=Paper=
{{Paper
|id=Vol-3156/paper44
|storemode=property
|title=Mathematical Fundamentals of Structural and Entropic Analysis of Digital Data Flows
|pdfUrl=https://ceur-ws.org/Vol-3156/paper44.pdf
|volume=Vol-3156
|authors=Nataliia Vozna,Andriy Segin,Igor Pitukh,Artur Voronych,Luybov Nykolaychuk
|dblpUrl=https://dblp.org/rec/conf/intelitsis/VoznaSPVN22
}}
==Mathematical Fundamentals of Structural and Entropic Analysis of Digital Data Flows==
https://ceur-ws.org/Vol-3156/paper44.pdf
Mathematical Fundamentals of Structural And Entropic
Analysis of Digital Data Flows
Nataliia Voznaa, Andriy Segina, Ihor Pitukha, Artur Voronychb, Lyubov Nykolaychukc
a
West Ukrainian National University, 11, Lvivska Str.,Ternopil, 46009, Ukraine
b
Ivano-Frankivsk National Technical University of Oil and Gas, 15, Karpatska Str., Ivano-Frankivsk, 76019,
Ukraine
c
Nadvirna Vocational College by National Transport University, 177, Soborna str., Nadvirna, 78400, Ukraine
Abstract
Areas of entropy application and structural analysis for solving a wide range of information
problems in the field of states monitoring for control objects are identified. Mathematical
bases of existing algorithms for entropy estimation of stationary random processes are
presented. Criteria of structural complexity are systematized for microelectronic tools, which
allow to compare the system characteristics of different structures for operating devices and
specialized processors in the computer architecture. The most priority modern architectures
of interactive CPS i nterms of the emergence and parallelization coefficient for data flows are
defined. The principle of data encryption based on the entropy method of signals
manipulation method is proposed. On its basis, the priority structures of crypto protection of
data are offered. These structures are used for the reception and decoding of crypto-protected
entropy-manipulated signals. The proposed structures are characterized by the limit
characteristics of maximum speed and minimum time and structural complexity.
Keywords
Entropy, structures, Specialized processor for entropy estimation, cryptographic protection,
entropy-manipulated signals. 1
1. Introduction
In modern cyberphysical systems, the volume of digital data streams is growing significantly and
algorithms for their processing are being developed. One of the effective ways of data processing for a
wide range, such as digital data research, encoding and encrypting data, transmitting information, etc.
became entropic analysis. Hartley and Shannon formulas are most often used to estimate the entropy of
digitized processes [1-2]. However, entropy analysis needs further development in terms of improving the
theoretical foundations, practical implementation algorithms and specialized processors for their
calculation [3-5]. In addition, it is necessary to improve the criteria for determining the complexity of
cyberphysical systems using certain algorithms and technical means.
2. Justification of the relevance of entropy and structural analysis
There are a number of approaches and algorithms for entropy characteristics estimation of data flows.
Based on them, appropriate specialized processors have been developed to calculate entropy estimates [6].
IntelITSIS’2022: 3rd International Workshop on Intelligent Information Technologies and Systems of Information Security, March 23–25,
2022, Khmelnytskyi, Ukraine
EMAIL: nvozna@ukr.net (N. Vozna); andriy.segin@gmail.com (A. Segin); pirom75@ukr.net (I. Pitukh); archy.bear@gmail.com (A.
Voronych); lmnykolaychuk@gmail.com (L. Nykolaychuk)
ORCID: 0000-0002-8856-1720 (N. Vozna); 0000-0002-3556-248X (A. Segin); 0000-0002-3329-4901 (I. Pitukh); 0000-0003-0701-917X
(A. Voronych); 0000-0002-7733-4573 (L. Nykolaychuk)
©️ 2021 Copyright for this paper by its authors.
Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
CEUR Workshop Proceedings (CEUR-WS.org)
1
These specialized processors have different parameters of hardware complexity of their implementation,
the time of calculating the final result, the accuracy of entropy estimation and others.
