With the development of low-orbit satellite Internet systems (NSIS), issues of ensuring effective operation in conditions of intentional interference come to the fore. One of the solutions is related to the use of systems using both OFDM methods and generators implementing frequency hopping (HF). Obviously, the more complex the algorithm for selecting operating frequencies, the more efficient the operation of the microwave. In the article, it is proposed to use the SPN cipher "Grasshopper" as a generator for selecting operating frequencies. As a result, the CCF system will have a high resistance to calculating operating frequency numbers by electronic warfare systems. However, failures and failures may occur during the operation of the SSC. To prevent their consequences, it is proposed to implement an SPN cipher using polynomial modular codes of residue classes (PMCC). One of the transformations in the "Grasshopper" is a nonlinear transformation that performs the substitution operation. It is obvious that the creation of a new mathematical model for performing a nonlinear transformation using MCCS will ensure the operation of the SPN-cipher-based RF generator in conditions of failures and failures.

Keywords: low-orbit satellite Internet systems, the Grasshopper SPN cipher, nonlinear transformations, modular codes of residue classes, mathematical model, fault tolerance, frequency hopping, polynomial modular code of residue classes

Orthogonal Frequency Division Multiplexing –OFDM) multiplexing technology is quite promising in wireless communication systems. Simultaneous use of multiple subcarriers allows for a relatively high information transfer rate. The use of mathematical models of discrete wavelet transformations instead of the fast Fourier transform (hereinafter FFT), allows you to increase the speed of signal processing by using modular codes of residue classes (hereinafter MKV). At the same time, these codes can be used to increase the noise immunity of systems with OFDM. It is known that block turbo codes (hereinafter referred to as TC) are widely used to combat packets of errors that occur when transmitting signals over a communication channel. The article presents a developed method for constructing modular turbocodes based on a system of residual classes (hereinafter MTKSOC). Obviously, the use of MTCS entails changes in the structure of the system with OFDM. Therefore, the development of a method for constructing a modular turbo code of SOC and a structural model of an interference-resistant system with OFDM using MTXOC is an urgent task. The purpose of the article is to increase the level of noise immunity of systems with OFDM, using wavelet transformations implemented in MKV instead of FFT, through the use of modular turbo code SOC.

Keywords: modular codes of residue classes, residual class system, modular turbo code of residual class system, error correction algorithm, structural model, multiplexing, orthogonal frequency division of channels

One of the directions that makes it possible to increase the efficiency of low-orbit satellite Internet in conditions of destructive influences is the use of OFDM systems that support the frequency hopping mode. It is obvious that the effectiveness of countering interference generated by electronic warfare (EW) is largely determined by the algorithm for selecting operating frequencies. In this paper, it is proposed to implement a block of SSF based on the SPN cipher "Grasshopper", which provides high resistance to the selection of the operating frequency by the SREB. However, in the event of failures and failures in the operation of such a unit, the transmitter and receiver operating in the microwave mode will not be able to establish information transmission. To solve this problem, the article proposes to use polynomial modular residue class codes (PMCCS). However, the analysis of the well-known error correction algorithms in PMCS has shown that they cannot be used to increase the reliability of the SPN-based CCF unit.

Keywords: Keywords: OFDM systems supporting frequency hopping, pseudorandom number generation methods, Grasshopper SPN cipher, polynomial modular residue class codes, error correction algorithm

The prospects for building a satellite Internet based on a low-orbit constellation of satellites (LOСS) are due to the fact that only it allows for stable and reliable communications anywhere on the planet. Therefore, the technology of satellite industrial Internet of things IIoT has become widespread in the oil and gas industry, many of whose facilities are located in the Far North. However, when using satellite low-orbit Internet (SLOEI), new threats and attacks arise against it. Among attacks on LOCS, a special place is occupied by attempts to impose unauthorized content on SLOEI subscribers. This situation can be prevented using a satellite identification system. For the effective operation of the “friend or foe” challenge-response system for LOCS, a copy-resistant authentication protocol with zero knowledge disclosure was developed. This property was achieved by reducing authentication time through the use of modular deduction class codes (MDCC). The use of parallel VDCC codes has led to a change in the method of satellite identification, which entails a revision of the principles for constructing a request-response system. Therefore, the development of a block diagram of a satellite identification system operating in the MDCC for a low-orbit satellite Internet system is relevant.

