=Paper=
{{Paper
|id=Vol-3200/paper2
|storemode=property
|title=Using the Sum of Real Type Functions to Encrypt Messages
|pdfUrl=https://ceur-ws.org/Vol-3200/paper2.pdf
|volume=Vol-3200
|authors=Viktor Avramenko,Mykyta Bondarenko
}}
==Using the Sum of Real Type Functions to Encrypt Messages==
https://ceur-ws.org/Vol-3200/paper2.pdf
Using the Sum of Real Type Functions to Encrypt Messages
Viktor Avramenko1, Mykyta Bondarenko2
1,2
Sumy State University, Rymskogo-Korsakova st. 2., Sumy, 40007, Ukraine
Abstract
This paper presents a symmetric key cryptosystem using the sum of real type functions which
allows to increase the cryptographic strength. Both transmitter and receiver choose Key
Functions with the same argument, the interval for setting the argument, and the step for
changing it. The symbol of the transmitted message is encrypted in an array where each
element is the sum of Key Functions with random amplitudes. This sum includes those Key
Functions for which the corresponding bit is one. Decryption uses disproportion functions. The
system is suitable for encrypting both discrete and continuous messages.
Keywords 1
Cryptosystems, disproportion functions, function of real variable, key functions, encryption,
decryption, text messages
1. Introduction systems, but it’s not promising due to the
intensive development of the quantum computers
[7], which will significantly affect the
Widely used cryptosystems are based on the
cryptographic strength of existing cryptosystems
set of integers. They implement symmetric and
[8]. The ordinal brute force has complexity O(2k),
asymmetric encryption algorithms. In symmetric
meanwhile Grover's quantum algorithm [9]
systems, the same key is used for both encryption
reduces it to O(2k/2) [9].
and decryption. The most famous symmetric
Implementing quantum algorithms will also
systems are AES [1] and GOST 28147-89 [2, 3].
reduce the robustness of asymmetric systems. The
To hack such a system, an enumeration of
RSA system uses the super polynomial
possible keys is required. The brute-force
computational complexity of the factorization of
complexity is O(2k), where k is the key length in
natural numbers. At the same time, there is a
bits. For symmetric systems, if the
quantum algorithm whose complexity is
communication channel is open, there is a
polynomial O(n3) [10]. It means the cryptographic
problem of secure key transmission. This problem
strength of asymmetric systems can be reduced as
does not exist for asymmetric open key systems.
a result of the implementation of Shor's quantum
In these systems, the most widely used algorithms
algorithm for computing the discrete logarithm. In
are RSA and El-Gamal [4, 5]. The RSA algorithm
[11], Shor's algorithm is given for the group of
is based on the computational complexity of the
points of an elliptic curve over the field GF(p)
integer factorization problem. El-Gamal's
with complexity O(n3). Implementing quantum
algorithm is based on the difficulty of computing
algorithms will also reduce the robustness of
the discrete logarithm, especially over a group of
asymmetric systems. A method for increasing the
points of an elliptic curve [6]. For breaking
crypto resistance of the system under these
asymmetric cryptosystems, there are
conditions is proposed in [12]. Along with the
cryptanalysis methods which are faster than full
search for ways to hack cryptosystems, methods
search. This circumstance makes it necessary to
for detecting signals of means of secretly
use longer keys compared to keys in symmetric
III International Scientific And Practical Conference “Information
Security And Information Technologies”, September 13–19, 2021,
Odesa, Ukraine
EMAIL: vv.avramenko@cs.sumdu.edu.ua (A.1);
nikbond97@gmail.com (A. 2)
ORCID: 0000-0002-6317-6711 (A. 1); 0000-0002-8849-7378
(A. 2)
© 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)
obtaining information are also being developed The cryptosystems [15-18] in the process of
[13]. computer modeling have shown high
The above analysis shows that one should look cryptographic strength when trying to guess the
for other ways to create cryptosystems. In parameters of keys functions, even if their form is
particular, to complicate the selection of keys known. To further complicate the work of
using the simple enumeration method, one should cryptanalysts, the task is to develop a
switch from using integers to real ones. It is cryptosystem that could combine the advantages
known [14] the set of real numbers has a higher of the systems considered in [15-17] and the
cardinality compared to the set of natural system [18]. So, it’s necessary to develop the
numbers, so one can expect the cryptographic algorithms for encryption and decryption of
strength of a cryptosystem based on real numbers analog and discrete messages, using several
will be higher. The possibilities of creating functions of a real variable as keys without the
cryptosystems using one or more functions of a necessity to calculate derivatives.
real variable as keys are considered in [15-18].