1. The method of estimating entropy using a centred autocorrelation function, takes into account
statistical relationships between data. It is described by the expression as follows:
1
( )
m
1
I x ( R) = n E log 2 D x2 − R xx
2
( j ) , (1)
2 m j =1
where, n is the sample volume; E • is the integer function with rounding to a larger whole; j = 1, m are
shifting parameters of time counts; m is a number of autocorrelation reference points; Dx is a dispersion;
Rxx ( j ) - is an autocorrelation function.
The centred autocorrelation function Rxx ( j ) is bounded by asymptotics given by expressions (2) [6].
n
1
R xx ( j ) = xi x i − j ; xi = xi − M x ; j = 0, m. (2)
n i =1
2
1
n n
1
where, M x = xi is a selective mathematical expectation at intervals [1, n], Dx = xi is
n i =1 n i =1
a dispersion, the graph of which is shown in the Fig.1 [6].
The probability entropy function I x (R) , which is calculated on the basis of the autocorrelation
function Rxx ( j ) , is shown in Fig.2.
Figure 1: Asymptotes of autocorrelation function Rxx ( j )
I x (R)
Dx2
I x (R)
C 2j
mj
Figure 2: Estimate of correlation entropy I x (R) , where С 2 = D x2 − R xx
2 ( j) .
j
Based on the described approach of entropy estimation using the correlation function Rxx ( j ) , the
structure of the special processor is developed. This structure is shown in Fig. 3 [6].
The specialized entropy estimation processor shown in Figure 3 consists of: x(t ) – input analog
signal; 1 – synchronizer; 2 – ADC; 3 – digital data centring module, xi = xi − M x ; 4 – multiplication and
2
squaring module С 2j = ( xi − xi −1 ) ; 5 – multi-bit shift register; 6 – generator of adjugate squares C 2j ; 7 –
a group of adders; 8 – pyramid adder; 9 – binary logarithmic function encoder; I x (R) – output code.
x(t)
2 3 4
I x (R)
1 5 ... 6 ... 7 ... 8 9
Figure 3: The structure of the special processor for determining entropy, taking into account
statistical relationships using a correlation function
This approach to entropy estimation using the autocorrelation function R xx ( j ) has the following
disadvantages:
і) the need to perform a data centring operation, which leads to an increase in computational time;
іі) the presence of the operation of accumulation of the products sum for squares of the centred values.
The consequence of these shortcomings is the considerable hardware complexity structure of the
specialized processor for entropy estimation and significant time costs, which lead to low performance.
As a result, such structural implementation of specialized processor for entropy estimation is
characterized by and low performance.
2. The next way for entropy estimation uses the equivalence correlation function Fxx ( j ) . This formula
of entropy estimation has next form (3):
1 m 2
M x − Fxx ( j ) ,
I x ( F ) = n E log 2 2 (3)
m j =1
where, E • is integer function with rounding to a larger integer number; Fxx ( j ) – autocorrelation
equivalence function.
Asymptotic characteristics of the equivalence function Fxx ( j ) are described by expressions (4) [6].
1 n
Fxx ( j ) = Z ( xi , xi − j ) , j = 1, m , (4)
n i =1
Its graph is presented in Fig. 4.
Fxx ( j)
Fxx (0) = M x
Mx
Fxx () = 0
n =128
m = 32
m
0 j
Figure 4: Graph of equivalence function Fxx ( j ) and its asymptotes
The entropy estimation I x (F ) based on the correlation equivalence function is displayed in
Fig. 5.
Developed structure of the specialized processor based on the entropy estimation (3) using the
equivalence function Fxx ( j ) is presented in Fig. 6 [6].
