=Paper=
{{Paper
|id=Vol-3826/paper4
|storemode=property
|title=Algorithms for reliable permutation transmission protocols in noisy communication channels
|pdfUrl=https://ceur-ws.org/Vol-3826/paper4.pdf
|volume=Vol-3826
|authors=Emil Faure,Alimzhan Baikenov,Artem Skutskyi,Denys Faure,Olga Abramkina
|dblpUrl=https://dblp.org/rec/conf/cpits/FaureBSFA24
}}
==Algorithms for reliable permutation transmission protocols in noisy communication channels==
https://ceur-ws.org/Vol-3826/paper4.pdf
Algorithms for reliable permutation transmission
protocols in noisy communication channels⋆
Emil Faure1,2,*,†, Alimzhan Baikenov3,†, Artem Skutskyi1,†, Denys Faure4,†
and Olga Abramkina5,†
1
Cherkasy State Technological University, 460 Shevchenko ave., 18006 Cherkasy, Ukraine
2
State Scientific and Research Institute of Cybersecurity Technologies and Information Protection, 3 M. Zaliznyaka str., 03142
Kyiv, Ukraine
3
Almaty University of Power Engineering and Telecommunications named after Gumarbek Daukeyev, 126 Baitursynov str.,
050013 Almaty, Kazakhstan
4 Odesа Polytechnic National University, 1 Shevchenko ave., 65044 Odesa, Ukraine
5 International University of Information Technology, 34A Manasa, 050040 Almaty, Kazakhstan
Abstract
The existing approaches to frame synchronization of non-separable factorial code, as well as the reliable
transmission of its codewords, form the basis for creating a protocol for reliable permutation transmission
in conditions of intense channel noise and, accordingly, of a high probability of bit error. This study
considers a simplex data transmission system. For such a system, algorithms for frame synchronization of
permutations, as well as reliable transmission of permutations have been developed, providing processing
of fragments of bit sequences with a permutation length of M. A key feature of the proposed approaches is
that they are designed for situations where the initial moment of the transmitter’s syncword transmission
is unknown. It has been shown that to ensure the required level of false synchronization, the number of K
blocks, each consisting of l fragments, needs to be increased. An assessment of the probabilistic indicators
of the process of transmission and reception of information has been performed. Computer simulation
modeling has been carried out, confirming the theoretical results.
Keywords
permutation, synchronization, error correction, security, reliability, factorial coding, protocol, data
processing algorithm 1
1. Introduction conditions for the code frame synchronization using the
operating signal.
The theory of non-separable factorial data coding [1, 2] At the same time, modern conditions dictate the need [3,
allows using permutations as a transport mechanism in 11–14] to achieve high-reliability indicators in difficult
communication systems with short packets [3–5], and also signal propagation conditions [15–18]. Three-pass
to implement joint protection of transmitted data from cryptographic protocols [19–22], in particular, based on
communication channel errors and unauthorized access [6]. permutations [23], deserve special attention in this context.
Paper [1] shows that the codewords of a non-separable Previously conducted studies on the possibility of using
factorial code belong to a subset of the set of permutations non-separable factorial coding in conditions of a high
of length M . The permutation elements are encoded probability of bit error in a communication channel made it
by a fixed-length binary code with a codeword length possible to develop:
lr log 2 M . Then the syncword length is equal to
Methods of frame synchronization for non-
n lr M . separable factorial codes [24–27].
Due to the redundancy of the information carriers, Method for reliable permutation transmission in
permutations, used, and non-separable factorial codes allow short-packet communication systems [28].
detecting and correcting communication channel errors [7–
10]. In addition, the permutation structure creates The developed approaches and methods are effective. At the
same time, the frame synchronization methods are based on
knowledge of the initial moment of the syncword bits
CPITS-II 2024: Workshop on Cybersecurity Providing in Information 0000-0002-2046-481X (E. Faure);
and Telecommunication Systems II, October 26, 2024, Kyiv, Ukraine 0000-0002-6490-3159 (A. Baikenov);
∗
Corresponding author. 0000-0002-8632-1176 (A. Skutskyi);
†
These authors contributed equally. 0009-0002-9741-6282 (D. Faure);
e.faure@chdtu.edu.ua (E. Faure); 0000-0003-0137-1252 (O. Abramkina)
a.baikenov@aues.kz (A. Baikenov); © 2024 Copyright for this paper by its authors. Use permitted under
Creative Commons License Attribution 4.0 International (CC BY 4.0).
