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Abstract
The secure key generation and distribution (SKGD) are unprecedentedly important for a modern secure communication system. This paper proposes what we believe to be a novel scheme of high-speed key distribution based on interference spectrum-shift keying with signal mutual modulation in commonly driven chaos synchronization. In this scheme, delay line interferometers (DLI) are utilized to generate two low-correlation interference spectra from commonly driven synchronous chaos, and then a 2 × 2 optical switch can effectively change the relationship between the two interference spectra in post-processing by shifting the states of the switch. The signals then undergo electro-optic nonlinear transformation through a hardware module, which includes a signal mutually modulating module (SMMM) and a dispersion component. This optimization significantly enhances the entropy source rate of synchronized chaos from both legitimate users. Moreover, thanks to the introduction of DLIs and electro-optic nonlinear transformation module, the key space of the proposed scheme is remarkably improved. In comparison to traditional chaotic drive-response architectures, the scheme effectively suppresses residual correlation. A 6.7 Gbit/s key distribution rate with a bit error rate below 3.8 × 10−3 is experimentally demonstrated over a 40 km single-mode fiber (SMF).
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1. Introduction
With the advancement of optical fiber communication and the growing computing power, the risks associated with data transmission have become increasingly alarming. This has led to a surge in global incidents of information leaks and security breaches. Hence, there is an urgent need to enhance the security of optical fiber communication. To safeguard the communication networks against eavesdropping, secure key distribution plays a pivotal role. It involves the generation and sharing of secure keys for information encryption among legitimate communication users. This field of research has garnered significant attention from an increasing number of researchers due to its critical importance in ensuring secure communication networks. However, in line with the principle of one-time-pad data encryption, it is desired to get a higher speed in key distribution to address information encryption during transmission as optical communication rates increase [1].
At present, algorithm-based key distribution is a widely used secure encryption method, but its security has been questioned due to the rapid advancement in computing power [2,3]. As a method that can circumvent the risks associated with supercomputers, secure key distribution based on physical principles has been proposed to employ in data encryption, such as quantum key distribution [4–6], non-cloning function-based key distribution [7], fiber laser-based key distribution [8–11], and channel noise-based key distribution [12–14]. However, these methods are subject to certain limitations inherent in their principles, resulting in a key rate that can only reach up to the magnitude of Mbit/s and falls short of meeting the current optical communication rates.
The utilization of chaotic signals in secure communication has garnered considerable attention due to their numerous intentional properties, such as their noise-like nature, aperiodic nature, and broad bandwidth. The application of chaotic signals in modulation schemes, such as coherent chaotic shift keying [15,16] and noncoherent differential chaotic shift keying [17], seeks to employ chaotic signals in transmission to enhance resistance to multipath fading and nonlinear distortion [18], while simultaneously exhibiting superior anti-interference capabilities and a low probability of intercept [19]. The aforementioned characteristics of chaotic signals render the decoding process of the modulated data particularly challenging for an eavesdropper, thereby enabling the establishment of secure communication channels. Another application of chaotic signals in secure communication lies in their ability to serve as a broadband physical entropy source, which enables the generation of high-speed correlated random bits at rates of up to Tbit/s through their wideband and noise-like characteristics [20–23]. Based on the characteristics described above and the phenomenon of chaos synchronization, key distribution methods utilizing chaotic synchronization have emerged as an effective means to achieve high-speed data transmission rates. Kanter et al. first proposed a scheme using chaos synchronization between two mutually injected lasers to achieve masking and exchanging secure keys [24], and then Porte et al. demonstrated an experiment over back-to-back transmission realizing a key distribution at the rate of 11 Mbit/s [25]. Nonetheless, this approach entails the transmission of key information over public channels, thereby introducing the potential risk of information leakage. To solve the above problem, a series of key distribution schemes were proposed, employing a random source as the commonly driven signal injected into the response lasers deployed in two communication parties to achieve chaos synchronization. Subsequently, by introducing the keying-code technique to control parameters in the private hardware module succeeding the response laser, the operational states of the two authorized users can be altered. Legitimate users can exchange their keying codes to sift the bits obtained in the time window of two communication parties with identical keying codes. These filtered bits can be used as the shared keys for securing data between two legitimate parties [26]. The security is ensured by the fact that it is extremely challenging for eavesdroppers to acquire key information, as counterfeiting lasers with parameters perfectly matched to legitimate users poses a formidable obstacle. The keying-code technique proves to be an effective means of enhancing security, as it increases the complexity for eavesdroppers attempting to set up and simultaneously operate multiple configurations to intercept signals for each set of keying codes. Based on keying techniques and commonly-driven-source induced high-quality chaos synchronization, various schemes such as vertical-cavity surface-emitting lasers with polarization-shift keying, hybrid optical chaos source with phase-shift keying, and Fabry-Perot lasers with mode-shift keying and chaotic self-carrier phase modulation with time-delayed shift keying were proposed for key distribution [27–30]. However, due to the inherent limitations imposed by the laser relaxation oscillation frequency, the rate of entropy source generation is severely constrained, resulting in a relatively low-rate key distribution. Furthermore, a scheme has been proposed, that involves using an oscillator as the response end to bypass the limitations of relaxation oscillation. However, no experimental demonstration of this approach has been presented [31]. It is still necessary to confirm the quality of synchronization between the two communication sides and elevate the effective bandwidth and other performances under the real experimental system. To generate two low-cross-correlation chaotic signals needed for the keying process, most of the former schemes have to equip two identical hardware modules with parameters controlled by keying code generators, leading to an increment in the complexity of these schemes. Moreover, with fewer parameters available for the hardware module that serves as the crucial key, these schemes cannot effectively improve the security of the system, where eavesdroppers are entirely likely to counterfeit the setup to obtain the correct information.