3
I x (F )
M x2
I x (F )
0 m j
Figure 5: Entropy estimation I x (F ) for correlation equivalence function.
x(t)
2 5 6 7
I x (F )
1 3 4 5 6 8 9 10
Figure 6: Structure of a specialized processor for entropy estimation based on function Fxx ( j )
The following notations are used in Figure 6: x(t ) – input analog signal; 1– synchronizer; 2 – ADC; 3
– multi-bit shift register; 4 – a group of logical elements "AND"; 5 – counters; 6 – square generators; 7 –
encoder; 8 – a group of adders; 9 – pyramid encoder; 10 – a binary logarithmic encoder; I x (F ) – output
code for entropy estimation.
The advantages of this method of entropy estimation and the corresponding specialized processor are:
i) a lack of centring and multiplication operations;
ii) Using the operation of comparing Z ( xi , xi − j ) of values xi and x i − j ;
iii) As a result of points i) and ii) the simpler algorithm and higher performance of the specialized
processor of entropy estimation;
iiii) 4 times reduced the required sample volume n 128 of input digital data with calculating the m-
points of autocorrelation function.
The analysis of entropy estimation algorithms [6-8] and corresponding structural solutions allows to
develop single-crystal specialized processor and widely use them in telecommunication systems and
networks [9], as digital receivers of entropy-manipulated signals. It is also advisable to extend the
functionality of such specialized processors by parallel outputting of entropy estimation codes and
intermediate results of centred values calculations x i , mathematical expectation M x , dispersion Dx and
estimated values of autocorrelation functions R xx ( j ) , Fxx ( j ) , which are integral characteristics of
entropy as it shown in Fig.7.
Figure 7: Entropy and her integral characteristics
Determination of entropy estimation I x (H ) is carried out according to the formula of C. Shannon [1],
which is based on the probability distribution of events:
m
I x (H ) = − ( pi log2 pi ) . (5)
i =1
Ni
where pi = – probability of appearance of i -event; m – a number of statistically independent
N0
events, N 0 – the total number of options.
4
It is more practically convenient to calculate the probability entropy according to the algorithm [6].
Since the N 0 N i calculation of the logarithmic function is performed according to formula (6):
N
log 2 i = (log 2 N0 − Log 2 Ni ) , (6)
N0
Thus, the calculation of probabilistic entropy when N = 256 will be performed according to the
expression:
256
1
I x (H ) = N i (log 2 N 0 − log 2 N i ) , (7)
N 0 i =1
The graph of the entropy calculation results according to Shannon's formula in the decimal number
system with the sample volume n = 256 and the total number of random messages m = 256 are shown
in Fig.8.
144
128
112
96
80
64
48
32
16
0
1 17 33 49 65 81 97 113 129 145 161 177 193 209 225 241
Figure 8: The graph of the probabilistic entropy estimation in decimal number system
You can see the following properties of estimating the probability entropy according to Shannon's
formula as a result of computer modelling of the corresponding calculations and from the graph shown in
Figure 8:
i). The entropy value I x ( H ) = 0.5 corresponding to the equal probability of independent events is
observed in two cases when, for the given experimental conditions, the probabilities pi = 64 and
p i = 128 ;
ii). The maximum value of the entropy estimate I x ( H ) = 0.530737 is observed when pi = 94 ;
iii). The characteristic of the estimated entropy I x (H ) in the range of 1 pi 255 is asymmetric in
contrast to the known traditional graphs of entropy estimates, which are symmetric in relation to the point
of maximum entropy estimate.
For N i values that correspond to the whole binary digits, there is symmetry of the same values of
entropy estimates.
Thus, when N i = 16 and N i = 192 I x ( H ) = 0.25 ;
when N i = 32 and N i = 175 I x ( H ) = 0.375 ;
when N i = 64 and N i = 128 I x ( H ) = 0.5 .
It is obvious that the form of the graph of entropy estimation values according to Shannon's formula is
conditioned by the graphical representation of the logarithmic function, with the argument defined in the
range from 1 to 255.
5
Since modern digital electronics is based on binary codes, for convenience, Table 1 shows the results of
calculating the logarithmic function in decimal and binary number system.
