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Article

A Construction Method for the Random Factor-Based G Function

by
Yongxin Feng
,
Jiankai Su
and
Bo Qian
*
College of Information Science & Engineering, Shenyang Ligong University, Shenyang 110159, China
*
Author to whom correspondence should be addressed.
Appl. Sci. 2024, 14(22), 10478; https://doi.org/10.3390/app142210478
Submission received: 23 September 2024 / Revised: 31 October 2024 / Accepted: 9 November 2024 / Published: 14 November 2024
Browse Figures
Figure 1
<p>Frequency transfer relationship.</p> "> Figure 2
<p>Frequency state transition grid diagram.</p> "> Figure 3
<p>Construction diagram of AES algorithm.</p> "> Figure 4
<p>Schematic diagram of iterative decomposition.</p> "> Figure 5
<p>Structure diagram of the RFGF.</p> "> Figure 6
<p>The inverse G function algorithm.</p> "> Figure 7
<p>Frequency sequence power spectrum.</p> "> Figure 8
<p>Frequency statistics results of the equal distribution test.</p> "> Figure 9
<p>Frequency statistics histogram of two-dimensional continuity test by four methods.</p> "> Figure 10
<p>Single-frequency path statistics.</p> ">
Versions Notes

Abstract

:
In consideration of the prevailing methodology for constructing G functions, there are certain limitations such as fixed change rules and restricted flexibility when producing frequency-hopping sequences. This paper introduces a novel construction method for the Random Factor-based G function (RFGF). This approach incorporates random factors to dynamically divide the frequency set into equal intervals and randomly selects the frequency hopping frequency within each subset. This effectively reduces the correlation between adjacent frequency-hopping frequencies, enhancing the randomness of the sequence and the system’s anti-interference performance. Furthermore, this method utilizes chaotic sequences to scramble data information, further strengthening the security of the information. The experimental results demonstrate that the frequency-hopping sequence generated by this proposed G function construction method outperforms the sequence generated by the time-varying iterative decomposition in terms of randomness, uniformity, and two-dimensional continuity. Specifically, under the same parameter conditions, the two-dimensional continuity is improved by 36.87%.

1. Introduction

As the communication spectrum becomes increasingly crowded, secure communication to avoid interference and interception has become a hot topic recently [1,2]. Spread-spectrum technology expands the frequency spectrum of the signal to a wider range, making it less susceptible to interference or interception during transmission. It has been widely used in military and civilian secure communication [3,4].
The differential frequency-hopping technique is a specific form of spread-spectrum communication. It changes the signal frequency in continuous time intervals, which provides more flexibility and unpredictability in the communication signal [5,6,7]. Since differential frequency-hopping technology offers high interference resistance and security, this wireless communication technology has extensive application potential in both military and civilian fields [8].
In recent years, machine learning research has rapidly developed in multiple fields, becoming an important driving force for technological progress and innovation [9,10,11]. However, in differential frequency hopping communication systems, the construction method of the G function remains a fundamental and critical aspect [12,13]. A large number of construction methods have been proposed by scholars around the world. Fahey S et al. [14] proposed a method for controlling the G function by cascading multiple pseudo-random number generators and dynamically adjusting the period, number, delay, and other parameters of the generators. The performance of the sequence improves with an increase in the number of pseudo-random number generators; however, the optimal number of generators has yet to be determined. Furthermore, Rui Che et al. [15] proposed a method for generating high-order differential frequency-hopping sequences, which entails a trade-off between greater computational and storage resources and the randomness and security of the frequency-hopping sequence. Concurrently, this scheme requires the incorporation of a trusted entity to authenticate the user cluster, thereby exacerbating the overall complexity of the system. Yin et al. [16] developed an encryption scheme for pseudo-random sequences using logical chaotic mapping, which improved two-dimensional uniformity but had a negative impact on other system performance. Meanwhile, Liu et al. [17] proposed an enhanced differential frequency hopping (EDFH) framework that integrates training signals into traditional transmission processes and designs a mixed signal matched filter (CMF) to optimize the receiver’s processing of user and interference signals. Ning et al. [18] proposed a G function design scheme based on iterative decomposition. This scheme differs from the previous one in that it modifies the frequency transfer relationship over different time periods by decomposing the frequency pair set. However, this approach encounters a challenge in accurately decoding the de-mapping algorithm, which in turn increases the bit error rate.
The existing G function is constrained by predefined conditions, resulting in limited flexibility in generating frequency-hopping sequences. To address this limitation, this paper proposes the Random Factor-Based G function. By introducing a random factor, the method dynamically adjusts the position of frequencies within subsets, breaking the fixed frequency-hopping pattern. This approach enhances system adaptability and improves anti-interference capabilities.
The main contributions of this paper can be summarized as follows:
  • We propose the RFGF, which significantly enhances the two-dimensional continuity of frequency-hopping sequences and improves the anti-interference capabilities of communication systems.
  • The RFGF increases the unpredictability of frequency-hopping sequences by introducing random disruptions within each subset’s hopping patterns. This approach ensures that frequency selections are more random rather than sequentially ordered.
  • The effectiveness of RFGF is demonstrated via simulations. It can achieve better sequence performance as compared to other schemes.
The rest of this paper is structured as follows. Section 2 provides a review of relevant literature and theoretical background. In Section 3, the principles and limitations of commonly used G functions are discussed. Section 4 presents the detailed methodology, including the design and implementation of the RFGF. The experimental results are analyzed and compared to existing methods in Section 5. Finally, Section 6 concludes the paper by summarizing the findings and suggesting directions for future research.

