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Abstract
This paper proposes a high-security multidimensional data protection system based on the Hartley algorithm-driven chaotic scheme. We utilize the fast Hartley algorithm instead of the fast fourier computation, and we employ chaotic sequences generated by the multi-winged chaotic system to achieve chaos-driven 3D constellation mapping, effectively integrating the chaotic system with the stochastic amplitude modulator. We reduce the signal’s peak-to-average power ratio (PAPR) by deploying a random amplitude modulator. Simultaneously, this approach enhances the security of the physical layer of the signal. The PAPR reduction can reach up to 2.6 dB, while the most robust and stable modulator scheme can gain 2 dB. Finally, in the Hartley frequency domain, the signal's frequency is disrupted, providing the entire system with a key space of 10131 to resist violent cracking and thus improving the system's overall security. To validate the feasibility of our scheme in comparison to conventional IFFT-based encrypted 3D orthogonal frequency division multiplexing, We achieved a transmission rate of 27.94 Gb/s over a 2 km multicore fiber. Experimental results show that since the random amplitude generator effectively reduces PAPR, our proposed encryption scheme increases the forward error correction threshold range by 1.1 dB, verifying that our proposed scheme has highly reliable security performance.
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1. Introduction
Modern communication technology is advancing rapidly, and information technology has become deeply ingrained in daily life. The widespread use of intelligent terminal devices has significantly increased the demand for channel capacity in communication systems. Meanwhile, traditional time, frequency, and spatial multiplexing techniques are approaching their limits [1]. Orthogonal frequency division multiplexing (OFDM) stands out as a multicarrier modulation technique recognized for its resistance to channel dispersion and high spectral efficiency [2,3]. In optical communications, intensity modulation and direct detection orthogonal frequency division multiplexing (IMDD-OFDM) systems are extensively employed in passive optical networks (PONs) owing to their straightforward and cost-effective configuration [4–6]. However, a significant drawback of OFDM systems in optical communications is the high peak-to-average power ratio (PAPR). The elevated PAPR in optical OFDM signals can result in nonlinear distortion, particularly at high transmit power levels, due to the inherent nonlinearity of the transmission fiber. Additionally, excessively high PAPR in optical OFDM signals might cause nonlinear effects in devices like Mach-Zehnder modulators (MZM), Digital-to-analog converters (DAC), Analog-to-digital converters (ADC), and optical fibers. This, in turn, could result in intermodulation among subcarriers, introducing nonlinear distortion.
To address the challenges associated with optical OFDM signals, researchers have proposed several techniques to effectively mitigate the PAPR in optical OFDM systems [7–15]. Two primary methods are selective mapping (SLM) and partial transmission sequence (PTS), while clipping is a simpler method achieved by truncating the peaks of the OFDM signal.In the SLM method, the transmitted data sequence is directed to multiple independent scramblers utilizing different phase rotations to reduce the PAPR before transmission. However, both methods encounter high computational complexity. Moreover, to generate a genuine OFDM symbol meeting the IFFT input requirements, the signal must exhibit Hermitian symmetry (HS), commonly known as discrete multitone modulation (DMT) [16]. This results in only half of the subcarriers being utilized to carry data.Furthermore, to guarantee HS, the DC and Nyquist frequencies are both set to zero. Similarly, employing the fast Hartley transform (FHT) for generating OFDM symbols is a viable method. With FHT modulation, bit sequences are mapped to real constellations, ensuring that the resulting OFDM symbols are real-valued, and all N subcarriers support data symbols [17–19]. This technique effectively satisfies the Hermitian symmetry condition. We use a method that combines a random amplitude modulator with a fast Hartley algorithm, which can completely accommodate the impact of signal distortion, thereby greatly improving the fault tolerance rate.