Table 1 shows the results of calculations of the logarithmic function of the products pi log 2 pi with
the number of registered random events N i corresponding to integers N i = 2 k , k = 0, 8 .
Table 1
Value of entropy N i (log 2 N 0 − log 2 N i ) for all Ni = 1 N
log 2 N 0 Ni log 2 N i N i (log2 N 0 − log2 N i )
8 1 0 1 (8- 0 ) = 8
8 2 1 2 (8- 1 ) = 14
8 4 2 4 ( 8 – 2 ) = 24
8 8 3 8 ( 8 - 3 ) =40
8 16 4 16 ( 8 – 4 ) = 64
8 32 5 32 (8 - 5) = 96
8 64 6 64 ( 8 – 6 ) = 128
8 128 7 128 ( 8 – 7 ) = 128
8 256 8 256 ( 8 – 8 ) = 0
As a result of the entropy calculation I x (H ) in the decimal and binary number systems, the numerical
values of estimation I x (H ) are obtained, which are presented by the informative fragments in Table 2.
Table 2
Value of entropy in binary and decimal system for all N i = 1 256
Ni(10) Ix(H)(10) Ni(2) Ix(2)
1 8,0 00000001 1000,000000000
2 14,0 00000010 1110,000000000
3 19,2451125 00000011 10011,011000000
4 24,0 00000100 11000,000000000
5 28,39035953 00000101 11100,011000000
6 32,490225 00000110 10000,011000000
7 36,348555 00000111 100100,010000000
8 40,0 00001000 101000,000000000
9 43,47067499 00001001 101011,011000000
… … … …
12 52,98044999 00001100 110100,1110000000
15 61,39664107 00001111 111101,011000000
16 64,0 00010000 1000000,00000000
… … … …
31 94,41991438 00011111 1011110,01000000
32 96,0 00100000 1100000,00000000
… … … …
63 127,4313648 00111111 00111111,01000000
64 128,0 01000000 10000000,00000000
… … … …
88 135,5700176 01011000 10000111,10000000
… … … …
94 135,8637172 01011110 10000111,10000000
6
Continue of table 2
… … … …
100 135,6143810 01100100 10000111,10000000
101 135,520602 01100101 10000111,10000000
128 128,0 10000000 1000000,00000000
… … … …
192 11000000 1001110.10000000
… … … …
224 43,15249746 11100000 101011.00100000
225 41,89923198 11100001 101001.11100000
… … … …
234 30,33465562 11101010 11110,01010000
235 29,01851756 11101011 11101,00000000
… … … …
240 22,34623705 11110000 10110,01010000
241 20,99366997 11110001 10100,11110000
… … … …
248 11,35931502 11111000 1011,01011000
249 9,959518915 11111001 1001,11110000
250 8,553928834 11111010 1000,10001000
251 7,142567958 11111011 111,00100100
252 5,725459278 11111100 101,10111000
253 4,302625602 11111101 100,010011000
254 2,87408956 11111110 100,010011000
255 1,439 11111111 1,01110000000
256 1,0 100000000 1,00000000
Since the logarithmic function is irrational, it is clear that with a limited number of digits, its value can
be calculated only with a certain accuracy, which is limited by the number of decimal places. Accordingly,
in the decimal number system its value will be displayed more accurately than in binary with the same
number of digits. Limiting the accuracy of only the integer part of number of the logarithmic function in
the binary number system is quite sufficient, given the method of entropy estimation.
3. Theory and structural characteristics of wireless bus and 2D topologies
cyber-physical systems
Estimates of hardware and time complexity are traditionally used to assess the system characteristics of
cyber-physical systems (CPSs) components [10-12]. At the same time, these estimates do not take into
account the current level of micro- and nano-electronics technologies in the crystal environment. The
structural and technological complexity of such crystals, which contain transistors and the connections
between them, is almost the same. There are many other estimates of the complexity of microelectronic
computing modules in the CPS design process [11-15]. It is advisable to use the following more extensive
estimates of the system characteristics of CPS components, among which the most important is the
structural complexity [10]. Table 3 shows the criteria of structural and functional-informational complexity
of microelectronic components and structures of CPS [10].
