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. i0   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 i0   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 nv 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  106 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 v7   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  0.5l  l1 1    1 l  l1 1  One way to reduce the probability of false synchronization   l1 1 l1 1 j    j  l i0   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  0.5l  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 * vw   (9) threshold value Ptrue _ min , and that correct synchronization     Cdij  v   n v  w  j 1  v  dij  d lim  w0  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)  nv  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 fragments containing bits of the source’s synchronization Conference on Acoustics, Speech and Signal word. Since 8160 – 5640 = 2520, n = 24, in 90% of cases, the Processing (ICASSP) (2018) 6608–6612. doi: reliable reception procedure for the permutation, which 10.1109/ICASSP.2018.8461650. follows the synchronization procedure and is initiated by [4] C. Feng, H. Wang, Secure Short-Packet the fact of establishing synchronization, will utilize the Communications at the Physical Layer for 5G and shifted boundaries of the data block used for transmitting a Beyond (2021). doi: 10.48550/arXiv.2107.05966. single permutation (synchronization was completed earlier, [5] C. Feng, H.-M. Wang, H. V. Poor, Reliable and Secure the recognition of the permutation began sooner, while the Short-Packet Communications, IEEE Trans. Wireless synchronization words are still being transmitted in the Commun. 21(3) (2022) 1913–1926. doi: channel). 10.1109/TWC.2021.3108042. Such desynchronization leads to a decrease in the [6] R. Aleksieieva, et al., Software Tool for Ensuring Data efficiency of the permutation recognition procedure, which Integrity and Confidentiality Through the Use of needs to be compensated for with additional procedures. 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