DEVISING METHODS TO SYNTHESIZE DISCRETE COMPLEX SIGNALS WITH REQUIRED PROPERTIES FOR APPLICATION IN MODERN INFORMATION AND COMMUNICATION SYSTEMS

Global trends in growing threats to information and cybersecurity determine the need to devise and implement new models, methods, and technologies for managing telecommunication networks, information security, services, and quality of service. There is a need to build methods of information exchange, methods to synthesize new classes of complex discrete signals-data carriers with the required ensemble, correlation, and structural properties. Among the main directions of improving the indicators of information security, noise immunity, and secrecy of ICSs, there are areas that are associated with the use of channels with high-frequency redundancy, significant spatial, structural, energy, and time secrecy. To ensure frequency redundancy at the physical level, discrete signals have been widely used, in which the manipulated parameters change at strictly fixed intervals. The law for changing these parameters is set by discrete sequences that fully determine the properties of discrete signals. The researchers’ efforts are aimed at finding signal ensembles whose characteristics are approaching the boundary of “dense packaging” [1]. Such signals have an ideal periodic auto-correlation function (PACF) and a periodic mutual correlation function (PMCF), and have a significant volume. A widespread criterion for such an approximation is a minimax criterion that focuses on the synthesis of the enDEVISING METHODS TO SYNTHESIZE DISCRETE COMPLEX SIGNALS WITH REQUIRED PROPERTIES FOR APPLICATION IN MODERN INFORMATION AND COMMUNICATION SYSTEMS

the principles of building telecommunication systems and selecting signals that could be applied to them. That makes it possible to make design decisions on the construction of such systems, to evaluate the performance indicators of such systems, primarily those that are determined by the properties of the signals used in the systems. At the same time, there is no information about the possibility of practical implementation of methods of synthesis of signal systems. Study [4] addresses purely theoretical issues of optimal reception and processing of signals. The approach used in the cited study is focused on the application of information and communication signals, which are based only on the linear laws of construction. This, in turn, does not make it possible to resolve the issues inherent in modern systems for which it is critical to ensure noise immunity and information security. Paper [5] describes the principles of building modern wireless digital communication systems and gives practical recommendations for the use of types of signals (OFDM), methods of encoding, synchronization, data transmission. It has been shown that the introduction of information exchange methods and certain classes of signals could contribute to resolving issues of ensuring information and frequency efficiency, noise immunity of signal reception, the speed of data transmission. At the same time, no solutions are given to solve the problems of ensuring information security (cryptographic stability, imitation protection) at the physical level of a telecommunication network. Work [6] provides an in-depth analysis of the classification and methods of synthesis of a large number of signals. However, there are no specific solutions for the possible practical implementation of these methods and existing restrictions when applying certain signals for many applications of communication systems. Free from these restrictions is work [7]. It shows the possibility of solving the issues of ensuring information security, noise immunity at the level of the source of complex signals. It is shown that the basis for the construction of signals should be nonlinear rules, and data exchange in the system should be based on a dynamic change of conformity: messagecomplex signal. However, the cited work does not provide practical methods for constructing such signals. Paper [8] describes approaches to determining the requirements for the properties of signals, the use of which in information systems would make it possible to provide the required indicators of protection against influences from intruders. However, the cited paper does not contain a detailed description of the properties of the signals offered and does not give an assessment of the performance indicators of the systems. Study [9] reports the description and characteristics of the block symmetric encryption standard. The encryption algorithm defined by this standard is used in the implementation of the proposed method of synthesis of complex nonlinear discrete cryptographic signals as a source of pseudo-random process. Paper [10] describes the requirements for the initial sequences of cryptographic information conversion algorithms. It is these requirements that are used to formulae requirements in the synthesis of nonlinear signal systems. At the same time, the cited work does not provide methods for the synthesis of signals as a physical data carrier in communication systems. One of the signal classes offered in the work must have properties that are inherent in random sequences of characters. It is [11] that defines the criteria (requirements) for such sequences. semble by minimizing the maximum values of side peaks on a set of all unwanted correlations. Works [1,2] describe the boundaries for the rms and maximum (peak) values for autoand mutually-correlation functions. Specifically, the fundamentally achievable values of the maximum side peaks of the periodic autocorrelation function max a R ("dense packaging" limits) for the assigned period of the sequence N are determined from the following ratio: The above limit values establish the criteria for the synthesis of sequence sets (signatures). Ensembles with values corresponding to the specified boundary are optimal and are termed minimax. For an ideal hypothetical ensemble, R max is zero, and for any real ensemble, the minimum value of the correlation function can serve as an adequate measure of its proximity to the ideal.