An alternative method for transmitting information over a communication channel is based on the real Hartley transform (HT) [20]. R. V. L. Hartley introduced the continuous Hartley transform in 1942 [21]. This approach serves as an alternative to address certain challenges encountered by the conventional Fourier transform while ensuring the efficient channel information transmission. The fast computation of the discrete Hartley transform is typically slightly faster than the fast Fourier transform, especially in specific application scenarios such as OFDM systems. This acceleration in computation speed expedites the data processing efficiency and reduces the probability of errors, which is crucial for high-speed communication systems. The Hartley transform can be converted to the Fourier transform for Nth addition and multiplication, which is faster than post-processing in real numbers. By expressing Fourier coefficients as The fast Hartley transform (DHT), the discrete Hartley transform provides convenient computational expressions. Moreover, the accuracy of the fast Hartley transform is generally slightly higher than that of the fast Fourier transform [22]. Additionally, this increase in speed is essential for high-speed communication systems like OFDM systems, enabling quick data processing with a reduced number of errors, making the fast Hartley transform advantageous for communication systems. To enhance the channel capacity of communication systems, multi-core fiber (MCF) has been proposed as a space division multiplexing (SDM) fiber, showing great promise. The design concept of MCF aims to boost the capacity of each fiber by increasing the number of cores within it. This technique has been extensively studied [23,24]. In addition, with the rapid development of society, every industry has a higher impact on communication quality. The requirements for and information security are also getting higher and higher. The security of communication and information is not only related to the privacy of users, but also has an impact on the security of enterprises and even countries. Therefore, it has become very necessary to improve the security performance of communication systems. Especially for PON As network access users increase, security issues become particularly important. Illegal optical network units (ONUs) may disguise themselves as legitimate ONUs, and when optical line terminals (OLTs) transmit downlink data in the form of broadcasts, these illegal ONUs may Downlink data signals will be stolen or tampered with. Therefore, it is necessary to impose physical layer security encryption schemes on passive optical networks (PON) [25,26]. Traditional upper-layer encryption has some limitations. For example, the data header is easily exposed during data transmission, which will pose a major security threat. Physical layer encryption plays a key role in ensuring transmission security by encrypting data signals at high speed and low latency, while minimizing the impact on network systems. As a supplement and improvement to the upper-layer encryption mechanism, physical layer security encryption technology has broad research prospects and application potential. Presently, physical layer digital domain chaotic encryption is of significant interest in high-speed multicarrier optical communication systems [27]. Physical layer encryption, widely used in digital signal processing (DSP) for its low cost, high flexibility, and compatibility, is based on chaotic systems with traits such as traversal, pseudo-randomness, and parameter sensitivity, making it ideal for data encryption [28]. This approach effectively protects transmitted information from brute-force attacks. Liu et al. proposed a security enhancement technique based on symbol-level and bit-level chaotic scrambling for OFDM-PON at the physical layer [29–32]. L. Deng and his coworkers experimentally validated that the fixed-point digital chaos algorithm [33] and fractional-order Fourier transform chaos technique [34] meet the requirements of low implementation complexity and high-security performance for OFDM-PON. Combining digital chaotic encryption with specific fields holds promises for achieving higher security and improved system performance [35].
This paper presents a novel multidimensional data protection and chaotic driving scheme based on the Hartley transform, applied to OFDM in optical communications for the first time, facilitating the transition from 2D to 3D constellations. Our approach employs a multi-wing chaotic model for the 3D constellation mapping module's chaotic driving and successfully integrates the random amplitude modulator's chaotic input. Compared to the traditional OFDM 3D chaotic encrypted modulation scheme, our method significantly improves the system's BER and transmission security performance while effectively reducing the PAPR of signals. In our experiments, we implement hexadecimal modulation at the transmitter side, showcasing the successful implementation of our proposed Hartley transform-based multidimensional data protection and chaos-driven scheme. Comparative analysis with the conventional OFDM 3D chaotic encrypted modulation scheme reveals a 1.1 dB performance improvement in our approach, accompanied by a substantial 2.6 dB reduction in the signal's PAPR performance. The most stable random modulation scheme similarly attains a 2 dB improvement. This experiment substantiates the optical transmission security of the signal in the Hartley frequency domain and affirms the feasibility of our proposed scheme.