So, in [15], characters from the ASCII code 2. Mathematical formulation of the
table are encrypted by the sum of 10 functions of
a real variable, which are keys. Each key-function problem
is preceded by a coefficient, which, depending on
the character being encrypted, is equal to zero or The message that is encrypted is a sequence of
one. The amplitudes of these functions are random T numeric character codes from the ASCII table
for each new symbol. The resulting sum of the (or numeric values of the pixel brightness
values of the functions is transmitted over the components in the case of a graphic image
communication channel. On the receiving side, transmission). Each of them is encrypted using
fragments of key functions are recognized, which one-dimensional arrays of length N values. These
are represented in the received encrypted signal. arrays are obtained using one and the same step h
This allows you to decrypt the symbol transmitted of changing the argument of m Key Functions of
at the current time using the disproportion the real variable. In this case, the value y(j, i) of
functions [19-22]. the matrix y(T, N) has the form:
In [16, 17], a variant is proposed when , ,
(1)
symbols for transferring binary codes are encoded
with the help of three key functions of a real where:
variable. "1", "0", "space", "new line" are j is the number of the character in the transmitted
encoded. Any other character is recognized as a message;
new line. For unauthorized access to the fq(i) = fq(ih), (i = 1, 2, … N > m), (q = 1, 2, … m)
intercepted message, you need to select the type - an array of values of the q-th Key Function;
and parameters of the key functions. kqj - coefficients that are generated during
In [15-17], the disproportion functions over the encryption of the j-th element and can be either
first-order derivative were used. In this case, it is equal to zero or represent random numbers which
necessary to apply numerical methods for are unknown to the recipient.
calculating the current values of the first Key Functions can be either continuous or
derivatives. The need for these calculations led to discrete. These functions should be the same for
the fact that the ciphertext significantly exceeded the transmitting and receiving sides and have the
the length of the encrypted message. same numbering. Also, the step h of changing the
A completely different encryption principle argument of the Key Functions should be the
was proposed in [18]. One function of the real same. An encrypted message in the form of a
variable is used as the key. The disproportion matrix y(T,N) is transmitted over an open
function of the numerical representation of the communication channel. The task is to decrypt the
encrypted process is calculated with respect to the message using the matrix received at the receiving
key function. The obtained values of the end. To solve it, the integral disproportion of the
disproportion function are an encrypted message first order is used [23].
and are transmitted over the communication
channel. To avoid calculating the derivatives, the
integral disproportions of the first order is used 3. Disproportion functions
[23].
One of the first publications in which In this case, y(x) and f(x) are represented by
disproportion functions were proposed was [19]. one-dimensional arrays. If the approximate values
In particular, the disproportion with respect to the of the integrals in (6) are calculated using the
n-th-order derivative of the function y(x) with trapezoid formula, then for the one and the same
respect to x is described by the expression: step h for y(x) and f(x), disproportion (6) takes the
1 following form (7):
(2)
@ ⋅ , @ ,
(7)
!
Here the @ symbol is chosen to denote the
operation of calculating disproportion. The
symbol "d" stands for "derivative". The order is 4. Encrypting and decrypting
indicated in parentheses. The left part (2) reads "et messages
d n y with respect to x". The order n ≥ 1 is an
integer. If for any value of x, the function y(x) has
The transmitting and receiving sides must have
the form y = kxn, then disproportion (2) is equal to
the same system of m Key Functions of the real
zero regardless of the value of the coefficient k.
variable, their numbering, the interval of changing
For the case when n = 1,
the argument and step h of its change. The number
(3)
@ , of elements N of the one-dimensional array
corresponding to the encrypted character must
For defining the functions parametrically, also be set. These can be both characters from the
when, x = φ(t), y = ψ(t), where t is a parameter, ASCII table, and components of pixel brightness
disproportion (3) is described by the expression when transmitting color graphic images. Each of
(4) them is represented by an integer. The required
@ , number of Key Functions depends on the
maximum value of this number. For example, to
encrypt characters from the ASCII table, m = 8
For ψ(t) = kφ(t) disproportion (4) is equal to Key Functions are required. They can be either
zero in the entire area of existence x = φ(t), continuous or discrete. If the Key Functions are
regardless of the value of k. In [19], the case was continuous, it is necessary to calculate N elements
considered when of one-dimensional arrays of their values,
⋯ (5) changing the argument from the initial xmin to the
, final xmax value with a step h. When encrypting
where f1(x), f2(x), … fm(x) are known characters from the ASCII table or the pixel
functions; k1, k2, … km are coefficients whose brightness, their numerical representations differ
values are unknown. by one. In these cases, the step h of changing the
It is shown that the disproportion functions argument must be equal to one.
allow calculating the values of the unknown An m-bit binary code corresponds to each
coefficients in (5) from the data obtained for the encrypted character. Each bit in this code is
current value of the argument. This opportunity is associated with a specific number of the Key
used to create cryptosystems [15-17]. Function. If the bit is zero, the value of the
In practice, often the first derivative of the corresponding Key Function is also zero. If the bit
function does not exist or is equal to zero on some is equal to one, then a random value of the
interval. This excludes the possibility of using amplitude of the corresponding Key Function is
disproportions over the first-order derivative (2- played. The character to be encrypted is
4). In this case, it is advisable to use the integral represented by the sum (1).
disproportion of the first order [23]. This
disproportion of the function y(x) with respect to 4.1. Encrypting messages
f(x) has the form:
(6) 1. The following is a character encryption
@ , algorithm:
2. Calculate arrays of N > m values of Key
where h - is the preset time interval. In the discrete Functions: fq(x), q = 1, 2… m.
representation of signals, this is a time 3. Enter the encrypted j-th character and
quantization step. calculate its cipher in the form of values of the
one-dimensional array y(j, i), i = 1, 2... N , @ , (13)
,
according to (1). , 1 , ,
4. Repeat this point for all characters of the ,
message of length T. , 1 ,
5. A sequence of T arrays is an encrypted
message transmitted over an open communication Taking into account that the disproportion of
channel. F21(j, i) with respect to F21(j, i) is equal to zero, we
get:
4.2. Decrypting messages F j, i k j, i , (14)
Third level: The disproportion of F0121(j, i)
Pre-compute the arrays fq(i) = fq(ih),
with respect to F3121(j,i) is calculated in the
(q = 1, 2, ... m), (i = 1, 2, ... N > m), of Key
following way
Functions and to receive T one-dimensional
, @ , (15)
arrays y(j , i), j = 1, 2, ... T, i = 1, 2, ... N over the ,
communication channel. Further, in order to , 1 ,
simplify the description of the decryption process, , 1 ,
an example is given when only three functions are ,
used in the cryptosystem - the keys: f1(x), f2(x), 0,
f3(x). In this case m = 3. Accordingly, the j-th
It is equal to zero because, as can be seen from
character of the message is encrypted as
(14), there is a proportional relationship between
, , (8)
F0121(j, i) and F3121(j, i). This fact allows
1, 2, … 3, calculating from (14) k3j and k2j, k1j for the j-th
The process consists of m = 3 levels in message symbol.
accordance with the number of Key Functions. , (16)
First level: It is necessary to calculate the array ,
of disproportions (7) y(j,i) with respect to any of
the Key Functions , for example, f1(i):
, @ , (9) , , (17)
,
, 1 , ,
,
1
where i = 2, 3…N. , (18)
,
Also calculate the disproportions (7) of the
remaining key functions with respect to f1(i): Depending on which of these coefficients are
, @ , (10) nonzero and which are equal to zero, the j-th
, 1 , , message symbol is decrypted. In practice, it must
, be taken into account that there are calculation
1 errors.
where r = 2, 3. Therefore, it is necessary to compare the
Considering that the disproportion of the disproportion (15) calculated at the last level in
function relative to itself is zero, we get: modulus not strictly with zero, but with an
, , , , (11) approximate number ɛ. For example, it could be
Second level: It is necessary to select any ɛ=10-4. In this case, if | F01213121(j, i) | <= ɛ, then it
disproportion from right-hand of (11), for should be assumed that it is zero.
example F21(j, i). It is used to calculate next The value of ɛ is determined during testing of
disproportions: the cryptosystem. Theoretically, this
, @ , (12) disproportion is equal to zero for all i = 2, 3, ... N,
,
, 1 , , but, taking into account the calculation errors, it is
, recommended to do calculations using formulas
, 1 ,
(16-18) for i, at which the modulus of
disproportion (15) is minimal.
4.3. An example of encrypting and select the form of eight Key Functions and the
values of their parameters.
decrypting characters from an ASCII
table Table 1
Encrypted and decrypted characters “Hello”
Eight Key Functions are used (m =8): y ‘H’ ‘e’ ‘l’ ‘l’ ‘o’
1. f1(x) = 1000 sin((α1 - β1)x) cos(wβ1x) 0
1
‐323.36050
257.702939
‐1096.0141
167.391848
‐872.47149
1051.01033
37.134528
532.400561
‐112.93721
427.740614
2. f2(x) = 1000 exp(0.1α2x) sin(wβ2x) 2 57.298613 175.907791 ‐408.37541 ‐216.26334 ‐116.65218
3 ‐165.32821 126.358160 ‐324.75198 ‐162.19800 ‐197.22270
cos((α2 + β2)x) 4 ‐186.82906 ‐394.77504 ‐929.02530 ‐439.94548 ‐449.01146
3. f3(x) = 1000 exp(-α3x) sin(wβ3x) 5
6
‐163.70378
37.446166
‐392.33753
299.880685
‐1059.2853
370.310135
‐385.85981
74.675455
‐170.75848
44.746439
4. f4(x) = 1000 cos((α1x - β1)x) sin(wβ1x) 7
8
‐110.70494
‐2.714026
426.787248
‐59.278796
‐218.90900
115.564604
‐238.68403
‐2.371129
‐314.75860
‐165.23427
5. f5(x) = 1000 exp(0.1sin(α2x)) sin(w 9 9.436954 152.916970 116.697501 ‐23.361501 281.220083
10 42.465400 ‐412.02347 ‐203.82595 84.424717 150.029168
cos(βx)) cos((α2+β2) x) 11 ‐24.295297 615.002251 349.117675 ‐42.371452 ‐231.55086
6. f6(x) = 1000 sin(-cos(α3x)) cos(w 12
13
101.570285
195.422364
95.754178
132.219196
543.764259
855.514365
227.452661
432.359247
‐122.68102
754.345004
sin(β3x)) 14
15
‐11.067358
214.445079
‐507.14921
264.682389
‐252.27430
659.731238
‐29.696351
467.369970
228.457327
‐63.039751
7. f7(x) = 1000 sin(wx + α1) exp(-β1x2)
8. f8(x) = 1000 cos(wγx2)
Table 2
where α1 = 1, α2 = 0.12, α3 = 0.5, β1 = 0.1,
β2 = 1.5, β3 = 0.7, γ = 0.5, w = 400 are constants. Encrypted and decrypted characters ‘A’
y ‘A’ ‘A’ ‘A’ ‘A’ ‘A’
A sequence of numbers corresponding to the 0 ‐597.135343 ‐762.540347 ‐609.489179 ‐1245.052456 ‐917.400855
transmitted characters from the ASCII code table 1 9.473961 ‐6.667141 ‐2.760240 ‐37.310266 ‐17.407304
is encrypted. Each character is encoded by a sum 2
3
274.695430
15.193014
368.229254
87.216540
291.933312
60.428145
625.796927
237.898008
451.736269
138.849052
of Key Functions 4
5
‐123.497929
‐58.830278
‐217.377766
‐107.584825
‐165.579209
‐81.548078
‐438.953490
‐221.367775
‐291.370618
‐145.669069
y x x x x (19) 6 ‐8.280530 ‐11.911812 ‐9.337867 ‐21.332769 ‐14.999968
7 199.488556 342.700766 261.876769 683.403833 456.291669
x x 8 ‐76.316724 ‐131.227388 ‐100.265637 ‐261.818697 ‐174.769533
x x 9
10
‐60.158608
‐125.104506
103.377062
‐215.011307
78.993047
‐164.292499
‐206.183725
‐428.868345
137.653657
‐286.313719
x , 11
12
214.641127
‐14.047252
368.892513
24.142272
281.874942
18.447384
735.803375
‐48.154847
491.224813
‐32.148341
where: 13
14
6.530344
‐223.052958
11.223364
‐383.349601
8.575899
‐292.921754
22.386440
‐764.640006
14.945262
‐510.476209
x = ih – argument; 15 169.891087 291.983039 223.107534 582.397666 388.810617
h = 1 – step of changing the argument.
i – is the ordinal number of the element of the Below is an example that illustrates the
one-dimensional array for each of the Key resistance of the system obtaining keys, even if
Functions, as well as the array y0, y1, ..., yN-1, somehow it was possible to find out the forms of
which is the character cipher; Key Functions. Suppose that the above sequence
N – a number of elements of each one- of characters is encrypted using functions (7), and
dimensional array. Based on the requirement of decrypted using the same kind of functions, but
N > m, the amount of array elements N = 16. the constant w was guessed incorrectly. Instead of
Table 1 shows the transmitted characters in the w = 400 was used w = 399.999 during decryption.
upper horizontal line. The corresponding ciphers In this case, the disproportion at the last eighth
are given in the form of arrays arranged vertically. level in absolute value exceeds the permissible
The decrypted characters are located horizontally deviation ɛ from zero. That is, decryption is
on the bottom line. impossible. Only if w = 399.9999, the message
It is obvious that the received message matches may be decrypted. This result shows that even
the transmitted one. It should be noted that the such a slight deviation of one of the parameters of
ciphers (arrays) of the adjacent symbols `t` are the Key Functions does not allow decryption of
completely different. the transmitted character.
The codes of the other adjacent identical
symbols in the message are given in Table 2. The 5. Requirements for Key Functions
above results indicate that the ciphers of the
adjacent identical symbols in the message differ
from each other. This circumstance greatly 1. The Key Functions must be of real type.
complicates the "hacking" of the cryptosystem. In 2. They can’t be constant and must not take
order to "crack" the message, it is required to zero values.
3. When using the key function, there
should be no situation where division by a number
close to zero occurs, which leads to an [1] National Institute of Standards and
unacceptable calculation error. For this purpose, it Technology, Specification for the
is recommended to test the cryptosystem for the ADVANCED ENCRYPTION STANDARD
entire alphabet of characters that will be used in (AES) (2001). doi: 10.6028/NIST.FIPS.197.
messages. [2] GOST 28147-89. Sistemy obrabotki
4. Check that the sum of two or more key informacii. Zashhita kriptograficheskaya.
functions does not coincide with any other of the Algoritm kriptograficheskogo
key functions. preobrazovaniya [Information processing
5. It is recommended to include all systems. Cryptographic protection.
parameters in the expression for each key Cryptographic Transformation Algorithm],
function. In this case, a change in the value of any 1990.
parameter leads to a change in all key functions, [3] A. N. Lebedev, Kriptografiya s «otkrytym
but not one or several of them only. klyuchom» i vozmozhnosti ee
6. Before sending an encrypted message, prakticheskogo primeneniya [Cryptography
first check what the decrypted message looks like with "public key" and the possibilities of its
in order to avoid errors that may occur as a result practical application], Zashhita informacii.
of not taking into account the previous points. Konfident 2 (1992)
[4] R. Rivest, A. Shamir, I. Adleman, A method
6. Conclusions for obtaining digital signatures and public-
key cryptosystems, Communications of the
ACM, 1978, 21(2):120-126.
A cryptosystem with symmetric keys is
doi:10.1145/359340.359342.
proposed. These keys are real variable functions
that satisfy the above constraints. They can be [5] I. D. Gorbenko., Y. I. Gorbenko, Prykladna
either continuous or discrete. The number of kryptolohiya [Applied Cryptology], Fort,
functions is equal to the number of binary digits NURE, Kharkiv, 2012, p. 878.
used to encrypt a character, for example, in an [6] D. R. Hankerson, S. A. Vanstone and A. J.
ASCII table. Each of the functions corresponds to Menezes., Guide to elliptic curve
a certain binary digit. The symbol of the cryptography, Springer, New York, 2003, p.
transmitted message is encrypted with a one- 311.
dimensional array. The elements of this array [7] T. D. Ladd, F. Jelezko, R. Laflamme, Y.
represent the sum of Key Functions with random Nakamura, C. Monroe, J. L. O’Brien,
amplitudes. This sum includes those Key Quantum Computing, Nature, 464 (2010)
Functions, for which the corresponding binary 45—53. doi: 10.1038/nature08812.
digit is equal to one. [8] P. G. Klucharev, Kvantovyj komp'yuter i
Decryption is performed using disproportion kriptograficheskaya stojkost' sovremennyx
functions. The possibility of encryption and sistem shifrovaniya [Quantum computer and
decryption of text information is shown. The cryptographic strength of modern encryption
given examples show the complexity the guessing systems], Herald of the Bauman Moscow
Key Functions and the cryptographic strength of State Technical University, Series Natural
the proposed cryptosystem. So, for example, a Sciences 2 (2007).
real-type constant, which equals 400 during [9] I.K. Grover, Quantum Mechanics Help in
encryption, to break the system by brute-force, Searching for a Needle in a Haystack, Phys.
you need to select with an accuracy of 10-4, but Rev. Lett. 79, 325 (1997): 326-328. doi:
there can be any number of such constants. It is 10.1103/PhysRevLett.79.325.
very difficult to find all the constants of the real [10] P. Shor, Polynomial-Time Algorithms for
type at the same time with high precision and thus Prime Factorization and Discrete Logarithms
hack the system, even with well-known formulas on a Quantum Computer, in: Proceedings of
of functions - keys. the 35th Annual Symposium of Foundations
It should also be noted that the codes of the of Computer Science, 1994. doi:
same adjacent symbols are not repeated, which 10.1137/S0097539795293172.
can be seen from Table 2. This also increases the [11] J. A. Proos, Shor’s discrete logarithm
cryptographic strength of the system. quantum algorithm for elliptic curves,
Faculty of Mathematics University of
7. References Waterloo, Waterloo, 2003, p. 35.
[12] S. Yevseiev, R. Korolyov, A. Tkachov, O. [21] V. V. Kalashnikov, V. V. Avramenko, N. Y.
Laptiev, I. Opirskyy, O. Soloviova. Slipushko, N.I. Kalashnykova, N.I., A. E.
Modification of the algorithm (OFM) S-box, Konoplyanchenko, Identification of quasi-
which provides increasing crypto resistance stationary dynamic objects with the use of
in the post-quantum period, International derivative disproportion functions, Procedia
Journal of Advanced Trends in Computer Comput. Sci., (2017) 108(C): 2100–2109.
Science and Engineering, 2020, volume 9, [22] V. V. Avramenko, A. Moskalenko,
pp. 8725-8729. Operative Recognition of Standard Signals in
doi:10.30534/ijatcse/2020/261952020. the Presence of Interference with Unknown
[13] O. Barabash, O. Laptiev, V. Tkachev, O. Characteristics, in: Proceedings of the
Maystrov, O. Krasikov, I. Polovinkin, The Second International Workshop on
Indirect method of obtaining Estimates of the Computer Modeling and Intelligent Systems
Parameters of Radio Signals of covert means (CMIS-2019), Zaporizhzhia, 2019.
of obtaining Information, International [23] A. P. Karpenko, Integral'nye harakteristiki
Journal of Emerging Trends in Engineering neproporcional'nosti chislovyh funkcij i ih
Research, 2020, volume 8, pp. 4133 – 4139. primenenie v diagnostike [Integral
doi:10.30534/ijeter/2020/17882020. characteristics of the disproportionality of
[14] A. N. Kolmogorov, S. V. Fomin, Ehlementy numerical functions and their application in
teorii funkcij i funkcional'nogo analiza diagnostics], Vestnik Sumskogo gos. un-ta.