The signals-physical data carriers used in ICSs have low structural secrecy and unsatisfactory ensemble characteristics. That does not make it possible to provide the required (especially for critical infrastructure objects) indicators of information security and noise immunity of data transmission within ICS. That is why improving the indicators of noise immunity and information security of ICSs based on the development of theoretical bases, methods, and tools for synthesizing new classes of optimal complex signals with the required ensemble, correlation, and structural properties is an actual task.

Literature review and problem statement
Study [1] gives a general description and analysis of the properties of different classes of signals, with the introduced indicators for the effectiveness of the functioning of multiuser communication systems. That makes it possible to comprehensively approach the issues of using certain signal systems to solve the tasks of ensuring the relevant indicators of the effectiveness of the functioning of such systems. Nevertheless, the cited study does not define how the problems of ensuring the necessary indicators of information security and secrecy of the functioning of systems could be solved. The reason for the inability to provide for the necessary (especially for ICSs operating at critical infrastructure facilities) indicators is the classes of broadband signals that are reported in this work. The evolution of views on ensuring the required indicators of the functioning of data transmission systems is described in work [2]. The work presents theoretical bases for the synthesis of a large number of signal systems, including characteristic signals. That makes it possible, when designing data transmission systems, to reasonably make decisions on the use of certain classes of signals as physical data carriers. Despite the obviously correct decision to use the nonlinear signal classes, the cited work does not provide a description of methods that would make it possible to algorithmize the processes of synthesis of signal systems to build modern software and hardware complexes for synthesizing, forming, and processing signals. Paper [3] reports When synthesizing the signals studied in the cited work, it is necessary to use algorithms for block data encryption, as well as take into consideration the requirements for the properties of random sequences of characters. The rules and requirements for such sequences (so-called random lookups) are given in paper [12]. However, this paper does not provide principles, limitations, practical methods for the synthesis of signal systems.
The systematization of the results of the above studies allows us to believe that existing approaches to the use of signals with a linear law of formation do not make it possible to provide the required indicators of noise immunity and information security. This issue can be resolved by devising theoretical bases, practical methods, and software and technical tools to synthesize, form, and process the systems of nonlinear discrete complex signals with the required properties.

The aim and objectives of the study
The purpose of this work is to synthesize, based on the devised methods, discrete complex signals with the improved ensemble, correlation, structural properties, which could improve the performance indicators of ICS operation, namely, the performance of signal synthesis, information security, noise immunity (noise immunity of signal reception and secrecy of functioning) of the system.
To accomplish the aim, the following tasks have been set: -to determine the mathematical dependence of elements and indexes of elements of the simple and extended Galois fields to devise a method of synthesis of complex signals; -to build models of the structure of complex nonlinear discrete signals in the finite fields and to investigate the structural secrecy, correlation, and ensemble properties of nonlinear discrete characteristic signals for estimating the indicators of noise immunity and information security of ICSs; -to define conditions for the functions of signal correlation, which are assigned by a set of systems of nonlinear parametric inequalitieş and which are used to devise a method for the synthesis of discrete complex cryptographic signals; -to investigate the properties of the synthesized classes of nonlinear discrete complex cryptographic signals for use in ICSs as a physical carrier of information.

The study materials and methods
The essence of our research hypothesis is as follows. Is it possible to devise methods for synthesizing the systems of complex discrete signals with the required (predefined) correlation, ensemble, structural properties, the use of which in modern ICSs could improve the performance indicators of such ICSs?