Here, we propose and experimentally demonstrate a novel method for secure key distribution based on interference spectrum-shift keying with signal mutual modulation in commonly-driven chaos synchronization, which implements delay line interferometers to generate dual signals and utilize 2 × 2 optical switches to shift the operating states during the keying process. Owing to the characteristic of DLIs, which divide the input into two different outputs with a certain low correlation, our scheme eliminates the requirement of two identical sets of hardware to generate these low-correlation signals, which can effectively decrease the complexity and difficulty in practical application and simplify the structure of the setup. Then, the 2 × 2 optical switches can be shifted between straight-through and cross-over states, allowing for the convenient alteration of the relationship between the two controlled signals in the post-processing stage. Moreover, the introduction of the mutual modulation module can remarkably enhance the effective bandwidth of chaos, which directly increases the entropy source rate for the subsequent key extraction process. Due to the diverse hardware parameters present in DLIs and mutual modulating modules, the hardware key spaces of the proposed scheme are significantly expanded, in consequence, it is difficult for an eavesdropper to replicate a setup with precisely matched hardware parameters. Compared with the relatively high correlation between the driving source and the final chaotic output in traditional drive-response structures, the residual correlation is effectively suppressed, which eliminates the leakage of key-related information in the public transmission link and results in the enhancement of security for key distribution. Ultimately, this method can achieve a high rate of 6.7 Gbit/s key distribution based on chaos synchronization.
2. Principle
The schematic diagram of the proposed key distribution based on the interference spectrum-shift keying chaos synchronization with mutual modulation is illustrated in Fig. 1. Two private hardware modules with matched parameters are deployed in legitimate users (Alice and Bob) to ensure high-quality chaos synchronization between both sides. The private hardware modules consist of a distributed feedback laser as the response laser, a delay line interferometer, a signal mutually-modulating module, and a dispersive component. The procedure for the key distribution is as follows: Firstly, a random common-drive source is utilized to drive the response laser of legitimate users to achieve chaos synchronization, resulting in generating highly correlated waveforms from the response lasers of legitimate parties. Then, characterized by their filtering capabilities and dual-output nature, DLIs are deployed to generate two interference spectra with low correlation. Thirdly, the two signals are injected into the SMMM, performing the mutual modulation between the two signals by employing a 2 × 2 optical switch (2 × 2 OS) to intentionally change the optical input and electrical input of a phase modulator. A photodiode (PD) is utilized to convert one optical signal into the electrical domain to modulate the other in the phase. Next, a dispersive module is employed to convert the phase scrambling into intensity perturbation, so that it can remarkably optimize the chaotic effective bandwidth to over tens of gigahertz, breaking the limit imposed by the relaxation oscillation frequency inherent in lasers [32]. The effective bandwidth is defined as the range from DC to the frequency where 80% of the energy is included [33,34]. Since the entropy source rate is closely correlated with the chaotic effective bandwidth, enhancement in bandwidth can significantly enlarge the entropy source rate, which is essential for increasing the key generation rate in the case of frequency-multiplication extraction [29,30]. In addition to the DFB lasers’ parameters in conventional key distribution scheme, the introduction of the DLIs and mutual modulation module with an extra of hardware parameters, including delay time of DLIs, delay time in mutual modulation module, modulation coefficient, and dispersion, can greatly enhance the hardware key spaces and the privacy of generated keys, which inevitably increases the difficulty for eavesdropper to obtain legitimate hardware parameters to achieve chaos synchronization for intercepting the confidential information.