Table 3
Criteria for structural and functional-informational complexity of microelectronic components and
structures of CPS
7
№ Analytical expression
1. Petri.
Vk
kc =
bn + bm
k , n, m is number of vertices, unidirectional and bidirectional edges
2. Quine.
n m
SK = X i + Y j
i =1 j =1
n, m is the number of inputs and outputs of the structure respectively
3. M. Kartsev .
n
single-level AC = Аi
i =1
AC is general assessment of hardware complexity; i, j, k are types of components or levels
of device structure.
4. S. Mayorov.
m n
two-level AC = Аij ,
j=1 i =1
m n l
three-level AC = Аijk
j=1 i =1 k =1
m, n, l is the appropriate number of different components types or levels of the structure of
the device
5. M. Cherkaskyi. Logarithmic structural complexity
E
S = − E log 2
n(n − 1)
where E is the number of elements of the adjacency matrix of the system; n is the number of
vertices of the graph
6. M. Cherkaskyi. Software complexity
F
P = − F log 2
nm
F ; n, m is the corresponding number of control signals, control inputs and time samples of
the time chart;
7. V. Glukhov.
m −1
L = ( gi + vi ) (1/ 2...3 / 4)m2 ; gi = xi + 1 , v j = m + di + 1
i =0
gi are lengths of horizontal, vi are the lengths of the vertical connections on the conditional
FPGA.
8. J. Martin. Structural complexity of the network structure
N S
К d = i ; Kd =
N0 G
Ni are numbers of connections, N0 is number of components; S is number of readings or
requests, G is number of records or data updates
9. Y. Nykolaychuk, I. Pitukh. Advanced assessment of network complexity
S G
K ed = i 0
S0 Gi
Si , S0 , Gi , G0 are the actual number of requests, the maximum possible number of requests, the
actual number of records or updates, the maximum possible number of records or updates in
the node of the matrix model, respectively
8
№ Analytical expression
10. N. Vozna. Criterion of complexity of multifunctional structure
n
kc = i Pi Pi (l , P, x, d , r, h, z, b, c, i, n, a, f )
i =1
Pi are informative parameters of structures attributes, i are weights of expert assessments
of structural complexity, n is the number of microelectronic structure components.
11. Y. Nykolaychuk, N. Vozna. Information and structural complexity
m
Ke = K
FC
kC
max ; FC =
fj
j =1
K is data level identifier; FC is information complexity of microelectronic structure.
12. N. Vozna. Information and functional complexity of inputs and outputs
n m
f j = i finput + i f output
i =1 i =1
f j are functional and informational characteristics; j , j are coefficients of informativeness
of input-output functions; m, n is number of inputs and outputs.
13. Y. Nykolaychuk, V. Hryga. Additive criterion for estimating the complexity of the data
ordering structure.
n
Kv = A + ; A = Ai ; Ai = AМ ;
1
Ai , i - respectively hardware and time complexity of the i-th microelectronic component;
14. A. Melnyk. Multiplicative normalized criterion of operating device complexity
KM = 1 Wk Tk max;
Wk are total equipment costs; Tk is duration of data processing;
It should be noted that the multiplicative normalized criterion of complexity of the operating device
proposed by Professor A. Melnyk [16] is the most informative assessment of maximizing the efficiency of
system characteristics of ADC components, vector, scalar and quantum supercomputers.
Systematized criteria (Table 3) for assessing the structural complexity of microelectronic components
CPS can increase the efficiency of comparing the system characteristics of different structures of operating
devices and specialized processors in the architecture of computing facilities.
This is especially true of the criteria presented (Table 3, No.11&14), which are the minimum
characteristics of the efficiency of the equipment use for processor operating devices and computer
memory.