a.skutskyi@ chdtu.edu.ua (A. Skutskyi);
highdensityarts@gmail.com (D. Faure);
o.manankova@iitu.edu.kz (O. Abramkina)
CEUR
Workshop
ceur-ws.org
ISSN 1613-0073
40
Proceedings
reception, which is not always possible. In addition, the joint The frame synchronization method proposed in [25, 26]
use of synchronization and reliable transmission procedures involves the sequential transmission of a syncword into the
in one protocol has not been studied. communication channel. For example, for M 8 , such a
The purpose of this study is to develop algorithms for a syncword is the permutation
protocol of reliable transmission of permutations for 000,001,111,011,010,101,100,110 , up to its circular
simplex data transmission systems with non-separable
factorial coding under conditions of high noise intensity in shift by a number of bits that is a multiple of lr 3 , bit
the communication channel. inversion, and the reverse order of their sequence.
Let us assume that high noise intensity results in the
2. Sliding window algorithm for a receiver being unable to determine the initial moment of the
transmitter’s syncword. In this case, the algorithm for
frame synchronization system identifying the boundaries of the syncword is modified
The first step of the protocol involves establishing frame slightly.
synchronization for the transmitted permutations. For this Recall that according to [26], the sufficient number of
purpose, a frame synchronization method [25, 26] will be accumulated fragments to ensure the minimum value of the
used. This method employs as a syncword a permutation with probability of correct synchronization Ptrue _ min is chosen as
the maximum value of the minimum Hamming distance from
the minimum value of l , at which the probability of correct
its binary representation to all its circular shifts.
The receiver accumulates K blocks of l fragments of synchronization for K 1 is not less than the specified
M symbols from the communication channel, followed by Ptrue _ min . Paper [26] denotes this value as lmax 1 . In this
majority [29, 30] and correlation processing [31–33] of the paper, we will denote it as lmax .
accumulated fragments. The values of K and l change Based on the above and the fact that the initial moment
according to the methodology defined in [26]. A pre- of syncword transmission is unknown, the receiver will use
established minimum threshold for the probability of a sliding window with a width of lmax fragments to search
correct synchronization Ptrue determines the sufficient
for synchronization (Figure 1).
number of accumulated fragments.
Figure 1: Diagram of the use of a sliding window consisting of lmax fragments
Thus, the receiver, shifting the sliding window 1 bit to the processing of lmax received fragments. However, the
right, continuously analyses lmax fragments received from receiver’s lack of knowledge about the initial moment of
the communication channel, attempting to establish frame syncword transmission leads to the following.
synchronization. It is evident that, in this case, the dynamic Since the receiver has to constantly “listen” to the
adjustment of K and l values is meaningless. channel, in the absence of a signal from the transmitter, only
The mathematical model of the syncword reception noise is present in the sliding window. Accordingly, the
process will also differ from that presented in [26]. probability of bit error is equal to 0.5
After the transmitter begins to transmit service signals
2.1. Probabilistic metrics of the frame for the clock (not considered in this study) and frame
synchronization system synchronization into the communication channel,
fragments with syncwords begin to appear in the sliding
The probabilities of correct and false synchronization window of the receiver synchronization system (Figure 2).
depend on the probability of bit error p0* after majority
Figure 2: Diagram of the stage of filling the sliding window with data from the source
Let there be L bits of the source syncword in the sliding shaded areas contain only noise bits (error probability is
window (Figure 2). To provide a clearer view of the majority 0.5), while the unshaded areas contain bits of the source
reception process of the accumulated bits, we represent the syncword (with an error probability of p0 ).
fragments in the sliding window as shown in Figure 3. The
41
Figure 3: Diagram of majority reception of accumulated bits
From the accumulated fragments, a refined sequence R is Cli l 0.5 lmax l1
computed by the majority, in which some errors (if any) are max 1
lmax l1
p0
* l1
l1 j
, (2)
j l
corrected. i0 Cl1 p0 1 p0
j j
It should be noted that the number of bits of the source max 1 2 i
syncword present in the sliding window of the receiver’s
L
synchronization system may not be a multiple of the where l1 is the number of complete fragments
n
codeword length lr log 2 M , as demonstrated in Figure
containing only bits of the source syncword (which may be
3. Therefore, the probability of bit error in the refined affected by errors).
sequence after majority processing of lmax received Estimates (1) and (2) are formed by replacing the
fragments can be estimated as follows: fragment that contains noise bits and bits of the syncword
l 3 with a fragment that contains only noise bits, as well as
for L n max : taking into account that p0 0.5 .
