decoding are studied. The energy efficiency of the transmission of discrete messages by M-ary orthogonal signals in the application of algebraic geometric codes is studied; the achievable energy gain from the use of noise-immune coding is estimated. It is shown that in discrete channels without memory it is possible to obtain a significant energy gain, which increases with the transition to long algebraic geometric codes constructed by curves with a large number of points with respect to the genus of the curve. It is established that the computational complexity of implementing algebraic geometric codes is comparable to other known noise-resistant codes, for example, the Reed-Solomon codes and others. Thus, high energy efficiency in combination with acceptable computational complexity of implementation confirm the prospects of algebraic geometric codes using in modern telecommunication systems and networks to improve the noise immunity of data transmission channels.

Nonlinear substitutions (S-boxes) are an important component of modern symmetric cryptography algorithms. They complicate symmetric transformations and introduce nonlinearity in the input-output relationship, which provides resistance to some cryptanalysis techniques. S-boxes generation can be done in a number of ways. However, the most promising are heuristic techniques. On the one hand, the generated S-boxes look like random settings, which complicates algebraic cryptanalysis. On the other hand, heuristic search makes it possible to achieve high nonlinearity and δ-uniformity, which complicates linear and differential cryptanalysis. In this article, we will explore the simplest local search algorithm for generating S-boxes. To assess the efficiency of the algorithm, the concept of a track of a cost function is introduced in the article. We optimize the algorithm parameters and show that with the chosen parameters it is possible to reliably form S-boxes with nonlinearity 104.