An important criterion for the structural complexity of network 2D architectures CPS is the criterion of
emergence proposed by J. Martin (Table 3, No.8).
2D network architectures CPS are classified: monopoly, hierarchical multilevel, ring, star-bus,
interactive hierarchical, star-ring with open atmospheric optical channels communication, hierarchical one-
level, bus, systolic, interactive monopoly, interactive multilevel hierarchical, ring-star, problem-oriented
dialog.
The multilevel hierarchical, ring, systolic, star-bus and star-ring structures are the most perfect in the
structure of CPS, which belong to the DCS [17, 18], in terms of functional and structural priority
characteristics. The system characteristics of complexity for the specified network architectures CPS are
calculated, according to the criterion of emergence of J. Martin [10] (Table 4-5).
Note that the most priority modern architectures of interactive CPS are structures (Table 5, No.3,4),
which are characterized by the highest level of emergence and parallelization of data streams and
processing.
Such 2D CPS structures are used as information systems for background monitoring of natural
protection areas.
The concept of the theory of formation and processing of interactive and dialog data in 2D architectures
of DCS is shown on Fig.9.
9
Table 4
Emergence of 2D network non-interactive CPS architectures
Modifications of DCS architectures
Multilevel hierarchical Star-bus
ke = 0,8 ke = 0,95 ke = 8,3 ke = 8,3
Table 5
Emergence of 2D network interactive CPS architectures
Modifications of DCS architectures
1. Multilevel hierarchical
ke = 1,7 ke = 1,8
2. Problem-oriented dialog system
ke = 2,8
3. Star-bus
ke = 4, 2
10
Continue of table 5
3. Star-bus
ke = 4, 2
4. Star-ring with open atmospheric optical communication channels
1 2 3
4 5 6
7 8 9
ke = 2,7
Figure 9: Structure and information functions of the formation and processing theory concept for
interactive data
The main result of the using such concept in practice is the substantiation of methods of traffic
organization and processing of information monitoring and dialog data in 2D structures of the CPS.
The developed concept is a basic tool for designing and improving the system characteristics of the
components of monitoring, dialog, cyber-physical and interactive computer systems.
11
4. Crypto-protected transmission of information in cyber-physical systems
based on entropy-manipulated signals
An important problem in the design of CPS for use in various industries, environmental and regime
areas is the effective cryptographic protection of information data flows from unauthorized access.
There are known fundamental limitations of Shannon, which relate to the reliable receiving of
manipulated signals against the background of noise [1, 3]. The essence of such restrictions is that the ratio
of the sign of the manipulated signal (amplitude, frequency, phase, energy, etc.) must exceed the
corresponding noise characteristic by 2 times according to the following statements:
Ps P (f ) P(f ) R ( j) Hs H cs
2; s 2; s i 2; xx s 2; 2; 2. , (8)
Pn Pn (f ) Pn ( fi ) Rxx ( j )n Hn H cn
where: Ps , R xx ( j ) s , H s , H cs – corresponding powers of amplitude, frequency, phase,
autocorrelation, noise, entropy and crypto-protected entropy, Pn , R xx ( j ) n , H n , H cn – corresponding
powers of noise characteristics.
It is shown the characteristics (Fig. 10) of reliable signal extraction against the background of noise and
interference depending on the distance of propagation according to the fundamental limitations of
C.Shannon.
Figure 10: Methods of signal manipulation in conditions of intense interference
It is shown (Fig. 10) that the most promising methods of signal manipulation in modern CPS are CEM
– crypto-protected multilevel entropic manipulation.
The structure of the device for determination of entropy according to the formula of probabilistic
estimation of entropy of C. Shannon [1, 3] is offered in a work [9].
S
H S = −k p j log p j , (9)
j =0
where k is a positive coefficient that takes into account the basis of the logarithm; pj is the probability
of the sj's state of information source; S is a number of independent states of information source.