2
Cli p0i 1 p0 l1 i Paper [34] defines that for M 8 and p0 0.4 , the
1
l1
p
* value of lmax 75 . For parameters M 8 and p0 0.4 , the
lmax l1 (1)
lmax l1
j l
Clmax l1 0.5
0
i0
j
graph showing the dependence of the estimated probability
max 1 2 i
of bit error in the refined sequence R on the value of
l 3
and for L n max : L 0;75 24 is presented in Figure 4.
2
Figure 4: The estimated probability of bit error in the refined sequence on the number of syncword bits in the sliding
window for M 8 and p0 0.4
42
The graph has a stepped nature due to the simplifying upper Proof.
estimates (1) and (2). At the same time, at L 0 , the value We will use Figure 3. Since the number of bits of the
is p0* 0.5 , and at L 1800 , the value is p0* 0.0396 . source syncword L in the sliding window is generally not
To calculate the exact value of the obtained probability L
a multiple of the fragment length lr M , in L n
of bit error in the refined sequence, the following statement n
can be used. cases out of n в the formation of a bit based on the majority
Theorem 1. The probability of bit error in the refined L
sequence R after receiving L bits of the source syncword principle involves 1 bits of the source syncword and
n
is equal to:
L L
lmax 1 bits of noise. Accordingly, in n L n
l 1 n
n
1. for L n max :
2 cases out of n the formation of a bit based on the majority
L
L principle involves bits of the source syncword and
p 1 l1
*
0 n
n
L
C p 1 p0 l1 i
i i lmax bits of noise. Consider that a bit error occurs
l1 l1 0 n
lmax l1
lmax l1
when the number of errors in the corresponding bits of the
j l
i 0 Clmax l1 0.5
j
max 1 2 i l 1
, (3) accumulated fragments is not less than the value of max
L 2
l1 . Then we can obtain the necessary expressions (3) and (4)
n
l 1
Cli 1 p0i 1 p0 l1 1 i for the bit error probability for L n max and
1
l1 1 2
l l 1
lmax l1 1
;
l 1
max 1
j l
i 0 Clmax l1 1 0.5
j
L n max respectively. ■
max 1 2 i 2
lmax 1 The graph showing the dependence of the probability of
2. for L n :
2 bit error in the refined sequence R on the value of
L 0,75 24 at M 8 and p0 0.4 is presented in
L
p0* l1 Fig. 5.
n
C 0.5 max 1
i l l 1
lmax l1 1 lmax l1 1
l1 1
l1 1 j
j l
i 0 Cl1 1 p0 1 p0
j j
max 1 2 i (4)
,
L
1 l1
n
Cli l 0.5 lmax l1
max 1
lmax l1
l1
l1 j
.
i 0
j l 1 2 i
Cl1 p0 1 p0
j j
max
Figure 5: The probability of bit error in the refined sequence on the number of syncword bits in the sliding window for
M 8 and p0 0.4
43
To estimate the probabilities of correct and false The graphs of functions (5) and (6) as a function of L at
synchronization, we will use the expressions defined in [26]: M 8 and p0 0.4 are presented in Error! Reference
dlim
Ptrue n, d lim , p0 , L Cnv p0* 1 p0*
v nv
source not found. and Figure 7.
v 0 (5) Figure 6 and Figure 7 show that the probability of
, correct synchronization increases from 0.0033 for L 0 to
Pfalse n, d lim , p0 , L 0.9997 for L 1800 , while the probability of false
dij v dij dlim C w p* v w synchronization decreases from 0.076 for L 0 to
n 1
v
n dij
Cdij
0
(6) 1.198 106 for L 1800 .
n w v
j 1 v dij d lim w 0 1 p0 *
For experimental confirmation of the dependencies
Ptrue 24,5,0.4, L and Pfalse 24,5,0.4, L , a computer
.
For the syncword 000,001,111,011,010,101,100,110 simulation modeling of the frame synchronization system
operation process was performed and the relative
expression (5) takes the form:
5
frequencies Wtrue 24,5, 0.4, L and W false 24,5,0.4, L of
Ptrue 24,5, p0 , L C v
24 p 1 p
* v
0
* 24 v
0
, and
both correct and false synchronization were determined. For
v 0
Error! Reference source not found. transforms to each value, 1000 tests were performed. The graphs of the
studied dependencies are presented in Figure 8 and Figure
v 7 C w p* v w
12 9.