The following assumptions apply to a given hypothesis: -methods of signal synthesis should be based on nonlinear rules; -there should be an opportunity to synthesize signals for any value of the signal period (the natural series of numbers); -signals must have high structural secrecy (to define the rule of its synthesis, it is necessary to know at least half of the signal symbols); -there should be a possibility of software and hardware implementation of signal synthesis methods.
Modeling is used as one of the forms of scientific research. The experimental component involves the use of the designed software complex, which implements the functions of signal synthesis, in accordance with the methods proposed in this work (given as a sequence of actions described mathematically, and which make it possible to achieve a certain result), as well as the study into the properties of signals obtained as a result of the synthesis (hereinafter, Complex). The list of main uses of the developed Complex includes: -the formation of discrete complex signals (CSs); -the calculation of correlation functions under the periodic and aperiodic modes of operation; -the calculation of statistical characteristics (mathematical expectation, variance, rms deviation, excess coefficient) of signal correlation functions under the periodic and aperiodic modes; -the calculation of the largest and smallest values for the side peaks of correlation functions, their number, and the comparison of their values with the optimal boundary for the corresponding correlation function; -the synthesis of discrete complex signal systems using the decimation process; -finding the parameters used in the synthesis of signals (the primary element of the field, Euler functions, mutually simple numbers with some given, etc.). Scientific research at the theoretical level involved the following: -the generalization of positions from the theory of groups, fields, rings, in order to employ this mathematical apparatus to devise methods for synthesizing the systems of complex signals in finite fields, approaches from the theory of cryptographic protection of information regarding the construction of block symmetric ciphers in the development of methods for synthesizing complex discrete cryptographic signals; -the comparison of known provisions from the theory of signal systems with the results of our research regarding the synthesis of signal systems and signal properties; -the mathematical formalization of the representation of the structure of an object to determine the steps of implementation of the devised methods of signal synthesis, which makes it possible to reproduce these methods and obtain the investigated signals.

Results of studying the methods of synthesis and
properties of nonlinear discrete complex signals

1. Determining the mathematical dependence of elements and indexes of elements of simple and extended Galois fields to build a method of signal synthesis
A series of requirements are put forward to discrete signals: good correlation properties, uniform spectrum, the permissible level of maximum peaks of the auto-and mutually correlated functions, large volume, the existence of duration values for a large number. We have considered N-positional codes (characteristic discrete signals, hereinafter referred to as CDS) with a two-level periodic auto-correlation function (PACF), the construction of which is based on the use of character ψ of the multiplicative group [2] of the field GF(p n ) for 4 2 1 n N x p = + = − and 4 1.
n N x p = = − It is shown in [2] that the volume of a signal system (M) is equal to the number of classes of non-inverse-isomorphic coefficients, which can be obtained by decomposing a multiplicative group into adjacent classes according to the class of automorphic coefficients; determined as M=Ψ(N)/2, where Ψ(N) is the Euler function. It is also known from [2] that the values of the maximum side peaks of the CDS periodic autocorrelation function take the values R µ ={-2, 2}, or R µ ={0, -4}. Analysis shows that CDSs exist for much higher duration values than M-sequences [1,2]. At the same time, CDSs, as well as M-sequences, are optimal in terms of PACF and are close to optimal in terms of the aperiodic autocorrelation function (AACF).
The method to form CDS [2] with a duration of N, which employs the table of elements and indexes of Galois field elements compiled in the theory of numbers, becomes difficult to implement already at n≥1 and N≥100. This is explained primarily by the fact that at the homomorphic mapping of the elements of field a i onto a set of symbols of discrete sequence when using the complex-significant function ψ(a i )=W i =-e jπUi , it is necessary to solve on average N/2 equations of the following form Θ is the j-th primary element of the field; P is the Galois field characteristic.
To solve the equations in form (1), the pre-calculated tables of elements and indexes of Galois field elements are used. The computational complexity of this method of CDS formation is determined from the ratio: where m t , ad t , w t , r t , com t is the time of multiplication, addition, writing, reading, and comparison operations, respectively.