2. Mechanisms of the G Function

Compared to the traditional frequency-hopping communication method that uses a pseudo-random code sequence to create the frequency-hopping sequence, the differential frequency-hopping signal utilizes the G function to generate the sequence [19,20,21]. This sequence dictates the order in which the signal jumps between frequencies, directly affecting the system’s ability to resist interference and the efficiency of data transmission. The current hopping frequency Fn is determined by the previous hopping frequency Fn−1 and the information data Xn being transmitted [22]. This relationship can be expressed as follows:
F n = G F n − 1 , X n
Figure 1 illustrates the transfer relationship between the various frequency points within the frequency set. Each node represents a frequency hopping point. The number of data bits (BPH) carried by the carrier in each hop allows each node to derive two BPH frequency point branches, which is called the fan-out coefficient [23]. With BPH = 2, there are four possible mapping routes available for each frequency hopping point. The current hopping carrier frequency point is determined by selecting one of the mapping routes based on the transmitted data information on each branch.
The receiver is responsible for identifying the frequency of each differential frequency-hopping signal and for demodulating the transmitted information data Xn. This is achieved by utilizing the inverse G function to demodulate the adjacent hopping frequencies Fn and Fn−1. The relationship expression is as follows:
X n = G − 1 F n , F n − 1
The G function in a differential frequency hopping system determines the frequency-hopping rules and the performance of the frequency-hopping sequence, which in turn affects the communication capability of the communication system [24,25]. Therefore, a good design of the G function is the key to achieving high-performance communication.

3. Analysis of the Principles and Limitations of Commonly Used G Functions

3.1. G Function Based on Congruence Theory

The G function based on congruence theory can be expressed as follows [26]:
F n = G F n − 1 , X n , N
where N represents the number of frequency points. This function generates the frequency-hopping sequence for the current hop based on the frequency-hopping sequence of the previous hop and the information data carried by the current hop. It uses a mapping rule and modulo N to generate the sequence.
The minimum free distance is the minimum Hamming distance between two frequency-hopping sequences, which is defined as the number of different elements between them. Since the sequence is generated within the modulus N, its period can be at most N elements. The mapping rule of Xn is fixed, so the difference in the sequence only affects the initial state. When the difference in the transmitted data is small, the generated sequence will have many identical elements in a short period. This can lead to overlap and repetition, especially when N is small.
While the G function method based on congruence theory is simple and easy to use, its low minimum free distance can result in high similarity between frequency-hopping sequences, which may compromise security and anti-interference capabilities.