2. Principles
Figure 1 illustrates the main schematic framework diagram of the high-security multidimensional data protection system based on the Hartley algorithm-driven chaotic scheme. The system comprises a three-dimensional constellation mapping module driven by chaos, a random amplitude modulator, and a multi-wing chaotic system. In the transmitter side of high-security multidimensional data protection system based on the Hartley algorithm-driven chaotic system, the pseudo-random binary sequence (PRBS) generated by DSP serves as the original input data. It undergoes serial/parallel (S/P) conversion to generate a matrix. A multi-wing chaotic system generates chaotic sequences to complete the three-dimensional constellation mapping driven by chaos. In this process, every four bits carry a set of 3D dimensional information, resulting in four sets of mutually distinguishable mappings. All signals with coordinate information then pass through a fast Hartley modulation module to complete the OFDM modulation. The two pairs of chaotic sequences generated by the multi-wing chaotic system, and the modulated OFDM signals undergo modulation by a random amplitude modulator to reduce the PAPR. This PAPR reduction process achieves random disruption of the signal amplitude and completes the chaotic scrambling of symbols in the Hartley frequency domain, realizing constellation masking. Subsequently, the signals with constellation masking are augmented with cyclic prefixes, and finally, the encrypted signals undergo parallel to serial (P/S) conversion to transform into single-channel real signals for transmission, completing the 3D-OFDM modulation.
Fig. 1. Schematic diagram of the high-security multidimensional data protection system based on the Hartley algorithm-driven chaotic scheme.
2.1 Hartley transform and random amplitude modulator
The Hartley transform can be converted to a Fourier transform for Nth addition and multiplication, which is faster than post-processing in complex numbers. If the source data is valid, the FHT may be more efficient than the FFT. DHT is an efficient transform algorithm for real sequences and is closely related to the fast Fourier transform (DFT). The DHT does not require operations with the imaginary part. This algorithm maps real-valued sequences to real-valued spectra by preserving some useful parts of the DFT. If $x(n),n = 0,1,2, \cdots ,N - 1$ is an N-point real sequence, then its DHT is defined as follows:
The corresponding IDHT is defined as:
Thus, recomputation of the DHT to the DFT can sometimes be avoided, e.g., by multiplying by a long number [36] The Hartley transform in the form of the positive and the negative can be written as a pair of transformations,where $cas\alpha = \cos \alpha + \sin \alpha$, the Hartley transform, is a real triangular transform with the property of self-inversion. As evident from the formula, modulation involves the Hartley inverse transform, and demodulation involves the direct Hartley transform. Signal processing algorithms for modulation and demodulation are precisely identical, thereby significantly simplifying the hardware and software implementation of the communication system. However, there are some differences. The code for the FHT is simpler. The inverse transformation formula is the same as the direct transformation formula, except that the factor 1/N is different. In contrast,an additional parameter must be introduced for the FFT, or a new function must be created to compute the inverse. When computing the FFT for real vectors, the half-length “integral” FFT is computed first, leading to post-processing that does not exist in the FHT. This provides additional code simplification for the FHT compared to the FFT and affects the efficiency for shorter length vectors. However, the longer the vector, the weaker the effect. The following is the basic principle of the base 2DIT-FHT operation. Figure 2 shows the flowchart of 8-point 2DIT-FFT computation and 8-point 2DIT-FHT computation, respectively.
Fig. 2. (a) 8-point base 2DIT-FHT computational flow diagram (b) 8-point base 2DIT-FFT computational flow diagram.
Compared to the FFT algorithm, the fast algorithm FHT for DHT reduces the computational effort by almost half [22]. The number of real multiplications for the N-point base-2 time-domain extraction fast DHT (base-2DIT-FHT) algorithm is:
The total count of actual additions in the N-point base 2DIT-FHT algorithm is:
where $M = {\log _2}^N$. According to Eq. (3), it is easy to know that the number of real multiplication of base 2DIT-FHT algorithm is about half of base 2DIT-FFT algorithm, while the FFT requires up to ${{MN} / 2}$ complex operations.Therefore, the FHT algorithm can reduce complexity and simplify communication. In this paper there are two different designs of random amplitude modulators. The random amplitude modulator works as follows: In this article, there are N subcarriers, and each subcarrier's minimum instantaneous signal transmission power and maximum instantaneous signal transmission power are calculated. Compare their modulus values to determine their relatively smaller power values, and then calculate the signal power ratio by comparing the transmission power of two instantaneous signals. Then multiply all the data on this subcarrier whose instantaneous power mode values are greater than the smaller power value by the calculated signal power ratio. The new data will be obtained to replace the original signal amplitude, to obtain a new amplitude value. At this point a threshold for the random amplitude modulator is introduced, the principle of which is given in the following Eq (5–6).In the second design, after comparing the comparison of the minimum and maximum instantaneous signal transmitting power of each subcarrier, all values greater than the instantaneous signal transmitting power at that moment are directly substituted with the relatively smaller power values. Both designs concentrate on reducing the PAPR of the signal from two perspectives. One emphasizes power control across the entire spectrum range, while the other emphasizes subcarriers to control power in each subcarrier finely. All the comparisons can be found in the experimental setup and results in section III.