Our analysis of expression (2) shows that the main time costs in the construction of CDS are related to the terms com N N t ⋅ + ⋅ During our study, an improved method of CDS synthesis was devised, which demonstrates a much lower computational complexity compared to the methods considered in [2]. CDS synthesis is based on the use of the smallest (by value) primary element of the field GF(P) and is assigned by statement 1. Statement 1. Let the character of the multiplicative group of the field be fixed with the following function ( ) Then, the mathematical dependence of elements and indexes of elements of a simple Galois field can be described in the following steps.
An array of elements is formed -the numbers A i , 0, 2 i P = − of the GF(P) field: A group of numbers of the field GF(P) is formed, which is shifted by values per unity, according to the following rule: An array of indexes X(i), 0, 2, i P = − is formed, the values of which are the corresponding elements of the field indexes i+1, ordered by the content with the address: An array of J(i) indexes is built, the values of which are the indexes of the array X(i), selected at the address H(i); The nature of the field elements is calculated by the rule given in [2]: Let φ(a i ), 1, 1 n i р= − be a Galois field GF(p n ) of power n expansion and the polynomial elements whose power does not exceed n, calculated above the field, GF(p n ), Ф k (x) and θ jrespectively, the k-th primary primitive polynomial and the j-th primary element of the field. The function of the characters of the homomorphic mapping of the elements of the field GF(p n ) onto the field GF(2), fixed by the function ψ(a i )= e jπui , and the polynomial element of the field a i is determined from the solution to equation and u i is the set of index numbers arranged in ascending order.
Find the analytical dependence of elements and indexes of elements for the case of an extended Galois field.
1) An array is formed of the indexes 1, − shifted by value, arranged in ascending order, and an array of polynomial elements a i of the field GF(p n ): 2) An array of H(i) polynomial elements of the field GF(p n ) is formed, shifted by values per unity relative to the values of the array A(i): 3) An array of indexes u i is written to the array X(i) at the addresses defined by the values of coefficients at the polynomial H(i) in the P calculus system. 4) An array of J(i), = − 1, 2, n i p indexes is formed, the values of which are the u i indexes, which are read from the array at the addresses that are set by the values of coefficients at the elements-polynomials H(i) in the P calculus system. 5) A two-character symbol is calculated for all values of the array of indexes J(i) As an example, Table 1 gives CDSs with the number of elements 256. The specified signal was obtained by implementing steps (4) to (7) from the devised method of signal synthesis. The Table 1 also shows the values of the maximum side peaks of the periodic autocorrelation function (PACF) for this signal.
Our analysis of data from Table 1 confirms that a given signal class accepts the values of PACF side peaks [2]. Numerical calculations involving computer simulation of expressions (4) to (7) for signals with a length from 16 to 4,000 elements demonstrated that these steps determine the method for synthesizing nonlinear discrete complex signals in finite fields.
Expression (11) takes into consideration that the time required to perform operations of writing t w and reading t r is equal. Our analysis of (11) shows that the time of CDS construction using the proposed method linearly depends on the period of the sequence that is formed, whereas for a known method (expression (2)) the dependence is quadratic.
The gain in the time of CDS synthesis using the devised methods, compared to the known method, is: for the period N=256, 25.5 times; for the period N=9,972, 1,039 times.
We have devised a method for constructing characteristic signals, which demonstrates a lower computational complexity compared to the method based on the use of difference sets [2]. The specified method for constructing the CDS isomorphism system is set by the following statement. Statement 2. If the sequence {W i }, 1, , i N = which is one of the CDS isomorphisms with the number of elements (period) N is subjected to a decimation operation with the decimation coefficient C, where C is a number that is mutually simple with N (C∊j(N)), then the resulting sequence {v i } is the isomorphism relative to the sequence {w i }.