3.2. G Function Based on M Sequence

The G function based on m sequence is expressed as follows [27]:
F n = G F n − 1 , X n , m
where m represents the m sequence. The current hopping frequency is not only related to the current information data and the previous hopping frequency, but also to the selected m sequence.
The frequency-hopping points are divided into b subsets, and each log2b bit in the m sequence is used to select a subset. Specifically, when b = 4, each two bits of the m sequence (00, 01, 10, 11) correspond to frequency subsets (P1, P2, P3, P4), respectively. The frequency-hopping sequence generated by this method uses subset transformation to randomly jump to different subsets, thereby increasing the frequency point interval and improving the randomness and uniformity of the frequency-hopping sequence.
However, the periodicity and specific correlation of the m sequence can lead to uneven frequency subset selection. This means that some subsets may be selected multiple times in a specific period, while the frequency points of other subsets are selected less frequently, resulting in uneven probabilities from a certain frequency-hopping point to other frequency-hopping points outside the subset. Additionally, if the number of frequency points in a subset is much larger than the number of frequency points actually used (known as the fan-out coefficient), the probability of going from a certain frequency-hopping point to other frequency-hopping points in the subset is also unequal.
Therefore, while the frequency-hopping sequence generated by the G function based on the m sequence does increase the randomness and anti-interference ability of the frequency-hopping sequence to a certain extent, it may also affect the two-dimensional continuity performance of the sequence due to the unequal probability distribution.

3.3. G Function Based on State Grid

The frequency state grid diagram is shown in Figure 2. The G function, based on the state grid [7], considers the current working frequency as a state. The transmitted data are processed using a linear and finite state shift register, resulting in log2N binary numbers. These binary numbers are then converted into N-base numbers, with each N-base number corresponding to N states of the frequency sequence. This creates a grid diagram of the system frequency, known as the frequency-hopping sequence. In the decoding process, the Viterbi decoding method is used, taking advantage of the correlation between frequencies. The more frequency points in the frequency set, the deeper the decoding depth, and the greater the difference in the metric value of each path. This allows for the identification of the path with the smallest path metric value, which can then be used to demodulate the data.
While this method has shown some success in improving the bit error rate performance of the receiving end decoding, it requires a large amount of system calculation and does not improve the fan-out coefficient. Additionally, the generated frequency-hopping sequence has limited variation and poor randomness performance.

3.4. G Function Based on AES Password

Figure 3 illustrates construction diagram of AES algorithm. In the process of generating a frequency-hopping sequence based on the G function of AES password [28,29], the initial frequency point and key are randomly determined. The initial frequency point is then converted into a 128-bit binary sequence, with the upper 122 bits set to zero. This sequence is then input into the AES algorithm along with the 128-bit key to obtain the output ciphertext. Next, the lower six bits of the ciphertext are taken and XORed with the data information to be transmitted. The resulting binary data are then converted into decimal frequency point information, which is then converted into a 128-bit binary sequence. This sequence is then input into the next level of the AES algorithm along with the key. This process is repeated to obtain frequency points for each level, creating a frequency-hopping frequency set.
Although this method has the advantage of variable and equal key length, its processing process is complex, leading to increased computational overhead and implementation difficulty. Additionally, the public nature of the algorithm means that if attackers obtain the key for a certain round, they can use this information to launch further attacks and compromise the security of the system.

3.5. The Time-Varying Iterative Decomposition G Function

The G function, based on iterative decomposition [18], expresses the relationship between the chaotic sequence Yn, the RS code RS, and the parameter controlling the fan-out coefficient of each sub-block Pn, as follows:
F n = G F n − 1 , X n , Y n , m , R S , P n
This function is built upon the G function based on the m sequence. The data information Xn is scrambled using both the m sequence and the chaotic sequence Yn. The frequency point interval is controlled by the RS code, while the frequency hopping subset is controlled by the m sequence. This method divides all possible frequency transfer pairs into blocks, allowing for changes in the frequency transfer relationship over time.
Figure 4 is a schematic diagram of this iterative decomposition process. This approach ensures the time-varying nature of the G function and improves the performance of the frequency-hopping sequence to a certain extent. However, it does not specify the optimal decomposition order or decomposition sub-block method for different channel environments. Additionally, the de-mapping result of this method is not unique, requiring an additional receiving algorithm, which may hinder practical application.