Keywords: imitation resistance, zero-knowledge authentication protocol, modular codes of residue classes, block diagram of the low-orbit satellite identification system

Frequency multiplexing (OFDM) methods have become the main basis for most outbred systems. These methods have also found application in modern systems of low-orbit satellite Internet (LOSIS). For example, the StarLink system uses OFDM transmission systems that use a signal frame consisting of 52 channels to transmit data. One way to increase the data rate in OFDM is to replace the Fourier transform (FT) with a faster orthogonal transform. As such, the modified wavelet transform (MWT) of Haar was chosen. The Haar MVP allows to reduce the number of arithmetic operations during the orthogonal signal transformation in comparison with the PF. The use of integer algebraic systems, such as Galois fields and modular residue class codes (MCCR), makes it possible to increase the speed of a computing device that performs orthogonal transformations of signals. Obviously, the transition to new algebraic systems should lead to changes in the structure of OFDM systems. Therefore, the development of structural models of an OFDM transmission system using the Haar MWP in the Galois field and the ICCM is an urgent task. Therefore, the aim of the work is to develop structural models of wireless OFDM systems using a modified integer discrete Haar transform, which can reduce the execution time of the orthogonal signal transformation. And this, in turn, will lead to an increase in the data transfer rate in the SNSI.

Keywords: orthogonal frequency multiplexing, modification of the Haar wavelet transform, structural models of execution of the Haar MVP, Galois field, modular residue class codes

This article explores the LTE-R group, in which the OFDM system has a special place, and considers the possibility of developing new methods for detecting error-correcting coding to test high data transmission under rate estimation conditions. The problems associated with the transmission of large amounts of information in conditions of high speed of movement of trains and the volatility of the environment are considered, as well as the features of interference associated with the railway infrastructure. As noise-immune codes, modular codes are used, which, unlike criminal BCH codes, are arithmetic.

Keywords: LTE-R standard, OFDM system, modular codes, noise immunity, error hit interval, BCH codes, error packet, error rate

To improve the efficiency of low-orbit satellite Internet systems (LOSIS) and ensure high apeed of data exchange, systems supporting orthogonal frequency-division multiplexing (OFDM) technology are widely used. The use of OFDM methods makes it possible to expand the bandwidth of the radio channel based on the increase in spectral efficiency. In order to do this, OFDM systems use orthogonal signal transformation based on fast Fourier transformation (FFT). To reduce the time spent on signal processing, the article proposes to use discrete wavelet transformations of signals (DWT) implemented in modular residue classes codes (MRCC). The scientific novelty of the work lies in the fact that based on the integration of methods for constructing discrete wavelet transforms and methods for developing modular codes, mathematical models of DWT implemented in the MRCC will be created, the use of which will reduce the execution time of orthogonal signal processing in OFDM systems. This result will be achieved by parallelizing arithmetic operations on the modules of the modular code. At the same time, the use of low-bit operands makes it possible to switch from performing multiplication, addition, and subtraction operations in the MRCC to retrieving data from LUT tables. Thus, the development of mathematical models of discrete wavelet transformations implemented in modular codes of residue classes is an urgent task.