The decimation procedure means selecting each C-th character of the code and recording the characters obtained in this way so that   To form the isomorphism of a characteristic code, it is necessary to implement (in accordance with (13)) N operations over sampling elements at the appropriate decimation coefficient and N addition operations. The time of execution of these operations is denoted, respectively, t d and t r. Then the computational complexity of the decimation method can be evaluated using the following expression: Our analysis of operations performed during the implementation of the method of difference sets shows that the computational complexity of the latter T ds is determined from the following ratio: where t r , t com , t m -the time of execution of reading, comparison, multiplication operations, respectively. The results of computer simulation and the calculations carried out using (14), (15) show that the proposed method significantly exceeds, in terms of performance, the known method of difference sets. Thus, for the signal period of 2,380 elements, the win in time exceeds 28 times.

2. Building a model of the structure of complex nonlinear discrete signals in finite fields and investigating signal properties
It is known that the probability of imposing false messages (signals) by the enemy is determined, among others, by the ensemble properties of the signals used (system volume, the spectrum of values of the signal period for which signals can be synthesized). Table 2 gives the generalized data on the number of signal length and signal system volume values for M-sequences and characteristic discrete signals. Our analysis of the above analytical ratios, as well as data in Table 2, indicates that CDSs are better compared to the widely used M-sequences, Legendre sequences, and others.
It is known that the secrecy of ICS functioning largely depends on the complexity of determining by the station the counteraction to the law of discrete signal modulation (the structural secrecy of the signal system used). We shall evaluate the structural secrecy of nonlinear discrete characteristic signals. To this end, we shall state and prove the statements that determine the relationships between the elements of a finite field. Statement 3. Let We shall illustrate by an example the possibility of constructing ((P-1)/2+i) field elements, provided that the first (P-1)/2 elements are known.
Let the characteristic of the field be P=13, the primary element of the field is Θ=2.
Record the elements of a given field: The comparison of the corresponding elements of the field given in (17) with the field elements (18) shows that these elements are identical. Given the specified property of a Galois field, the nature of the elements of the field or the CDS symbols that are built in the field are likely dependent. This dependence is determined by statement 4. Statement 4. Let the character of the elements ψ(a i ) of the field (CDS characters in the field GF(p n ) be determined from the following ratio and the U i field element indexes are found by solving the following equation We shall illustrate that Statement 4 holds using an example.
Let the characteristic of the field GF(p n )) be P=13, and the primary element of the field be Θ=2. The CDS isomorphism in this field is The result is the same if one applies (4) to establish the CDS symbol dependence. Using statement 3 makes it possible to determine ((P-1)/2+i) CDS symbols ( ) at known first (P-1)/2 symbols. In this case, only the first and ((P-1)/2+i)-th CDS symbols are not defined, but the ((P-1)/2+i)-th CDS symbol is determined by the encoding rule [2]. For CDS, the number of symbols K accepting "1", is K=N/2. This means that the first CDS symbol can be determined if P−2 signal symbols are known. It is easy to verify that the statements also hold for the case of the extended Galois field GF(p n ), that is, for the case when n>1.
The relationships among the elements and characters of field elements identified and described in statements 3 and 4 make it possible to increase by at least twice the speed of CDS formation devices.
It is known [2] that the statistical characteristics of correlation functions include the mathematical expectation of outliers (m u ); the mathematical expectation of the modules Using the developed specialized software, we studied the correlation properties of complex nonlinear discrete signals in finite fields. Table 3 summarizes the statistical characteristics of different correlation functions of the most widely used discrete sequences. Our analysis of data in Table 3 indicates that the statistical characteristics of the correlation functions of CDSs are not inferior to the similar characteristics of other signals given in this Table.

3. Defining conditions for signal correlation functions assigned by a set of systems of nonlinear parametric inequalities
Based on the study of the algebraic structure of the systems of nonlinear parametric inequalities (SNPE), we have stated and solved in a general form the problem of the synthesis of a new class of complex nonlinear discrete signalscryptographic signals (CSs). CSs should be understood as a set of sequences (vectors) of characters of a certain alphabet, which have the required (specified) structural, ensemble, and correlation properties.
The synthesis of such signals is based on the use of random or pseudo-random processes, including algorithms for cryptographic information transformation [7][8][9].