4. The Random Factor-Based G Function

To enhance the irregularity and reduce the correlation between adjacent hopping sequences, this paper proposes a construction method for the RFGF. This method involves using chaotic sequences to scramble the transmitted data and incorporating random factors to control the selection of frequencies within the subset. This results in increased randomness and disorder in frequency changes. Additionally, a corresponding inverse G function is designed to recover the baseband data.

4.1. The Positive G Function Algorithm

Figure 5 illustrates the structure diagram of the RFGF.
The RFGF determines the current hopping frequency Fn through the previous hopping frequency Fn−1, the current hopping data information Xn, the m sequence, the chaotic sequence Yn, and the random factor R. The relationship between the variables can be expressed as follows:
F n = G F n − 1 , X n , Y n , m , R
Firstly, it encodes the baseband data xn in order to generate the transmission data Xn.
y n = − y n − 1 2 − y n − 1 + 1
Secondly, the chaotic mapping formula, Equation (7), is utilized to produce a chaotic sequence, which is then measured using a threshold function. Next, the suitable m sequence is chosen and used to perform an XOR operation with the quantized chaotic sequence Yn in order to generate the perturbation sequence. This perturbation sequence is then used to scramble the transmission data Xn, resulting in a scrambled sequence called Zn.
S n = 2 0 z 1 + 2 1 z 2 + … + 2 h − 1 z h
where z1 is the highest bit in every h bits zn, z2 is the second highest bit, and zh is the lowest bit. h is determined by the number of subsets J. Taking J = 4 as an example, the Sn mapping relationship is shown in Table 1.
In the end, K denotes the number of frequency points within each subset, and K = N/J. R is a truly random number with a value range of [0,K − 1]. The sequence Sn is calculated in accordance with Formula (6) and substituted into the mapping relationship Formula (6) in order to ascertain the frequency corresponding to the current hop.
F n = F n − 1 mod K + S n mod J * K + R
In this expression, the operations required for each layer calculation (including modulo, addition and multiplication) can be considered constant-time operations, with a complexity of O(1). The time required for calculation will increase linearly with the input size n. So, the time complexity of this recursive relationship is O(n).
The performance of frequency-hopping sequences generated by the RFGF is mainly affected by the values of parameters J and K. In general, as J increases, the number of frequency subsets increases, and the hopping rules between subsets become more complex. Additionally, as K increases, the range of variation in the random factor expands, and the frequency selection within the subset becomes more unpredictable. The random generation of a random factor not only increases the time-varying nature of the G function structurally but also enhances the overall performance of the frequency-hopping sequence.

4.2. The Inverse G Function Algorithm

The construction process of the G function must satisfy both determinism and reversibility. That is to say, the receiving end is able to restore the complete original information through a certain algorithm when receiving the correct frequency-hopping sequence. In accordance with the design principle of the sending end, this paper proposes a corresponding the inverse G function algorithm, which restores the transmitted data through the difference between two adjacent hop subsets. The specific algorithmic process is illustrated in the accompanying Figure 6.
The difference between the frequency subsets of the adjacent frequency-hopping sequences is calculated by Formula (10) in order to obtain the sequence Sn′.
S n ′ = F n − F n − 1 mod N / K
Subsequently, the inverse mapping relationship of Sn′ should be employed in order to obtain Zn′. To illustrate, if we take J = 4, the inverse mapping relationship is shown in Table 2.
An XOR operation must be performed on Zn′ and the disturbance sequence, as illustrated in Formula (8), in order to obtain the transmission data Xn′.
X n ′ = Z n ′ ⊕ Y n ′
Ultimately, the baseband data Xn’, which are transmitted, can be obtained through decoding when the encoding method is taken into account.