The device is characterized by a high level of parallelization of information processing, has a regular
microelectronic structure and contains: 1 – ADC; 2 – information input of the device, 1.1 – group of model
resistors, 1.2 – comparators with paraphrase outputs (direct and inverse), 1.3 – logic elements AND-NOT,
3 – binary counters, 4 – synchronizer; 5 – encoders, 6 – pyramidal adder, 7 – device output.
In each channel of the device the counter (3) accumulates the sum of identical values of digital samples
pj, and at the output of the tabular encoder (5) the product code pjlog2(pj) is formed. At the end of the
cycle of sampling n-digital samples at the output of the pyramidal adder (6) the source code of the
estimated entropy of the information source is formed.
The functional limitation of such device is the delay of the calculation process in the encoders (5) and
the adder (6), which reduces the speed of the device. Therefore, the structure of the entropy estimation
device (Fig. 11) is proposed [19], which is characterized by increased speed by parallelizing the processes
of accumulation of the sum of probabilities pj and parallel encryption and estimating the initial sum of
entropy according to the expression:
12
com + c
H = (10)
x
t + e + ,
com , c , t , b , - respectively delays of the comparator, counter, trigger, encoder and adder.
Figure 11: Device for entropy estimation
Structure of such a specialized processor [19] for receiving entropy-manipulated signals is proposed,
which is shown in Fig.12.
Each channel of such device uses an n-bit jk-counter (3), the calculation results of which are registered
by the memory register (5) on D-flip-flops. At the same time, in the process of calculating the product
pjlog2(pj) and determining their sum by the pyramidal adder (7), the accumulation of new probability
estimates pj in synchronous jk-counters (3) is carried out.
Patent [19] presents the results of comparing the hardware and time complexity of the two devices for
entropy estimation at a sample size of m = 256, bit counts k = 8 and bit encoder codes h = 11.
Probability entropy detection devices are important components of telecommunication systems in the
CPS structure, which provide an appropriate level of encryption of information data flows. The principle of
data encryption based on the entropic method of signal manipulation, which provides noise-like formation
of bit “0” and “1” bits is proposed. This modifies the structure of the entropy estimation device, which can
receive and decode a bit-oriented stream of crypto-protected data with protection against unauthorized
access.
It is shown an example (Fig. 13) of such a modified probability entropy determination structure [19],
which is used to receive and decode crypto-protected entropy-manipulated signals.
The proposed method of crypto-protected entropy-manipulated is characterized by wide possibilities
that require fundamental theoretical and experimental research, as well as a large amount of computer
modelling.
Wide range of possibilities of methods of cryptographic protection of entropy-manipulated signals by
hashing of streams {pi} and the possibility of their logical processing with logical elements "OR", delays
and logical elements "AND".
In addition, multiplication by log2pi, log2pj, log2piz provides additional opportunities to increase
cryptographic protection.
Then we can selectively summarize the individual S i log 2 S j to generate individual bits or quasi-
ternary bits H S .
13
Figure 12: The structure of the specialized processor for entropy estimation
Figure 13: The structure of the modified device for entropy estimation
5. Acknowledgements
Thus, the analysis of existing entropy estimates is carried out and a new theoretically substantiated
approach is proposed, taking into account correlation relationships. The results of entropy characteristics
and properties of digital components of cyber physical systems are investigated and given. The prospects
of entropy analysis and its use for the analysis of digital data flows are shown. High-performance
structures of specialized processors for determining probability and correlation entropy are proposed.
Improved structures of data cryptographic algorithms based on entropic signal manipulation. The
generalizations of approaches of complexity estimations definition of cyber physical systems components
are generalized and own criterion of structural estimation is offered and the mathematical apparatus of its
definition is formalized.
14
6. References
[1]. C. Shannon A mathematical theory of communication ACM SIGMOBILE Mobile
Computing and Communications Review, vol.5 n.1, Jan 2001 pp. 3-55.
[2]. Harré MS. Entropy, Economics, and Criticality. Entropy. 2022; 24(2):210.
https://doi.org/10.3390/e24020210.