Fig. 1. Schematic diagram of the proposed key distribution scheme based on interference spectrum-shifting keying chaos synchronization with signal mutual modulation in commonly-driven synchronization.
In the SMMM, dynamic keying of the interference spectrum is controlled by two independent random binary codes CA and CB, which are generated from random number generators deployed in legitimate users respectively. Thanks to the 2 × 2 optical switch shifting the state of the optical path between straight-through and cross-over states, the two distinct signals generated in DLIs can mutually modulate each other. In this scheme, output from DLI’s upper arm is denoted as α, and output from the lower arm is denoted as β. When the binary code is set to ‘0’, the optical switch is in the straight-through state, enabling α to be converted into an electrical signal for the phase modulation of β, and this modulation state is denoted as S0. In contrast, when the binary code is set to ‘1’ at the moment, the optical switch is in the cross-over state, β is converted into an electrical signal to modulate α in phase, and this modulation state is denoted as S1. When the binary codes randomly shift between ‘0’ and ‘1’, the corresponding state will be dynamically switched. Only in the time slots when CA = CB, can the modulation states remain identical and the temporal waveforms are synchronous between two legitimate users. From the description above, dynamic keying of the interference spectrum can improve the security and enhance the randomness of the key distribution process. It is worth mentioning that the security performance of the proposed scheme can be greatly enhanced by cascading SMMMs [35]. More specifically, if the number of SMMM stages is set to N, an eavesdropper would have to construct 2N cascaded SMMM systems to intercept the correct information.
Subsequently, the interference spectrum shift keying chaotic signals are sampled and quantized to generate raw random bits with the dual-threshold quantization method. If the samples exceed the upper threshold, they will be quantized as bits ‘0’; conversely, if the samples fall below the lower threshold, they will be quantized as bits ‘1’; samples that fall between the upper and lower threshold will be discarded. Since the raw random bits generated in the time slots of chaos synchronization are identical in principle, the two legitimate users exchange control codes CA, B over the public channel and compare CA, B to sift the identical bits as shared keys.
To examine the security of the proposed scheme, we have conducted a comprehensive analysis of the security of such a scheme under various attack scenarios [29,30,35]. Given that the eavesdropper lacks the necessary hardware modules, it becomes significantly more difficult for her to establish synchronization with the legitimate users, so it becomes significantly more challenging to extract the confidential key by direct detection of the common driving signal, thereby ensuring the key distribution system. Given the potential for sophisticated eavesdroppers to extensively search through all hardware parameters over an extended period of time, followed by the counterfeiting of a compatible hardware module through a brute-force attack, dynamic keying chaos synchronization becomes imperative to introduce an additional layer of enhanced security. Through the random keying of CA and CB, it becomes challenging for an eavesdropper to concurrently select the same keying codes as CA and CB, despite having access to the corresponding hardware modules. Take the scenario where CA randomly selects the keying codes as “00100100…” and CB is “01011010…”. An eavesdropper may attempt to randomly select its own codes as “11011101…”. Notably, for Alice and Bob, the chaotic synchronization can only be achieved and the key distribution executed once the keying codes of CA and CB coincide. Hence, the first and final bit of the keying sequences work for secure key distribution. On the other hand, the eavesdropper's random sequence will necessarily diverge from CA and CB, making any attempt to extract key information impossible.
3. Experimental setup
The experimental system configuration is illustrated in Fig. 2. A super-luminescent diode (SLD) is utilized as the driving source, and after being filtered and amplified, its output is divided into two beams by an optical coupler. Both of these beams are then transmitted over a transmission link consisting of standard single-mode fiber and dispersion compensation fibers before being unidirectionally injected into the response lasers at the two legitimate parties. To achieve chaotic synchronization between two legitimate users, the wavelengths, relaxation oscillation frequencies, and injected powers of the DFB lasers should be settled in suitable values, as described below. The linewidth and center wavelength of the filter utilized to filter the output of the SLD is set to 0.5 nm and 1549.30 nm respectively, with an injected power of 400 µW. The wavelengths of all the lasers in our scheme are set to 1549.22 nm, and the bias currents of DFB1 and DFB2 are set to 1.30 and 1.31 times the threshold current, respectively.