5. Simulation Results

This paper utilizes randomness, uniformity, two-dimensional continuity, and non-deterministic encryption performance indicators to simulate and compare the RFGF with the time-varying iterative decomposition G function (TVID-GF), the G function based on the scrambling algorithm (SA-GF), and the G function based on the m sequence, RS code, and chaotic sequence (MRCS-GF). It also analyzes the varied performance in generating frequency-hopping sequences. The simulation parameters are as follows: the sequence length is 100,000, and the frequency hopping points are 64.

5.1. Randomness

Randomness refers to the unpredictable nature of the appearance of each frequency point in a frequency-hopping sequence, which reflects the irregularity of the sequence during the hopping process [30]. The construction rule of the G function based on the m sequence improvement requires that all subsets be traversed in one cycle, which is a pseudo-random structure. This constraint leads to the order and high predictability of frequency selection. In contrast, the RFGF is not subject to this restriction. It allows some subsets not to be traversed in one cycle but requires that all subsets must be covered in multiple cycles. Such flexibility causes the generated frequency-hopping sequence to have a more obvious advantage in randomness.
The power spectrum of the generated frequency-hopping sequence is typically employed to assess the randomness of the G function [31]. A flatter power spectrum indicates superior random performance of the G function and a greater degree of signal interception difficulty. The power spectra of the frequency-hopping sequences generated by the G function using the four methods are presented in Figure 7.
In order to facilitate a more intuitive comparison of the flatness of the power spectra of the frequency-hopping sequences generated by the four methods, a statistical analysis was conducted on the standard deviations of the power spectrum value. The standard deviation is an indicator that reflects the degree of dispersion of the data and is employed here to evaluate the degree of dispersion of the power spectrum observations in relation to the mean value. The statistical results are presented in Table 3.
A smaller standard deviation corresponds to a flatter power spectrum of the frequency sequence, indicating that the hopping sequence has better randomness. As illustrated in the table, the RFGF method exhibits significant advantages in power spectrum standard deviation, with a value of 2.3897, which is significantly lower than the minimum value of 2.4084 of other methods. This result indicates that the power spectrum of the RFGF method is flatter, reflecting its superior random performance.

5.2. Uniformity

In terms of mathematical probability theory, uniformity can be defined as the condition in which the number of occurrences of each frequency point is approximately equal. This indicates that each frequency point can be selected with equal probability. It has been demonstrated that enhancing the uniformity of the sequence can lead to an improvement in the anti-interference performance of communication systems [32]. In this paper, the frequency statistics results of the χ2 test method and the equal distribution test in statistics are employed to assess the uniformity of the sequence. The χ2 test method is employed to ascertain the extent of discrepancy between the actual observed value of the statistical sample and the theoretical estimated value. For the purposes of this analysis, the variable “ki” will be used to represent the number of frequency points present in the frequency-hopping sequence, while the variable “M” will be used to represent the total number of samples. The test equation for frequency uniformity can be expressed as follows:
X 2 N − 1 = ∑ i = 0 N − 1 k i − M / N 2 M / N
When the calculated value of χ2(N − 1) from the actual measured data is less than the theoretical value, the original hypothesis is confirmed, indicating that the current frequency-hopping sequence exhibits good uniformity. For N = 64, the theoretical value is 82.2447 [18]. The mean results of the frequency-hopping sequences generated by the G function of the four methods following multiple χ2 tests are presented in Table 4.
As illustrated in the table, the outcomes of the χ2 test for the frequency-hopping sequences generated by the G function of the four methods are all below the theoretical value and all satisfy the uniform distribution assumption. Amongst the methods presented, the RFGF method, as proposed in this paper, yielded the lowest result. In order to ascertain the uniformity of the frequency-hopping sequence, the results of the equal distribution test of the frequency-hopping sequence generated by the G function of the four methods are subjected to statistical analysis. The theoretical value (TV) is calculated to be 1/64. Frequency statistics results of the equal distribution test are shown in Figure 8.
A more intuitive representation of the frequency distribution can be observed in Table 5. The RFGF method has the smallest range, 0.00157, indicating that the maximum and minimum probabilities of each frequency point are closer together. This suggests a more uniform overall distribution, demonstrating that the RFGF method achieves the best performance in frequency distribution uniformity.

5.3. Two-Dimensional Continuity

In the context of frequency-hopping communication, two-dimensional continuity is defined as the characteristic of adjacent frequency point changes, which is related to the regularity of hopping. The two-dimensional continuity of a frequency-hopping sequence can be evaluated by analyzing the transition probability between different frequency points in the sequence [33]. If the frequency points’ transition in the frequency-hopping sequence exhibits good continuity, then the number of occurrences between different frequency points will tend to be balanced, thereby increasing the difficulty for an adversary to predict the frequency hopping pattern.
It is expected that a two-dimensional continuity will be observed in an ideal differential frequency-hopping sequence, as follows:
P ( f i , f j ) = 1 / N 2 , f i , f j ∈ Ω ∈ { f 0 , f 1 , ⋯ , f N − 1 } 0 ≤ i , j ≤ N − 1
where P(fi,fj) represents the joint probability density function, with fi denoting the current hopping frequency, fj the previous hopping frequency, and Ω the frequency set. It is assumed that each frequency point is sent with equal probability.
P ( f i ) = 1 / N
In the case of the RFGF,
P ( f j | f i ) = 1 J · K
In accordance with the Bayesian formula, the joint probability density of the G function can be expressed as follows:
P ( f i , f j ) = P ( f i ) ⋅ P ( f j | f i ) = 1 N · J · K
In the case where N = J·K, the probability of occurrence of a particular outcome, P(fi,fj), may be expressed as 1/N2. This value meets the ideal two-dimensional continuity requirement. Figure 9 illustrates the two-dimensional continuity test frequency statistical histograms for the four aforementioned methods of frequency-hopping sequences.
In order to gain a more intuitive understanding of the uniformity of the two-dimensional continuity of the frequency-hopping sequences generated by the four methods, the standard deviation of the number of frequency point pairs is calculated. A larger standard deviation indicates a greater degree of discrete occurrence of the number of frequency point pairs.
As evidenced in Table 6, the frequency-hopping sequence generated by the G function of the RFGF method exhibits a lower standard deviation in two-dimensional continuity performance when compared to other methods. This indicates that the frequency point pair distribution is more uniform and demonstrates diminished correlation between the adjacent hops. The standard deviation of the RFGF method is reduced by 2.9141 in comparison with TVID-GF, which indicates that the two-dimensional continuity performance is enhanced by 36.87%.

5.4. Non-Deterministic Encryption Performance

In the context of non-deterministic encryption, the use of the same encryption strategy on the same original data under identical encryption conditions will result in a distinct ciphertext. This performance can be expressed by using the same G-function to send the same data, resulting in different frequency-hopping sequences and reducing its predictability. Typically, the encryption process incorporates some degree of randomness, either through the inclusion of a random factor or the introduction of random elements. It is possible that the ciphertext may vary even when the input data and the encryption algorithm remain constant, due to the presence of random elements. In comparison to deterministic encryption algorithms, non-deterministic encryption algorithms exhibit greater randomness, which renders ciphertext more difficult to crack and enhances the security of communication systems. The following table illustrates the theoretical values of different transmission paths when transmitting N identical data.
The RFGF approach proposed in this paper, which is based on a random factor, has the distinct advantage that the transmission information only affects the changes in the subset. The system is capable of randomly selecting the frequency points within the subset for hopping. Given that a subset comprises K frequency points, there are K potential options for each generation of the frequency-hopping sequence. In contrast, the remaining three G function methods exhibit reduced flexibility when generating frequency-hopping sequences. The RS code is the only method that can be adjusted without affecting the transmission process; however, the range is limited to two states, 0 and 1. Consequently, when transmitting N bits of the same data, the number of different path choices is only 2N. In conclusion, the G function based on the random factor markedly enhances the non-deterministic encryption performance of the frequency-hopping sequence by augmenting the randomness of frequency selection.
As can be observed in Table 7, an increase in the K value results in a greater number of distinct paths being generated from the same data set, thereby enhancing the non-deterministic encryption performance of the resulting frequency-hopping sequence. When K is greater than 2, the RFGF is observed to exhibit superior security characteristics in comparison to the others.