Keywords: OFDM systems, discrete wavelet transformation, parallel coding, modular residue classes code

As the number of low-orbit satellite constellations (LEOs) increases, the probability of destructive impact from alien spacecraft (SC) will increase. One of the most effective impacts on the NHS is the setting of relay interference, which is an intercepted and delayed signal. This problem can be eliminated by using the "friend or foe" identification system for the spacecraft. At the same time, in order to reduce the probability of selecting the correct transponder signal by a foreign spacecraft, it is proposed to reduce the time spent on calculating the status of the satellite through the use of parallel calculations using codes of the polynomial residue number system (PRNS). A characteristic feature of these codes is the independent and parallel execution of calculations based on the bases of the PRNS. However, this property of the PRNS code can also be used to correct errors that occur during the operation of the identification system. In this case, the algorithm must perform this procedure at a lower time cost. Therefore, the modification of the error correction algorithm, which makes it possible to provide a higher speed of searching and correcting errors, is an urgent task. The purpose of the work is to reduce the time spent on the execution of the procedure for detecting and correcting errors during the operation of the identification system, by modifying the correction algorithm based on the Chinese residual theorem

Keywords: satellite identification system, codes of the polynomial residue number system, positional characteristic, error detection and correction algorithms

Low-orbit satellite communication systems (LowSCS) are successfully used to organize stable communication in Northern latitudes. With the help of theLowSCS, effective management and monitoring of the process of production and transportation of hydrocarbon raw materials is provided. This approach allows us to ensure minimal costs for the extraction and delivery of oil and gas from fields located on the shelf of the Arctic Ocean. As the number of countries engaged in the development of Arctic Ocean deposits increases, so does the number of LowSCS groupings. To prevent the possibility of interception and imposition of a delayed command to control the intruder satellite, it is necessary to increase the information secrecy of the LowSCS with the help of "friend-foe" identification systems for the spacecraft. At the same time, to ensure high imitability in such systems, it is proposed to use zero-knowledge authentication protocols. To increase their efficiency, the article suggests using the codes of residual number system (RNS). The novelty of this idea is that the use of parallel SOC codes will reduce the time spent on performing arithmetic operations implemented in authentication protocols. And this, in turn, will lead to an increase in the information secrecy of the NSSS, so the probability of selecting the correct response signal by the intruder satellite decreases. Therefore, the purpose of the article is to develop authentication protocols for low-orbit spacecraft based on parallel codes of residual class systems, the use of which will reduce the time for satellite identification.

Keywords: satellite identification system, zero-knowledge authentication protocols, residue number system

One of the effective ways to increase the information secrecy of a constellation of low-orbit spacecraft (LVS) is the use of a satellite authentication system (SAS). In order to reduce the time spent on determining the status of the satellite, as well as to increase the fault tolerance of the identification system, in a number of works it is proposed to use the codes of the polynomial system of residue classes (PSCS). This paper presents a methodology for constructing a fault-tolerant authentication system for a spacecraft, which is based on exchange operations implemented in PSKV codes. Using exchange operations, that is, changing the number of informational and redundant bases, it is possible to ensure the possibility of maintaining the operational state of the identification system in the event of a sequence of failures, as a result of which the level of information secrecy of the NSA will not be reduced.

Keywords: satellite authentication system, polynomial system of residue classes, exchange operations, method of building a fault-tolerant identification system

The use of spacecraft identification systems can increase the informational secrecy of low-orbit spacecraft constellations. However, existing methods of recognizing "friend or foe" do not provide high cryptographic strength, and authentication protocols with zero disclosure are time-consuming. This drawback can be eliminated through the use of Polynomial Residue Number Systems (PRNS) that allows parallel computing in the protocol. Therefore, the development of a high-speed authentication protocol with zero disclosure is an urgent task. The purpose of the work is to reduce the time required to calculate the status of the satellite through the use of PRNS

Keywords: authentication method, spacecraft authentication protocol, Polynomial Residue Number Systems

The paper presents the basic differential characteristics identified for the S-boxes of GOST 28147-89 algorithm, included in the data encryption standard GOST 34.12-2015, which comes into force from January 1, 2016. The detected properties can be used for analysis with such methods as differential cryptanalysis and impossible differential cryptanalysis.

Keywords: encryption algorithm, symmetric encryption, block cipher, S-boxes, differential cryptanalysis, impossible differential cryptanalysis, differential properties

`

Keywords: