=Paper=
{{Paper
|id=Vol-3829/short8
|storemode=property
|title=Enhancing information transmission security with stochastic codes (short paper)
|pdfUrl=https://ceur-ws.org/Vol-3829/short8.pdf
|volume=Vol-3829
|authors=Bohdan Zhurakovskyi,Sergei Otrokh,Mykhailo Poliakov,Oleksii Poliakov,Pavlo Skladannyi
|dblpUrl=https://dblp.org/rec/conf/cqpc/ZhurakovskyiOPP24
}}
==Enhancing information transmission security with stochastic codes (short paper)==
https://ceur-ws.org/Vol-3829/short8.pdf
Enhancing information transmission security
with stochastic codes ⋆
Bohdan Zhurakovskyi1,†, Sergei Otrokh1,†, Mykhailo Poliakov2,†, Oleksii Poliakov2,†
and Pavlo Skladannyi3,∗,†
1
National Technical University of Ukraine “Igor Sikorsky Kyiv Polytechnic Institute,” 37 Peremogy ave., 03056 Kyiv, Ukraine
2
National University “Zaporizhzhia Polytechnic,” 64 Zhukovsky str. 69063 Zaporizhzhia, Ukraine
3
Borys Grinchenko Kyiv Metropolitan University, 18/2 Bulvarno-Kudriavska str., 04053 Kyiv, Ukraine
Abstract
All known algorithms of cryptographic systems, which have the property of interference resistance, are
based on codes that detect and correct errors. This work proposes a study of stochastic codes for their
potential use in cryptographic system algorithms. For stochastic codes, there is a “copy” decoding
algorithm when two or more values of a code block of a stochastic code, including (n, n–1) is a code
with the detection of errors that are the same during their transmission, it is possible to carry out joint
decoding of the extended code with bug fixes. Furthermore, the number of errors that can be corrected
in a single block of the extended code is significantly higher than the total number of errors that can be
corrected in each block. To simplify the comparative analysis, we converted the given value Pq to the
probability of flipping the binary symbol P0. We estimated this probability for different degrees of error
grouping using the Portov model with the coefficient a.
Keywords
stochastic code, cryptographic protection, probabilities of distortion in the channel, error-correcting
codes, error bursts, decoding mode 1
1. Introduction The best-known public-key cryptosystem based on
algebraic coding theory is McEliece’s cryptosystem based
The introduction of modern information technologies on a class of error-correcting codes called Goppa codes.
into the everyday life of society has caused problems in The basic idea is to create a Goppa code and disguise it as
ensuring information security [1, 2]. One of the solutions a regular linear code. There is a fast algorithm for
to this problem is the widespread use of cryptography [3, decoding Goppa codes, but the general problem of finding
4]. At the moment, strict technological requirements are codewords of this weight in a linear binary code is an NP-
imposed on cryptographic algorithms not only in terms of complete task [8].
stability but also in terms of speed [5]. Analysis of the crypto resistance of this algorithm
The need to maintain the high performance of indicates that to ensure reliable protection of information,
automated systems after protection mechanisms are the in imum parameter values required are n = 1024 and
implemented has led to increased speed requirements. Ease k = 524. The protected properties of the algorithm are
of hardware implementation is necessary to reduce the contingent on the parameter t, which must be chosen such
cost of encryption tools, which will contribute to their that t>50. This value is optimal for channels because the
mass application and wider possibilities of embedding in error probability is only 10-4 [8]. For reliable cryptographic
portable equipment. Given the specific way that protection, it is necessary to obtain the decoding
information is presented in digital devices, blockciphersare complexity that would meet modern cryptographic
of particular interest. standards (of the order of 250). To ensure there is required
Their problem oriented use in the devices and systems decoding complexity in the analyzed cryptosystem, it’s
mentionedabovecanprovide effective protection against necessary to use 750-800 columns in the check matrix of the
cyberthr eats. Thus, the development of problem-oriented Goppa code [9].
encryption systems is an important and urgent task of As can be seen from the above analysis, meeting the
applied cryptography [6]. Codes that detect and correct necessary limit requirements for system parameters
errors are the backbone of all known cryptographic systems ensures fairly reliable cryptographic protection of
that possess interference resistance properties [7].
CQPC-2024: Classic, Quantum, and Post-Quantum Cryptography, August 0000-0003-3990-5205 (B. Zhurakovskyi); 0000-0001-9008-0902
6, 2024, Kyiv, Ukraine (S. Otrokh); 0000-0002-7772-3122 (М. Poliakov); 0000-0002-9355-7056
∗ Corresponding author. (O. Poliakov); 0000-0002-7775-6039 (P. Skladannyi)
†
These authors contributed equally.
© 2024 Copyright for this paper by its authors. Use permitted under
zhurakovskybiyu@tk.kpi.ua (B. Zhurakovskyi); 2411197@ukr.net Creative Commons License Attribution 4.0 International (CC BY 4.0).
(S. Otrokh); polyakov@zntu.edu.ua (М. Poliakov);
poliakov.job@gmail.com (O. Poliakov); p.skladannyi@kubg.edu.ua
(P. Skladannyi)
CEUR
Workshop
ceur-ws.org
ISSN 1613-0073
62
Proceedings
information. For instance, the durability of McEliece’s compared to the currently widely used error-detecting
system is. codes. The transition from correction codes to error
Demonstrated by the fact that despite multiple tempts detection codes can be explained by several main reasons:
to cryptanalysis, none of them have been successful. Despite
their interference resistance, several coding algorithms used Firstly, the greater computational complexity of
for detecting and correcting errors introduce artificial implementing an error-correcting codec.
information redundancy [10]. This can be a major drawback Secondly, the need to match the type and
of interference-resistant codes. This circuit stance leads to a parameters of the error-correcting code with the
significant increase in the ciphered text compared to the conditions of information transmission, that is the
original (in the McEliece system, by a fact or two). intensity and distribution law of errors in the used
Furthermore, the public key in the MacEliece and communication channel.
Niederreiter systems is quite large by modern standards, at Thirdly, the use of, as a rule, high-quality
219 bits [11]. channels, a high degree of development of the
Jam-resistant crypto-algorithms shave high necessary technical solutions for the
requirements for hardware [12], speed [13], memory, and implementation of the cyclic code in the developed
security. These requirements depend on the properties of microcircuits for connection with communication
the applied code algorithms that use artificial redundancy. channels produced by several companies and the
standardization of channel-level protocols, which
2. Statement of research problem include the implementation of the cyclic code [14].
2.1. Self-resistant coding in transmission Therefore, to consider the alternative of using codes
channels with error correction, it is worth looking for significant
reasons for such a transition. Let’s formulate the properties
The main works of С. Shannon [14], in which the tasks of
of error-proof code with error correction that allow us to
interference-resistant information transmission with any
talk about such an alternative, and then consider a possible
predetermined accuracy of information transmission are
option for building and using such a code. So, such code
formulated, proposes to use the principle of randomness of
should have the properties:
the used signals as a solution to these tasks. For
interference-resistant information transmission, it is The code has error detection and error correction
proposed to use random (n, k)-codes, formed by randomly modes, providing in both modes a guaranteed
selecting from 2n possible binary combinations of length n (predetermined) probability of decoding with an
2k combinations, each of which is identified with one of the error in an arbitrary communication channel and
information combinations of length k. Using this model of a reliable rejection of decoding when the error
signals for transmission over a communication channel, С. cannot be corrected.
Shannon proved a theorem about the possibility of The code has such a correcting ability and allows
transmitting information over a communication channel you to choose such parameters n and k that the
with a probability of error that depends on the parameters information transmission algorithm that uses
n and k, and which can be made arbitrarily small by them is characterized by no worse probabilistic-
choosing the appropriate values for these parameters. The temporal characteristics in comparison with the
proof of this theorem was of fundamental importance for use of alternative codes.
the creation of the theory of interference-resistant coding,
The code provides, in the error correction mode,
although it did not give constructive suggestions about the
the selection of a part of the correctly received
implementation of such a possibility [15].
symbols with a specified accuracy, even if the
In practice, a relatively small group of algebraic
error multiplicity exceeds the code’s correction
interference-resistant codes is used: Bowes-Choudhury-
ability.
Hockingham (BCH) codes, Reed-Solomon (RS) codes, and
The code allows you to decode several copies
convolutional codes. The most widely used cyclic codes
(identical in terms of the information content of
with error detection, are a partial case of BCH codes and are
the code blocks) of the block with an efficiency
used in standard X.25/2 protocols (LAP-B, LAP-M). RS codes
that exceeds the efficiency of decoding the source
with error correction in radio communication channels are
code with the detection or correction of errors.
being used. Convolutional codes are widely used in satellite
This property can be used to work in parallel
communication channels, which are characterized by the
channels when multiple transmissions of a
independent nature of errors. Codes with error correction
message on a single channel or in a channel with
are not widely used due to the complexity of implementing
feedback when processing copies after receiving a
error correction, and the high dependence of the probability
repeated block.
of a decoding error on the law of error distribution.
In the works on information theory and interference- Code encoding and decoding procedures contain
resistant coding, written in the 70s, codes with error only modulo two operations.
correction were considered. First, codes based on The coding method should have properties of the
С. Shannon’s random codes, then algebraic codes. This is randomness of signals at the encoder output,
explained by the achievement of higher characteristics which provide a joint solution to the problems of
when transmitting information with error-correcting codes,
63
ensuring interference resistance in C. Shannon’s reliability of message transmission is ensured due to
formulation. tamper-resistant coding, and information secrecy and
protection against unauthorized access—due to coding,
The implementation of such a statement of the task will which refers to non-cryptographic methods of
allow: information protection. With CRC, the information-
theoretic level of information protection is provided,
To expand the spectrum of used communication
which is determined by the level of uncertainty of the
channels according to the permissible level of
choice of an ensemble of code combinations
channel quality due to the use of channels of
corresponding to the transmitted message, for an
reduced quality.
attacker who carries out radio interception [20].
Ensuring the guaranteed probability of the level
specified by the consumer (10- 9, 10-18, 10-27) in
2.2. Construction and properties of error-
case of any type of distortion in the
communication channel.
correcting stochastic codes
To remove the problem of accuracy (probability) In the 1980s, work was started on the creation of a new
of information when creating global hyper- design of codes that fit into the structure of existing data
informational spaces under the condition of transmission networks, to increase the technical and
information transmission via almost any economic effect when transmitting information through
communication channels. communication channels of different quality [21]. The
To ensure a return to C. Shannon’s classic work resulted in the creation of designs and algorithms for
statement in solving the problems of interference coding and decoding q stochastic codes with error
resistance but within the framework of a single correction. These codes are based on the formation of binary
information transformation algorithm. codes for communication channels of varying quality [22].
The following estimates are valid for these codes,
Interference-resistant coding is effective among the confirmed by theoretical studies and test statistics of
known methods of increasing the reliability of message practically implemented complexes [23]:
reception, but its use in a complex interference a) the code provides a predetermined probability
environment caused by the active influence of radio- (guaranteed probability of a decoding error) both when
electronic warfare means is limited because in such detecting and when correcting errors, related to the
conditions it can lead to an increase in the number of selected length of the q-symbol and the allowed number
errors at the decoding stage (the effect of error of corrected errors and relative to the maximum possible
multiplication) [16]. In this case, it is advisable to use the number of corrected errors t associated with the code
majority coding principle, which allows you to avoid the distance of the original binary code d,
effect of multiplying errors.
The majority principle consists of the fact that an odd
𝑡 = 𝑑−2 (1)
number of times the same message is sent to the channel,
and on the receiving side, code combinations of the same
This property can be used in duplex and simplex
name (or binary digits of the same name) are compared
communication channels.
with each other. At reception, the code combination (or
b) in a system with feedback [15], which employs a
bit) that has been received the most number of times is duplex data transmission channel, the error correcting code
chosen [17]. provides the following benefits (see the tables below):
The disadvantage of majority coding is the
redundancy of information, which increases in An increase in the relative (effective) speed of
proportion to the number of repetitions of the same information transmission, in comparison with
message (bit), therefore, when using it, it is necessary to the use of error-detecting codes, in the entire
take into account the time limits on the transmission of range of possible channel quality (that is,
messages. always) [24].
It is worth noting that for telemetry systems, A higher probability of successful decoding of
monitoring of remote objects, control systems of the code block in case of error correction, about
unmanned aerial vehicles, and other special purpose the error detection mode; at the same time, the
systems, in addition to increasing the reliability of data transmission channel acquires the
information reception, an especially important task is to properties of a real-time channel (“tempo”
ensure the information confidentiality of message channel) [25], where information is transmitted
transmission. One of the approaches that allows solving with a much smaller number of repetitions,
such tasks is the use of Combined Random Coding (CRC) which maximally satisfies the requirements for
[18]. combining data transmission and speech in one
The method of combined random coding, which is channel (digital speech transmission is critical
proposed in [19], involves the use of a combination of to repetitions) [26].
interference-resistant coding and a pseudo-random
change of the ensemble of code combinations— c) the encoder output signal has the character of
stochastic coding of information. At the same time, high “white noise,” because not one randomly selected (n, k)
64
code is used, but an ensemble of codes, where a code That is, for the code (16, 15) at q=232, the number of
change occurs at each successive code block [27]. binary encoding (decoding) operations is 16 per block of
d) in the presence of two or more values in the length 16×32 = 512 bits.
receiver that are a priori the same before coding on the The probability of successful decoding of the code block
transmitting side of the code blocks (first transmission (Pr(1)) from the first transmission and the effective speed
and repetition on request in the feedback system or [37] can be calculated using the following formula:
multiple transmission of the block in single-channel and 𝑘∗ 𝑁
𝑅 = (4)
multi-channel simplex systems—“copies” of blocks) 𝑛∗ 𝑁
there are algorithms for decoding copies that make it where Nr and Nt are the number of received and
possible to significantly increase the reliability of transmitted blocks, respectively.
message delivery in conditions of intense interference in For stochastic codes, there is a “copy” decoding
communication channels [27]. algorithm, when for two or more values of a code block of
As a result, it is claimed that the considered a stochastic code, including (n, n–1)—a code with the
construction of codes has a scope that coincides with the error detection that is the same during their
scope of the application of information systems and transmission, it is possible to carry out joint decoding of
telecommunications technology in general. the extended code with error correction [35, 36]. At the
Below are the main properties of error-correcting same time, the number of errors corrected in the block of
stochastic codes with a guaranteed probability of a the extended code significantly exceeds the number of
decoding error [28]. errors corrected in total in each block [38, 39], for
The code base is selected q = 232, which means, the example, if the source code corrects t = 2 errors, then
binary length of the q-symbol is 32 bits, and the number when the source block is repeated 2 times in an extended
of such symbols in the block is n and k. block, at least 6 twisted q-symbols are corrected, with
The probability of an error [29] in decoding stochastic three repetitions—at least 10 symbols, etc. At the same
q-codes does not depend on the type and nature of time, the guarantee of the reliability of the decoded
distortions and is mainly related to the value of q as in the information is preserved [40].
error detection mode (n, n-1)—code (with one redundant The copy decoding mode is most promising in simplex
symbol), and in error correction mode [30]. With the radio channels, particularly those with low-quality
selected base q, the probability of an error after decoding communication channels and intense radio interference. It
does not exceed any type of twists is also effective in duplex channels that employ joint
decoding of previously decoded and repeated blocks [41].
The temporal (or pace) characteristics of the code depend
𝑃 <𝑞 =2 < 10 (2)
on two factors: the effective transmission speed Ref and the
probability of the block being successfully delivered in the
The number of corrected errors t is related to the code
first (or subsequent) transmission [42].
distance d of the original binary code by the ratio t = d-2
and approximately corresponds to the number of
corrected errors of the Reed-Solomon code with the same
2.3. Comparative characteristics of
parameters n and k [31]. stochastic codes with error
Note that these codes correct errors in a probabilistic correction, and obtained results
sense. Specifically, errors are always corrected in multiples of hardware and software tests
of 1 (when the 1 q-symbol is twisted), while errors in
multiples of 2 or more are corrected with a controlled We conducted bench tests of stochastic codes using a
probability that depends on the code. It is important to note software simulator of communication channel errors. The
that decoding errors or failures are still possible, but in results of the set tests are presented in Tables 1, 2, and 3.
practical implementation, these probabilities can be reduced
Table 1
to desired values. The decoding and encoding of the
The results of bench tests of stochastic codes obtained
stochastic codes use only binary operations with q-symbols
using a software simulator of communication channel
[32, 33]. Since the number of decoding operations does errors
not depend on q, as q increases, the number of operations
per 1 bit of transmitted information decreases [34]. The Code
number of decoding operations with error correction per Р0
Рq
block of q-code can be of the order of magnitude of bn α=0 α = 0,3 α = 0,5
binary operations with length operands 1/2 0,02142 0,05942 0,11532
Channel quality
𝐿 = − log 𝑞, (𝐿 = 32), (3) 1/4 0,00895 0,02511 0,04958
where the coefficient b = 5–10. In the calculation of 1 1/8 0,00416 0,01173 0,02332
bit of transmitted information, the number of operations 1/16 0,0020 0,0057 0,0113
decreases by L times and has a value of less than 1 op/bit 1/32 0,0009 0,0028 0,0056
[35]. In the error detection mode for (n, n–1) q-code, the 1/64 0,0005 0,0014 0,0028
number of encoding and decoding operations is minimal 1/128 0,00025 0,00069 0,00139
and is n binary operations with operands of length L [36].
65
During the tests, different values of the probability of
twisting in the q-symbol channel (Pq) were used. The values
were chosen randomly and ranged from once every two
symbols (1/2) to once every four symbols (1/4), and soon. To
simplify the comparative analysis, we estimated the
probability of twisting the binary symbol (P0) for different
degrees of error grouping based on the given value of Pq. We
used the Purtov model with the coefficient “a” to estimate
this probability. Specifically, we considered three different
values of “a”: 0 for independent errors, 0.3 for weak
grouping in the leading channel, and 0.5 for strong grouping
in the radio channel.
Figure 3: Dependence of the effective speed (Ref) on the
probability of distortion in the q symbol channel Pq for codes
(8,2), (16,11), (15,11), (16,7) and (32,26)
Figure 1: Channel quality for different degrees of error
grouping with coefficient α
Figure 4: Dependence of the probability of reception of the
block Рr (1) value of the probability of distortion in the
channel of the q-symbol Pqfor codes (4,3), (8,7), (16,15),
(32,3) and (8,4)
Figure2: Dependence of the effective speed (Ref) on the
probability of distortion in the q symbol channel Pq for codes
(4,3), (8,7), (8,4), (16,15), and (32,3)
Figure 5: Dependence of the probability of reception of the
blockРr (1) on the value of the probability of distortion in
the channel of the q-symbol Pqfor codes (8,2), (16,11),
(15,11), (16,7) and (32,26)
66
Table 2
Dependence of the effective speed (Ref) on the probability of distortion in the q symbol channel (Pq)
Channel quality
Code
Рq 1/2 1/4 1/8 1/16 1/32 1/64 1/128
Effective speed Ref
(4,3) 0.0319 0.1711 0.3339 0.4647 0.5009 0.526 0.0319
(8,7) 0.3740 0.0573 0.2524 0.4825 0.5817 0.588 0.6852
(16,15) 0.0048 0.0076 0.1336 0.3206 0.5073 0.646 0.7012
(32,31) 0.0057 0.0082 0.0096 0.1772 0.3026 0.446 0.7351
(8,4)
0.0635 0.2681 0.3815 0.3982 0.4062 0.406 0.4062
(t = 2)
(8,2)
0.1051 0.1533 0.1559 0.1562 0.1562 0.156 0.1562
(t = 2)
(16,11)
0.1024 0.1157 0.4383 0.5627 0.6215 0.641 0.6406
(t = 2)
(15,11)
0.1085 0.1139 0.2597 0.5483 0.6345 0.680 0.6802
(t = 2)
(16,7)
0.0925 0.3539 0.3869 0.3906 0.3906 0.391 0.3906
(t = 4)
Table 3
Dependence of the probability of reception of the block Рr (1) value of the probability of distortion in the channel of the
q symbol Pq
Channel quality
Code
Рq 1/2 1/4 1/8 1/16 1/32 1/64 1/128
The probability of receiving a block Pr(1)
(4,3) 0.0175 0.3684 0.6315 0.8070 0.8594 0.9298 0.962
(8,7) 0.0785 0.2634 0.3428 0.600 0.7142 0.8380 0.963
(16,15) 0.0874 0.1739 0.2222 0.3777 0.5333 0.7333 0.899
(32,31) 0.0038 0.0074 0.0096 0.1772 0.3026 0.6874 0.735
(8,4)
0.2970 0.695 0.9350 0.948 0.9735 0.9805 0.997
(t = 2)
(8,2)
0.6816 0.9825 0.9961 0.9981 0.9984 0.9989 0.999
(t = 2)
(16,11)
0.1818 0.2308 0.6615 0.7692 0.9538 0.9673 0.972
(t = 2)
(15,11)
0.1598 0.2763 0.4769 0.8153 0.9076 0.9153 0.951
(t = 2)
(16,7)
0.2190 0.8571 0.9810 0.9905 0.9917 0.9946 0.998
(t = 4)
3. Conclusion Technology Trends on the Smart Industry and the
Internet of Things, vol. 3149 (2022) 107–117.
Our results demonstrate that codes with natural [2] V. Buriachok, V. Sokolov, P. Skladannyi, Security
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