A CS synthesis problem is to be understood as the problem of constructing the subsets of sequences ( ) , q l W 1, , q N = 1, , l L = whose totality forms a system of signals of the alphabet of dimensionality М k =N×L. Each subset meets the conditions for structural, ensemble, correlation properties, the spatial and time complexity of signal generation. The synthesis of CSs is based on the use and analysis of the periodic and aperiodic correlation functions.
The mathematical model of the process of synthesis of this signal class takes the form given below.
1. Ensuring the conditions for meeting the requirements for the structural and ensemble properties, the possibilities to form a subset of CSs with permissible time and spatial complexity, including using keys.
2. The construction of CSs W q , whose periodic auto-correlation functions (PACF) satisfy the system of nonlinear parametric inequalities (SNPE): At 1=L for all 1, q N = (21) produces a convolution with the value of L 1 1 , 3. The construction of CDS pairs W q and W p , whose functions of mutual correlation (MCFs) meet the requirements determined by a set of SNPE (21), and also meet the requirements for the butt functions of mutual correlation (BFMC) of CDS pairs W q and W p with butt discrete words W qp and W pq :  We shall analyze the systems of nonlinear parametric square inequalities (hereinafter, systems) (21) to (27), using the model introduced.
Systems (24), (25) at 1=L for all 1, q N = must yield the full convolution with a value of L, that is, 1 1 , and (25) produces 1 1 , Systems (23), (25), and (27) at 1=L for all pairs W q and W p give the value of the mutual correlation function at a zero offset value in the following form, respectively: We shall analyze systems (21), (22) for the existence of solutions and independence. Directly from (21), we obtain that for each q of CS W q there are L unknowns 1 2 , ,..., .
q q q L W W W To find them, according to (21), one can build a system of L−1 independent SNPE. Further, using (22), we obtain another expression but an equation, in this case. The peculiarity of system (21) is that this system produces a convolution of each q of CSs with a value L. Based on (21), (22), when constructing each of the N subset of CS, one can build N independent SNPE, each of which would contain L-1 quadratic inequalities in the form of (21) and a formally one equation, so that all of them will equal L.
Based on a detailed analysis, we have that all (23) to (27) SNPEs define different implementations of PMCF and BFMC of only two CSs -W q and W p . Therefore, the mathematical model of building two CS W q and W p is uniquely defined by five SNPEs in the form of (23) to (27), and, as has already been substantiated, by equation (30).
The above results of our analysis make it possible to determine the complexity of the model and, on its basis, the complexity of building a subset of N CS. When building one CS, it is necessary, depending on the permissible values Thus, taking into consideration the boundaries of the physical packing of the CS subset [4], there are opportunities to build the subsets of CS according to (22)  Taking into consideration the need to ensure cryptographic stability and structural secrecy of pairs or subsets of CS, algorithms of block symmetric transformation or other sources of random or pseudo-random sequences can be used as a source of discrete sequences.
Taking formulas (21) to (27) into account, we defined the steps for the method of the synthesis of discrete complex cryptographic signals: 1. Form random or pseudo-random discrete sequences. 2. Assess the statistical properties of potential CS. 3. Build the required number of potential CS W q according to system (21) and key data.
4. Find the pairs or subsets of CS W q and W p that meet requirements (23) to (27) using the method of "branch and bound" 5. Construct a matrix of states of mutual-correlation functions of all possible pairs of potential CS, which were selected based on the results of the previous step and have all the necessary properties.
6. Analyze the matrix of states and form the required number of subsets or pairs of CS according to (21), (23) to (27); select to a subset only those that meet the requirements.

4. Studying the properties of the synthesized classes of nonlinear discrete complex cryptographic signals
For a series of ICS applications, signals with high structural secrecy, the necessary correlation properties, and a significant volume (ensemble) are required. We shall evaluate the ensemble, correlation, and structural properties of a given signal system.