[Elements of function theory and functional 16(2000): 20-25
analysis], Science, Moscow, 1972. [24] Vitalii Savchenko, Oleksandr Laptiev,
[15] V. V. Avramenko, M. I. Zabolotny, A Way Oleksandr Kolos, Rostyslav Lisnevskyi,
of Data Coding, 2009. Patent No. 42957, Viktoriia Ivannikova, Ivan Ablazov. Hidden
Filled March 16th, 2009, Issued July 27th, Transmitter Localization Accuracy Model
2009. Based on Multi-Position Range
[16] V. V. Kalashnikov, V. V. Avramenko, N. I. Measurement. 2020 IEEE 2nd International
Kalashnikova and Kalashnikov Jr. V.V., A Conference on Advanced Trends in
Cryptosystem Based Upon Sums of Key Information Theory (IEEE ATIT 2020)
Functions, International Journal of Conference Proceedings Kyiv, Ukraine,
Combinatiorial Optimization Problems and November 25-27. pp.246 –251
Informatics, 2017, volume 8, pp. 31-38. [25] Valentin Sobchuk, Volodymyr Pichkur, Oleg
[17] N. I. Kalashnikova, V. V. Avramenko, V. Barabash, Oleksandr Laptiev, Kovalchuk
Kalashnikov. Sums of Key Functions Igor, Amina Zidan. Algorithm of control of
Generating Cryptosystems, in: ICCS 2019, functionally stable manufacturing processes
Chapter 23, Lecture Notes in Computer of enterprises. 2020 IEEE 2nd International
Science, vol. 11540, Springer, Cham, 2019. Conference on Advanced Trends in
doi: 10.1007/978-3-030-22750-0_23. Information Theory (IEEE ATIT 2020)
[18] V. V. Avramenko, V. Demianenko, in: Conference Proceedings Kyiv, Ukraine,
CEUR Workshop Proceedings, 2020, 2608, November 25-27. pp.206 –211.
pp. 661-674. doi: 10.15588/1607-3274- [26] Oleksandr Laptiev, Oleh Stefurak, Igor
2020-2-8. Polovinkin, Oleg Barabash, Savchenko
[19] V. V. Avramenko, Characteristic properties Vitalii, Olena Zelikovska. The method of
of disproportionality functions and their improving the signal detection quality by
application to solving diagnoses problems, accounting for interference. 2020 IEEE 2nd
Transactions of Sumy State University, International Conference on Advanced
SSU, Sumy, 2000, №16, pp. 24-28. Trends in Information Theory (IEEE ATIT
[20] V. V. Kalashnikov, V. V. Avramenko, N. I. 2020) Conference Proceedings Kyiv,
Kalashnykova, Derivative disproportion Ukraine, November 25-27. pp.172 –176.
functions for pattern recognition, in: Watada, [27] Mashkov V.A. and Barabash O.V. Self-
J., Tan, S.C., Vasant, P., Padmanabhan, E., Testing of Multimodule Systems Based on
Jain, L.C. (eds.) Unconventional Modelling, Optimal Check-Connection Structures.
Simulation, and Optimization of Geoscience Engineering Simulation. Amsterdam: OPA,
and Petroleum Engineering, pp. 95–104. 1996. Vol. 13, pp. 479 – 492.
Springer, Heidelberg, 2018. [28] S. Toliupa, N. Lukova-Chuiko, O. Oksiuk.
Choice of Reasonable Variant of Signal and
Code Constructions for Multirays Radio
Channels. Second International Scientific-
Practical Conference Problems of
Infocommunications. Science and
Technology. IEEE PIC S&T 2015. pp. 269
– 271.