In a random amplitude modulator, all the Hartley frequency domain information on each of its subcarriers due to its coordinates and the influence of the multi-wing chaotic system. should all have a maximum and minimum value and the modal values should be the same. After the data has been subjected to the fast Hartley operation, the maximum and minimum modal values on each of its subcarriers are not necessarily the same. The reason for the occurrence of non-identity may be due firstly to the fact that the modulation scheme involves an element of randomness, using pseudo-random sequences for spreading. In this scenario, it's worth noting that the same input data may yield different outputs during each modulation. If the binary data is quantized before modulation (meaning the analog signal is sampled into digital data), quantization errors may lead to minor inconsistencies. These inconsistencies can arise from quantizer setting variations, rounding errors, or quantization noise. In a typical OFDM system, the FFT is a transformation method employed to convert a signal from the time domain to the frequency domain. This transformation decomposes the signal into a series of components represented by sine and cosine functions, thereby revealing the spectral information of the signal in the frequency domain. FHT is a transformation similar to FFT, but it executes the Hartley transform, converting a signal from the time domain to the Hartley frequency domain. In contrast to the Fourier transform, the Hartley transform employs a combination of sine and cosine functions. The FFT typically performs a complex-to-complex transform, as it generates a complex spectrum, necessitating the handling of complex operations within the FFT algorithm. On the other hand, the FHT is a transformation method specifically designed for real signals, transforming them from the time domain to the Hartley frequency domain. Unlike the Fourier transform, the Hartley transform preserves both the amplitude and phase of the signal. Instead of altering the amplitude or phase, it represents the signal in the frequency domain using a set of coefficients of sine and cosine functions.
From a mathematical analysis standpoint, the Hartley transform serves as a frequency domain representation of a real-number signal, and its mathematical properties are similar to the Fourier transform. The formula for the Hartley transform is expressed as follows:
Complex and nonlinear chaotic sequences are generated using a multi-wing chaotic system in random amplitude modulators. Generate a random 1 and -1 sequence related to a chaotic sequence through linear operation, which is embedded in a random amplitude modulator. This results in a random sequence of positive and negative 1's associated with the chaotic sequence. The signal is then amplitude modulated and multiplied element by element with the random 1 and -1 sequence. This multiplication operation introduces a randomness modulation to the original data, causing each data point to undergo a unique random amplitude modulation. Consequently, the signal becomes complex, with its amplitude constantly fluctuating during transmission, ensuring randomness throughout the system. Given that the chaotic sequence comprises an unordered series of numbers, the signal inherits its extremely high complexity and nonlinear characteristics. A peak occurs when N subcarriers are added in the same phase, and its power can be N times the average power. The ratio of the maximum peak power to the average power of an OFDM frame is defined as the signal PAPR.
The theoretical limit of PAPR (in dB) can be derived from the above equation, given in [37]: $PAPR = 10{\log _{10}}N$ where xm is the OFDM frame that will be transmitted through the channel. The complementary cumulative distribution function (CCDF) is commonly used to evaluate the performance of PAPR reduction techniques with the following equation. This function gives the probability that the PAPR of an OFDM frame exceeds the threshold PAPR0.