[3]. Pan Q, Zhou D, Tang Y, Li X, Huang J. A Novel Belief Entropy for Measuring
Uncertainty in Dempster-Shafer Evidence Theory Framework Based on Plausibility
Transformation and Weighted Hartley Entropy. Entropy. 2019; 21(2):163.
https://doi.org/10.3390/e21020163.
[4]. J. Karmeshu. Entropy Measures, Maximum Entropy Principle and Emerging
Applications. DOI https://doi.org/10.1007/978-3-540-36212-8.
[5]. Lee J, Lee K. A Method for Neutralizing Entropy Measurement-Based Ransomware
Detection Technologies Using Encoding Algorithms. Entropy. 2022; 24(2):239.
https://doi.org/10.3390/e24020239.
[6]. Artur Voronych, Lyubov Nyckolaychuk, Nataliia Vozna, Taras Pastukh. Methods and
Specialized processors of Entropy Signal Processing. The Experience of Designing and
Application of CADSM’2019, Feb., р. 3/59-3/62, 2019.
[7]. H. Zhang, Y. Deng. Entropy measure for orderable sets. Information Sciences. Vol.
561, 2021, Pages 141-151. https://doi.org/10.1016/j.ins.2021.01.073.
[8]. W. Xu, L. Jiang, C. Li. Improving data and model quality in crowdsourcing using
cross-entropy-based noise correction. Information Sciences.Vol. 546, 2021, pp. 803-814,
ISSN 0020-0255, https://doi.org/10.1016/j.ins.2020.08.117.
[9]. Patent of Ukraine 117037, July 11, 2018. (in Ukrainian).
[10]. N. Vozna. Structuring multifunctional data: theory, methods and tools: Monograph
Ternopil: TNEU, 2018. 378 p. (in Ukrainian).
[11]. Dean, Walter. Computational Complexity Theory and the Philosophy of Mathematics†.
Philosophia Mathematica, 2019, 27(3):381-439.
[12]. Miller, R. Computable Transformations of Structures. In: Kari, J., Manea, F., Petre, I.
(eds) Unveiling Dynamics and Complexity. CiE 2017. Lecture Notes in Computer
Science, vol 10307. Springer, Cham. https://doi.org/10.1007/978-3-319-58741-7_9.
[13]. B. Barak, P. Gopalan, J. Håstad, R. Mekha, P. Raghavendra, and D. Steurer Making the
Long Code Short”, SIAM Journal on Computing, 2015, Vol 44, pp 1287-1324.
[14]. Okhotin, A., Salomaa, K.: State complexity of operations on input-driven pushdown
automata. J. Comput. Syst. Sci. 86, 207–228 (2017).
[15]. Dassow, J. (2017). Descriptional Complexity and Operations – Two Non-classical
Cases. In: Pighizzini, G., Câmpeanu, C. (eds) Descriptional Complexity of Formal
Systems. DCFS 2017. Lecture Notes in Computer Science, vol 10316. Springer, Cham.
https://doi.org/10.1007/978-3-319-60252-3_3.
[16]. Melnyk, A., Melnyk, V. Specialized Processors Automatic Design Tools-the Basis of
Self-Configurable Computer and Cyber-Physical Systems. IEEE International
Conference on Advanced Trends in Information Theory, ATIT 2019. Ukraine, Kyiv
2019, pp. 326–331. https://doi.org/10.1109/ATIT49449.2019.9030481.
[17]. Milazzo M, Musciotto F, Miccichè S, Mantegna RN. Analysis of the Structure and
Dynamics of European Flight Networks. Entropy. 2022; 24 (2):248.
https://doi.org/10.3390/e24020248.
[18]. Suchecki K, Hołyst JA. Hierarchy Depth in Directed Networks. Entropy. 2022;
24(2):252. https://doi.org/10.3390/e24020252.
[19]. Declarative Patent of Ukraine, Patent 123920, June 23, 2021. (in Ukrainian).
15