It should be noted that CS, unlike the known classes of signals used in various ICS applications, can be synthe-sized for any values of the period of discrete signals. The volume of the system of nonlinear CS is determined by the requirements for the system, in terms of such indicators of the effectiveness of the functioning of the ICS as the noise immunity of reception, secrecy, and information security of the system. Users (owners) of the system, based on these restrictions, need to make compromise decisions on the choice of an ensemble of nonlinear CS with the required properties. Table 4 gives data characterizing the correlation and ensemble properties of CS of different periods. Specifically, the period of CS; the limit values of side peaks of auto-correlation functions; and the number of signals that correspond to the limit values in the CS class; the smallest values of side peaks of different correlation functions, and their quantity.
Our analysis of the data given in Table 4 indicates that the values of maximum side outliers, the statistical characteristics of CS are not inferior to the corresponding characteristics of signals that are built using M-sequences. Thus, for the sequence period of 1,023 elements, the number of CS pairs, satisfying the limit value for the side petals of MCF of 100, is 5,293,538. For M-sequences, the number of pairs corresponding to this limit is 435, that is, the excess volume of the CS system is more than 10 4 times. Varying the boundary values of the level of the side petals of the corresponding correlation function, the task of achieving the necessary values of indicators of interference with the resistance of signal reception, interference and information security of the system can be solved.
To study the structural properties of CS, the procedures for testing the generators of random (pseudo-random) sequences, which are defined in [10,11], are used. The test results showed that the CSs meet the requirements for random sequences [12]: the unpredictability of symbols, irreversibility, randomness, equal probability, independence, etc. Thus, the structural properties of CS do not differ from the properties of random sequences.

Discussion of results of studying the methods of synthesis of discrete complex signals for use in modern information and communication systems
Our results indicate that the proposed methods to synthesize characteristic discrete signals (CDS) in finite fields can significantly increase the efficiency of the synthesis of these signals. Indeed, the winning time for the synthesis of nonlinear signals in finite fields using the devised method (statement 1), compared to the known method, for the CDS period of 256 elements, is 25.5 times, and for the period of 9,972-1,039.6 times. The method to synthesize the entire CDS system based on the decimation procedure (statement 2) was built. The results of computer simulation and the calculations carried out using (14), (15) show that the proposed method of synthesis based on a decimation operation surpasses a known method of difference sets in terms of performance. Thus, for the signal period of 2,380 elements, the win in time exceeds 28 times. The relationships among the elements and the characters of field elements identified and described in statements 3 and 4 make it possible to increase by at least twice the speed of CDS formation devices. Our study into the correlation properties of CDS has shown that the values of maximum side peaks, as well as statistical characteristics of various correlation functions (Table 1), are not inferior to similar characteristics of the noise immunity best-known linear signals. This, in turn, means that the indicators of noise immunity when receiving the signals offered will be no worse than when using known signal systems. At the same time, the structural properties of CDS, which are associated with the complexity of determining by the station of counteraction to the law of signal formation, have been significantly improved compared to known signal systems. That directly follows from statement 4. It should be noted that the use of these signals makes it possible to improve the indicators of information security, namely the probability of the enemy imposing false messages (signals). It is known that this indicator is determined, among other things, by the ensemble properties of the signals used (system volume, the spectrum of values of the signal period for which signals can be synthesized). Our analysis of the above analytical ratios, as well as data in Table 1, indicate that CDS, in terms of ensemble properties, are better compared to a series of sequences used, such as M-sequences, Legendre sequences, and others.
By studying the algebraic structure of the systems of nonlinear parametric inequalities, as well as by analyzing the periodic and aperiodic functions of correlation, we have built a mathematical model of the process of synthesis of discrete complex signals (21) to (27), as well as practical methods for synthesizing the nonlinear discrete complex cryptographic signals (hereinafter, CS). The synthesis of CS employs random or pseudo-random processes (including cryptographic algorithms), which makes it possible to create the sequences of symbols (signals) of a certain alphabet. Such sequences satisfy the requirements of irreversibility, randomness, unpredictability, and possess the required structural, ensemble, and correlation properties, as evidenced by the data given in Table 4. Table 4 Correlation and ensemble properties of CS