2.2 Multi-wing chaotic system and chaotic-driven three-dimensional constellation mapping
In the chaos-driven 3D constellation mapping module, random amplitude modulator, and Hartley frequency domain, the multi-wing chaotic model is used to complete the chaotic drive for both modules and the Hartley frequency domain. The equations of the multi-wing chaotic model are as follows:
The positive Lyapunov exponent implies that even if the difference between the initial values of two orbits is extremely small, the difference separates exponentially over time, thus making the system locally unstable and globally stable. Variation of chaotic attractor wing number with initial conditions multi-stability of a system holds when different attractors are obtained from systems with different initial conditions. The butterfly effect can be realized in systems with multiple stability in the time response and phase diagram. That is why estimating the system parameters of such chaotic systems is quite difficult. Therefore, this property of chaotic systems makes them suitable candidates for different chaos-based applications, and data encryption processing. When a signal encrypted by these complex chaotic trajectories (chaotic sequences) is received, decryption can only be accomplished by having the exact initial value (private key), thus ensuring high security. The three chaotic sequences X, Y, Z, (as shown in Fig. 3) generated by the multi-winged chaotic system are processed to generate the masking factors A, B, and C, which are used in the chaos-driven three-dimensional constellation mapping module, the random amplitude chaotic scrambling modulation module in the modulator. The specific rules are as follows:
Fig. 3. Model of a multi-wing chaotic system (a)x-y-z projection (b) x-y projection(c)x-z projection (d) y-z projection.
Three-dimensional constellation mapping is a method of mapping data into three-dimensional space and is commonly used in digital communication systems, especially in optical communications. Typically, the data transmitted is binary, such as 00, 01, 10, and 11. each binary data corresponds to a specific three-dimensional coordinate. The 3D coordinates are used to represent each data combination. In order to map data into a 3D constellation, each binary data needs to be mapped to a unique 3D coordinate. For example, 00 can be mapped to coordinates (1, 1, 1), 01 can be mapped to coordinates (-1, -1, 1), 10 can be mapped to coordinates (-1, 1, -1), and 11 can be mapped to coordinates (1, -1, -1). A chaotic sequence is first obtained using a multi-wing chaotic system. Next, we divide the chaotic sequence into four disjoint parts. Each is identified by the letters A, B, C, and D, respectively. This segmentation process can be based on specific algorithms or rules that ensure that each part contains a different part of the chaotic sequence. This segmentation process has a key impact on the nature of the chaotic sequence because each part represents a different fragment of the nonlinear sequence. Now, each letter (A, B, C, D) represents a different part of the chaotic sequence, and each partial sequence can be mapped to a coordinate point in 3D space. In this step, we use four mappings, applying different weights for each partial sequence or employing different constellations of mappings (e.g., mappings 1, 2, 3, and 4). In this way, the position of the letters determines how the data will be mapped, and each partial sequence will be mapped to a different coordinate point in 3D space, and the mapping method is determined by the bit data as in Fig. 4.
This paper has a total of N subcarriers; one subcarrier contains F symbols, each symbol carrying four bits of information. Convert all the bit data in parallel series to four lines of data information. It means the overall conveyed information is: N*F symbols information. Weight the data in each row. The first way: weight each line of bits, the first line is the heaviest, the fourth line is the lightest; the second way: the same weighting process for each line of bits, but the starting line is different, and then loop in order. The third way: a similar weighting method is used, but the starting row is different again, and then the cycle is in order; the fourth way: the same weighting method is used, starting from the fourth row, and then the cycle is in order. These four processing methods can produce different results depending on the data and the order of weighting. Based on this, the chaos-driven 3D constellation mapping module is formed. The final weighted numbers can be expressed as 16 different numbers from a-p, linking the 16 numbers to 16 3D coordinate points in 3D space. The chaos-driven 3D constellation mapping module thus formed, i.e., four bits determine one 3D coordinate point information (x, y, z) in 3D space. The principle of the chaos-driven 3D constellation mapping module is shown in Fig. 5 below.