5.5. Single Frequency Performance

In a differential frequency-hopping communication system, a single-frequency path is defined as a situation where a specific frequency is repeated twice or more consecutively. As the number of single-frequency paths increases, the system becomes increasingly susceptible to interference at a specific frequency point. Should the enemy impose significant interference on this frequency point, there is a risk of misjudgment of the frequency, which may result in a degradation or even destruction of the system performance. The number of single-frequency paths of the frequency-hopping sequences generated by different G function methods was enumerated, and the results are presented in Figure 10.
In particular, since there is no situation in MRCS-GF where Zn2 + 2Zn − 2RS ≡ 0 (modl), resulting in Fn ≠ Fn−1, the single-frequency path is zero.
The mean values were calculated and are presented in Table 8.
As can be seen from this table, the average single-frequency path of the frequency-hopping sequence generated by the RFGF is 1563.52, significantly lower than TVID-GF and SA-GF, indicating that it has better single-frequency characteristics and better resistance to single-tone signal interference.

5.6. L-Z Complexity Performance

Lempel–Ziv (L-Z) complexity is a method for measuring the complexity of a sequence, originally proposed by Abraham Lempel and Jacob Ziv. This complexity measurement method is employed to analyze the randomness and structural characteristics of sequences and is widely utilized in fields such as information theory, data compression, and bioinformatics [34].
The fundamental premise of Lempel–Ziv complexity is to quantify the complexity of a sequence by identifying non-repeating sub-sequences within the sequence. In particular, the Lempel–Ziv complexity of a sequence is defined as the minimum number of non-repeating substrings that are required to uniquely decompose the sequence. In communication systems, Lempel–Ziv complexity can be employed to objectively quantify the degree of structure and randomness in a frequency-hopping sequence, thereby facilitating an evaluation of its complexity and randomness. A higher L-Z complexity value typically indicates a greater degree of unpredictability and difficulty in repeating the sequence. Table 9 presents the L-Z complexity calculations for the frequency-hopping sequences generated by the four methods.
As shown in the table, the average L-Z complexity of the frequency-hopping sequence generated by RFGF is 34.1812, which is higher than the other three methods. This indicates that the sequence is more complex and random in structure and is more challenging for an adversarial entity or interference source to capture and predict.

6. Conclusions

The Random Factor-Based G function method, as proposed in this paper, involves the introduction of random factor to transform the traditional one-to-one mapping relationship into a one-to-many mapping relationship. This expansion of the scope of the G function mapping is achieved through the scrambling of data information. This method preserves the established construction rules in their entirety, elucidates the time-varying properties in great detail, and effectively addresses the issue of limited flexibility in frequency-hopping sequences.
In order to verify the effectiveness of this method, a corresponding de-mapping algorithm has been designed, and a detailed theoretical analysis and simulation experiments have been conducted on the performance of frequency-hopping sequences. The results demonstrate that the receiver is able to rapidly and accurately recover and demodulate the original data. In comparison to alternative methodologies, the frequency-hopping sequence generated by the Random Factor-Based G function exhibits enhanced characteristics, including increased randomness, uniformity, and two- dimensional continuity.
The RFGF, as proposed in this paper, demonstrates remarkable adaptability in complex communication environments, coupled with excellent anti-interference performance. However, the selection of J and K values may lead to different trends in the performance of hopping sequences, which may impose limitations in certain application scenarios. Therefore, future research should focus on the impact of different subset segmentation techniques on communication system performance in order to find the optimal balance between information transmission rate and anti-interference ability. This will help improve the practicality of RFGF and optimize its practical application effect.

Author Contributions

Conceptualization, Y.F.; data curation, J.S.; investigation, B.Q.; methodology, Y.F.; writing—original draft, B.Q. All authors have read and agreed to the published version of the manuscript.