Pfalse 24,5, p0 , L 19 C12v
12 0
v7
w 0 1 p0 * 24 v w
14
v 9
2 C14v C10w p0* 1 p0*
v w 24 v w
v 9 w 0
16
v 11
2 C16v C8w p0* 1 p0*
v w 24 v w
.
v 11 w 0
Figure 6: Estimated probability of correct synchronization on the number of syncword bits in the sliding window for M 8
and p0 0.4
Figure 7: Estimated probability of false synchronization on the number of syncword bits in the sliding window for M 8
and p0 0.4
44
Figure 8: Dependencies Ptrue 24,5,0.4, L and Wtrue 24,5, 0.4, L
Figure 9: Dependencies Pfalse 24,5,0.4, L and W false 24,5,0.4, L
A comparative visual assessment of Ptrue 24,5,0.4, L and P
The probability true _ final of false synchronization
Wtrue 24,5,0.4, L , as well as Pfalse 24,5,0.4, L and after receiving L bits of the syncword is estimated
W false 24,5,0.4, L , indicates the correctness of expressions from above by the sum of the probabilities of false
synchronization for all values of the accumulated
(5) and (6), as well as the adequacy of the developed model.
By analogy with [26]: syncword bits less than L :
Ptrue _ final n, dlim , p0 , L Ptrue n, d lim , p0 , L (8)
.
Ptrue _ final
The probability of correct Thus, the probability of false synchronization turns out
synchronization after receiving L bits of the to be unacceptably high. Figure 10 presents the results of an
syncword is estimated below by the probability: experimental study of the relative frequency of true Wtrue
Ptrue _ final n, d lim , p0 , L Ptrue n, d lim , p0 , L (7) and false W false synchronization for M 8 and p0 0.4 .
;
Figure 10: Relative frequency of correct and false synchronization on the number of syncword bits in the sliding window
for M 8 and p0 0.4
45
2.2. Parameters of the algorithm for L 1
p0* l1
reducing the probability of false K n
synchronization Cli l 1 0.5l l1 1
1
l l1 1
One way to reduce the probability of false synchronization l1 1
l1 1 j
j l
i0 Cl1 1 p0 1 p0
j j
is to increase the number of K blocks of l fragments. 1 2 i
(12)
The approach to increasing the K value [26] involves L 1
receiving K blocks consisting of l fragments of n bits. 1 l1
K n
For each block, the refined sequences Rk , k 1, K are Cli l 0.5l l1
1
l l1
independently calculated. If all sequences Rk , k 1, K l1
l1 j
,
j l
i 0 Cl1 p0 1 p0
j j
correspond to the same syncword shift, the decision device
1 2 i
of the frame synchronization system decides to establish
L
synchronization. where l1 .
According to [26, 34], the probability of false K n
To ensure that the probability of false synchronization
synchronization for K blocks of l fragments
does not exceed its threshold value Pfalse _ max , that the
Pfalse n, d lim , p0 , L, l , K
K
probability of correct synchronization is at least its
n dij p0
dij
n 1 v dij dlim C w * vw
(9) threshold value Ptrue _ min , and that correct synchronization
Cdij v
n v w
j 1 v dij d lim
w0 1 p
0 * occurs as quickly as possible, it is necessary to determine a
pair of K and l values, for which
monotonically decreases with the increase in both K and
Pfalse _ final n, d lim , p0 , L, l , K Pfalse _ max ,
l values. On the other hand, the probability of correct
synchronization Ptrue _ final n, d lim , p0 , L, l , K Ptrue _ min , and K l takes its
Ptrue n, d lim , p0 , L, l , K minimum value. In this case, Pfalse _ final n, d lim , p0 , L, l , K
Ptrue
K
n, d lim , p0 , L and Ptrue _ final n, d lim , p0 , L, l , K are estimated as follows:
K
(10)
nv Pfalse _ final n, dlim , p0 , L, l , K
dlim
Cnv p0* 1 p0*
v
v 0 L (13)
Pfalse n, dlim , p0 , j, l , K ;
also monotonically decreases with the increase in K , but j 0
increases with the increase in l . At the same time,
Ptrue _ final n, d lim , p0 , L, l , K
L 0; lr M l K , and the probability of bit error in the (14)
Ptrue n, d lim , p0 , L, l , K .
refined sequences Rk after receiving L syncword bits from
Let ltrue _ min K be the minimum number of fragments
the source can be estimated by modifying expressions (3)
and (4) as follows: in each of the K blocks, at which
Ptrue _ final n, d lim , p0 , Lmax K , ltrue _ min K , K Ptrue _ min ,
L l 1
3. for n : Lmax K lr M ltrue _ min K K . Then the task of
K 2
determining the pair of K ; l values consists of the
L 1 following:
p0* 1 l1
K n
5. n , dlim , p0 values are specified, and K 1 is
Cli p0i 1 p0 l1 i
1
l1 accepted;
l l
l l1
ltrue _ min K value is calculated to satisfy
1
j l
i 0 Cl l1 0.5
j 6.