3. Experiment setup and results
The practical efficacy of the multidimensional data protection and chaotic encryption system based on Hartley's algorithm, as depicted in Fig. 6, is experimentally validated over a 2 km weakly coupled seven-core fiber. The experimental setup employs a direct modulation and direct detection (IM/DD) system. At the optical line terminal (OLT), encrypted OFDM signals are generated offline using a digital signal processor (DSP). For comparison, the receiver is divided into legitimate and illegitimate receivers. The OFDM configuration includes 150 subcarriers, 128 symbols, 512 Fourier points for FFT, and a guard interval of 1/4 of the signal length. For digital-to-analog conversion of the encrypted 3D-OFDM signal, an arbitrary waveform generator (AWG, TekAWG70002A) with a sampling rate of 10 GSa/s is utilized. The light source is a continuous wave (CW) laser with a wavelength of 1550 nm and a power of 2 dBm. After passing through an electrical amplifier, the electrical signals are injected into the Mach-Zehnder modulator (MZM) to achieve intensity modulation and electro-optical conversion. Further amplification of the laser light is necessary through an erbium-doped fiber amplifier (EDFA) before the optical signals are coupled into the seven-core fiber via a 1:8 beam splitter and fan-in device. In the case of a legitimate optical network unit (ONU), the received optical power is adjusted by utilizing a variable optical attenuator (VOA). The received optical signal is then converted into an electrical signal through a photodetector (PD). The acquired electrical signal undergoes processing using a mixed signal oscilloscope (MSO, TexMSO73304DX) with a sampling rate of 50 GS/s. Following analog-to-digital conversion, the received data are decrypted using the same key employed by the transmitter. Without the proper key, unauthorized users attempting illegal access cannot obtain correct data through offline digital signal processing (DSP).
Fig. 6. High-security multi-dimensional data protection system based on the Hartley algorithm with chaotic drive scheme system.
Addressing the issue of limiting the instantaneous power range of a signal involves two distinct algorithms, both aimed at reducing the PAPR of the signal. The comparison of bit error rate (BER) performance is conducted from two separate perspectives. Specifically, the PAPR performance of the random amplitude modulator is analyzed for all designs in various scenarios using the CCDF. To assess our proposed High-security multi-dimensional data protection system based on the Hartley algorithm with chaotic drive scheme system, we compare the PAPR performance of 3D-OFDM signals with different threshold selections based on various random amplitude modulator designs, as illustrated in Fig. 7. The first algorithm concentrates on power control across the entire spectral range. In contrast, the second algorithm provides finer control within each subcarrier. The four algorithms under consideration are the full probabilistic replacement method (FPRM), the all-cropping method (ACM), the subcarrier probabilistic replacement method (SPRM), and the subcarrier cropping method (SCM).
Fig. 7. PAPR plots for different threshold selections with different randomized method amplitude modulator designs.
As observed in Fig. 7, after comparison, the PAPR performance of the signal is enhanced across all four approaches, accompanied by lower computational complexity. The approach focusing on power control operations in each subcarrier with finer control of power operations across the entire spectral range demonstrates substantial improvement and increased stability.Specifically, the performance of the full probabilistic replacement method improves by 0.6 dB, the full cropping method improves by 2.6 dB, and both the subcarrier probabilistic replacement and subcarrier cropping methods exhibit a 2.0 dB improvement. Importantly, there is no significant performance loss in the BER under the same received optical power. The stability of the proposed method under various approaches is verified, and the PAPR performance under the four methods is significantly enhanced, reaching up to 2.6 dB. The PAPR reduction performance is commendable and warrants affirmation.
The BER performance of 3D-OFDM signals, based on various random amplitude modulator designs with different threshold selections, is compared at different received optical powers, as depicted in Fig. 8. The figure illustrates that all four methods have successfully enhanced the BER performance of the signal, achieving lower computational complexity, albeit with varying degrees of improvement. In comparison to the other methods, the power control operation focusing on each subcarrier exhibits a 0.73 dB improvement, demonstrating a more stable performance. This shows the advantages of the SPRM scheme in resisting channel interference and bit error rate.
Fig. 8. BER performance of fast Hartley algorithm based 3D-OFDM system system with different random amplitude modulators.