Funding

This research was funded by the National Natural Science Foundation of China (Grant No. 61971291), the Natural Science Foundation of Liaoning Province (serial number: 2024-MS-113), the science and technology funds from Liaoning Education Department (serial number: JYTMS20230199), the Xingliao Talents Plan (serial number: XLYC2202013), and the science and technology funds from Liaoning Education Department (serial number: LJKZ0242).

Data Availability Statement

The original contributions presented in the study are included in the article, further inquiries can be directed to the corresponding author.

Conflicts of Interest

The authors declare no conflicts of interest.

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Applsci 14 10478 g001
Figure 1. Frequency transfer relationship.
Figure 1. Frequency transfer relationship.
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Figure 2. Frequency state transition grid diagram.
Figure 2. Frequency state transition grid diagram.
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Figure 3. Construction diagram of AES algorithm.
Figure 3. Construction diagram of AES algorithm.
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Figure 4. Schematic diagram of iterative decomposition.
Figure 4. Schematic diagram of iterative decomposition.
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Figure 5. Structure diagram of the RFGF.
Figure 5. Structure diagram of the RFGF.
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Figure 6. The inverse G function algorithm.
Figure 6. The inverse G function algorithm.
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Figure 7. Frequency sequence power spectrum.
Figure 7. Frequency sequence power spectrum.
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Figure 8. Frequency statistics results of the equal distribution test.
Figure 8. Frequency statistics results of the equal distribution test.
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Figure 9. Frequency statistics histogram of two-dimensional continuity test by four methods.
Figure 9. Frequency statistics histogram of two-dimensional continuity test by four methods.
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Figure 10. Single-frequency path statistics.
Figure 10. Single-frequency path statistics.
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Table 1. Sn mapping relationship table.
Table 1. Sn mapping relationship table.
Zn00011011
Sn0123
Table 2. Sn′ mapping relationship table.
Table 2. Sn′ mapping relationship table.
Sn′0123
Zn00011011
Table 3. Statistics of numerical standard deviation of power spectrum.
Table 3. Statistics of numerical standard deviation of power spectrum.
RFGFTVID-GFSA-GFMRCS-GF
Value2.38972.40842.41142.4296
Table 4. Statistics of Chi-square test results.
Table 4. Statistics of Chi-square test results.
RFGFTVID-GFSA-GFMRCS-GF
Value51.945352.688622.193758.5664
Table 5. Statistics of extreme values of frequency distribution.
Table 5. Statistics of extreme values of frequency distribution.
RFGFTVID-GFSA-GFMRCS-GF
Maximum0.016230.016460.016720.01664
Minimum0.014660.014740.014750.01465
Range0.001570.001750.001970.00199
Table 6. Statistics of the standard deviation of the frequency points.
Table 6. Statistics of the standard deviation of the frequency points.
RFGFTVID-GFSA-GFMRCS-GF
Value4.99057.904615.704442.5748
Table 7. Statistics of different paths for transmitting the same data.
Table 7. Statistics of different paths for transmitting the same data.
RFGFTVID-GFSA-GFMRCS-GF
ValueKN2N2N2N
Table 8. Statistics of average number of single frequency paths.
Table 8. Statistics of average number of single frequency paths.
RFGFTVID-GFSA-GFMRCS-GF
Value1563.521566.661569.080
Table 9. Statistics of L-Z complexity simulation results.
Table 9. Statistics of L-Z complexity simulation results.
RFGFTVID-GFSA-GFMRCS-GF
Value34.181234.098233.5205228.5316
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Feng, Y.; Su, J.; Qian, B. A Construction Method for the Random Factor-Based G Function. Appl. Sci. 2024, 14, 10478. https://doi.org/10.3390/app142210478

AMA Style

Feng Y, Su J, Qian B. A Construction Method for the Random Factor-Based G Function. Applied Sciences. 2024; 14(22):10478. https://doi.org/10.3390/app142210478

Chicago/Turabian Style

Feng, Yongxin, Jiankai Su, and Bo Qian. 2024. "A Construction Method for the Random Factor-Based G Function" Applied Sciences 14, no. 22: 10478. https://doi.org/10.3390/app142210478

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