Ptrue _ final n, d lim , p0 , Lmax K , ltrue _ min K , K
1 2 i
(11)
L 1
l1 Ptrue _ min , where Lmax K lr M ltrue _ min K K
K n
;
Cli 1 p0i 1 p0 l1 1 i
1
l1 1 7. if Pfalse _ final n, dlim , p0 , Lmax K , ltrue _ min K , K
l l 1
l l1 1
;
Pfalse _ max is satisfied, the pair of K ; ltrue _ min K
1
j l
i 0 Cl l1 1 0.5
j
1 2 i
values can be used as necessary parameters for the
L l 1 synchronization procedure;
4. for n :
K 2 8. if Pfalse _ final n, dlim , p0 , Lmax K , ltrue _ min K , K
Pfalse _ max , an algorithm for determining K ; l
transits to step 2, incrementing the value of K by
one.
46
We will compute the ltrue _ min K values for K 1, 2,3,... Table 1. In this case, we will assume Ptrue _ min 0.9997 ,
for the specified parameters and summarize the results in Pfalse _ max 0.0003 .
Table 1
Characteristics of the algorithm for reducing the probability of false synchronization
Characteristics Value
K 1 2 3 4
ltrue _ min K 75 81 83 85
Pfalse _ final n, dlim , p0 , Lmax K , ltrue _ min K , K 1 0.182 7.4 ·10-4 2.9·10-6
Pfalse _ final n, dlim , p0 , L, ltrue _ min K , K indicate the behaviour of the estimate (13) and demonstrate
Graphs of dependencies the numerical values of the estimate (13) given in Table 9.
Pfalse _ final
on L for various K , presented in Fig. 11,
Figure 11: Estimation of the probability of false synchronization on the number of syncword bits in the sliding window for
M 8 and p0 0.4 for different K values
Based on the values presented in Table 1, for the considered 2. Analogous to the procedure of frame
example with M 8 , p0 0.4 , Ptrue _ min 0.9997 , and synchronization, for each letter, the transmitter
sends and the receiver receives, accumulates, and
Pfalse _ max 0.0003 , it is sufficient to choose the K 4 value.
analyses l fragments of n-bit each.
Another potential method for reducing the probability 3. For each letter, the refined sequence Rj, j ∊ [1,N], is
of false synchronization could be to decrease the maximum computed using the majority processing of the
distance dlim to the syncword shifts used in identifying the received bits.
refined sequence R. However, reducing dlim leads to a 4. For each refined sequence Rj, the Hamming
decrease in the probability of correct synchronization, distances to the letters used by the source are
which necessitates increasing the accumulation coefficient calculated. If the distance does not exceed dlim, the
l. This approach may be effective; however, it is not jth symbol of the word W is associated with the
considered in this paper. corresponding alphabet letter.
5. If the received word is a permutation of the letters
3. Algorithm for reliable of the alphabet used by the source, and this
transmission of permutations permutation is used by the source, the word is
accepted, and the process of recognizing the next
Frame synchronization establishment initiates the next word begins.
stage of the protocol, the reliable transmission of
permutations. It should be noted that the function
The algorithm for reliable transmission of permutations Ptrue _ final n, d lim , p0 , L, l , K monotonically increases with an
based on method [28] consists of the following:
increase in L and, for example, at L 4432 the value is
1. The transmitter sends into the communication Ptrue 24,5,0.4, L,85,4 0.5 , while at L 5640 the value is
channel a permutation-word W consisting of N
Ptrue 24,5,0.4, L,85,4 0.9 . Considering that
symbols-letters Lj, 1 ≤ j ≤ N. Each letter is a
circular bit shift of permutation π of length M, Pfalse _ final 24,5,0.4, L,85, 4 2.9 10 , this leads to the
6
which has the maximum value of the minimum conclusion that there will be a 90% probability of correct
Hamming distance from its n-bit binary synchronization being established after receiving 5640
representation to all its circular shifts.
47
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