In Fig. 8, the BER performance comparison based on the two approaches with different random amplitude modulators is also presented. The factor-carrier probability substitution method excels in the BER performance comparison. It is chosen for all subsequent comparisons of the FHT operations, including comparing of different systems with random amplitude modulators. To study the effects of the chaos-driven 3D constellation mapping module, random amplitude modulator, and each module of the multi-wing chaotic system on the 3D-OFDM signals, as well as the impact of the adopted seven-core fiber transmission system, Fig. 9 displays the BER performance of the 3D-OFDM signal after 2 km of transmission in a seven-core fiber. Remarkably, the measured BER curves of the seven cores almost overlap at a range of 2 km, indicating an excellent performance of the seven-core fiber transmission system. The system's BER reaches 3.8 × 10−3. Even at this high BER, the difference in received optical power among the individual fiber cores is very small, less than 0.2 dB. This observation indicates that the BER performance of the seven fiber cores is maintained within acceptable limits under the FEC threshold requirement. As the optical power continues to increase, the BER of each fiber core gradually decreases, strongly verifying that the proposed 3D-OFDM transmission scheme exhibits excellent transmission performance in a seven-fiber transmission system. This finding emphasizes the system's robustness, providing consistent performance even under unfavorable conditions, which is crucial for various applications.
Simultaneously, to examine the impact of multicore multiplexing on 3D-OFDM signals using different modulator designs and to validate the superiority of the proposed high-security multidimensional data protection system based on the Hartley algorithm-driven chaotic scheme, we conduct a performance comparison of the proposed 3D-OFDM transmission scheme employing different demodulation algorithms at the receiver side. Specifically, the power control operation within each subcarrier is selected for comparison with the OFDM system based on the IFFT operation.The experimental results are depicted in Fig. 10.
Fig. 10. BER performance of 3D-OFDM system based on fast Fourier algorithm and fast Hartley algorithm.
In addition, the key space of the proposed encryption scheme is rigorously computed. After sensitivity testing, we calculate the critical space by adding tiny increments of 1 × 10^-N to a single initial value or parameter. Afterwards, experimental tests are performed to see if the data can be fully demodulated. Continuously adjust N until a high bit error rate node appears, which is one order of magnitude away from complete demodulation, which is the critical space. As shown in Fig. 11, the key consists of the initial value of the multi-wing chaotic system, the control parameters and the step size, i.e., {x, y, z, a, b, d, e, k}. Sensitivity analysis Fig. 12 evaluates the sensitivity of the proposed scheme to changes in the key. We change the values to 10-N accuracy in the analysis and plot the slight deviation trajectories in each direction. The BER value for the correct key is zero, and the parameter accuracy is very large, indicating that the encryption scheme has excellent sensitivity.If the step size is [1, 1*10−3], the key space can be experimentally calculated to be 1017 × 1017 × 1016 × 1015 × 1015 × 1015 × 1011 × 1011 × 1011 × 103 = 10131. Since the key space is so large, it takes a long time even to find the correct key, preventing prevents the hijacker from obtaining the key effectively.
4. Conclusion
To enhance the physical layer security and reduce the computational complexity of the system while improving the performance of the PAPR system, we propose a high-security multidimensional data protection system based on the Hartley algorithm-driven chaotic scheme. The computational complexity of the system is reduced by using the fast Hartree algorithm instead of the fast fourier computation, while the chaotic sequence generated by the multi-wing chaotic system to complete the three-dimensional constellation mapping of the chaotic drive, and then use the random amplitude modulator to complete the reduction of the signal PAPR, the maximum reduction of 2.6 dB and relatively more stable subcarrier replacement method also shows an increase of 2 dB. At this time, the physical layer security of the signal is implemented while reducing the PAPR. Finally, the frequency interference of the signal is accomplished in the Hartley frequency domain, which provides a key space of 10131 and improves the security of the system. To verify the scheme's feasibility and compare it with the conventional IFFT-based encrypted 3D-OFDM, a transmission experiment with a 2 km MCF is established. The experimental results show that our proposed encryption scheme has less impact on the BER performance of the signal within the FEC threshold compared to the conventional IFFT-based 3D-OFDM, and there is a 1.1 dB improvement compared to the conventional IFFT-based 3D-OFDM scheme, which verifies that our proposed scheme has a very reliable and implementable performance.
Funding
National Key Research and Development Program of China (2023YFB2805301); National Natural Science Foundation of China (62275127, 61935011, U2001601, 62205151, U22B2009); Jiangsu Provincial Key Research and Development Program (BE2022079, BE2022055-2); Natural Science Research of Jiangsu Higher Education Institutions of China (22KJB510031); Startup Foundation for Introducing Talent of Nanjing University of Information Science and Technology (NUIST).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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