Fig. 2. Experimental setup of the proposed key distribution scheme. DFB, distributed feedback laser; OC, optical coupler; MZI, Mach-Zehnder interferometer; VOA, variable optical attenuator; PC, polarization controller; SMF, single-mode fiber; DCF, dispersion compensated fiber; EDFA, erbium-doped-fiber-amplifier; CIR, circulator; PD, photo-detector; RNG, random number generator; AMP, amplifier; D, dispersion component; and PM, phase modulator.
At each legitimate party, the output from the response DFB laser is first divided by a DLI, with a free spectral range of 10 GHz, into two signals with the characteristic of low correlation in the temporal domain but complementary in the interference spectrum. In this experiment, DLIs are exploited to achieve the intended dual signal generation, the delay time of DLI is changeable for adjusting to different free spectral ranges. Afterward, the two distinct signals are processed by a 2 × 2 optical switch, shifting in straight-through or cross-over states. The lower output of the switch is utilized as an optical input of the phase modulator (PM). Whereas, the upper part, via a photodiode, is converted into an electrical signal for modulating the PM. Before that, the electrical intensity can be adjusted by a low-noise radio frequency amplifier (AMP) to take control of the modulation coefficient. The cutoff bandwidths and half-wave voltage of the PM is 10 GHz and 3.3 volts. And the bandwidths of PDs in the experiment are 40 GHz. After the interference spectrum shift keying and the mutual modulation of the two signals, a dispersion compensation fiber with a dispersion value of around -999 ps/nm is utilized as the dispersive module to convert phase modulation into intensity variation. Finally, an analog-to-digital converter in a real-time oscilloscope with 16 Gbit/s bandwidth and 100 GSa/s sampling rate sampling rate is employed to sample the chaotic signal, thereby generating the raw random bits.
4. Results and discussion
4.1 Chaos correlation
Figures 3(a)-(c) illustrate the temporal waveforms and optical spectra at the output of the two response lasers, indicating almost identical waveforms and spectra. As shown in Fig. 3(c), the correlation plot between the two temporal waveforms concentrates along a diagonal line, implying chaotic synchronization with a cross-correlation value of 0.975. Figures 3(d)-(f) depict the chaotic synchronization between the chaotic output from two DLIs’ upper arms for both Alice and Bob. The spectra of these upper arms exhibit almost identical comb shapes, thanks to the filtering mechanism of the DLIs. The temporal waveforms from the same arms of the DLIs are nearly identical, and the scatter plot demonstrates a concentration along a diagonal line, signifying chaotic synchronization with a cross-correlation value of 0.952 for a data length of 1$\mu s$. Figures 3(g)-(i) illustrate a low cross-correlation value between output from two DLIs’ different arms. The chaotic waveform and optical spectra exhibit noticeable differences, and the cross-correlation plot is present in an extremely decentralized state with a cross-correlation value of only 0.165. This low cross-correlation arises from the peculiarity of DLIs, which can effectively divide the input into two signals with different spectral power distributions.
Fig. 3. (a)-(c) the response lasers, (d)-(f) DLIs’ same arms, (g)-(i) DLIs’ different arms: optical spectra, temporal waveform and cross-correlation plot.
Figure 4(a) depicts the effect of DLIs’ delay time on the cross-correlation when both DLIs have the same delay time in their arms. The results show that the DLI delay time has no significant impact on cross-correlation in this case, maintaining high synchronization with a cross-correlation around 0.95. Figure 4(b) illustrates the effect of DLI delay time on cross-correlation when the DLIs have different delay times in their arms. When the delay is less than 100 ps, the resulting FSR would be larger and cause significant differences in energy distribution in the output interference spectrum of the two arms. This would lead to signal power ratio fluctuations and instability in the correlation between the arms. When the delay time exceeds 100 ps, the power ratio and cross-correlation remain stable at values of approximately 1 and 0.16, respectively. To achieve nearly identical modulation coefficients in different switch states and minimize cross-correlation between the DLIs’ different arms, we set the DLI delay time to 100 ps in the experiment. This setting corresponds to a power ratio of 0.993 and a cross-correlation value of 0.165 between the DLIs’ different arms.
Fig. 4. (a) Cross-correlation between two DLIs’ same arms versus DLIs’ delay time. (b) Cross-correlation between two DLIs’ different arms versus DLIs’ delay time and DLIs’ power ratio between armu (upper arm) and arml (lower arm).
In Figs. 5(a)-(b), as we can see, the chaotic temporal waveforms and optical spectra are remarkably similar under the correlation value of 0.932, when two optical switches operate in the same states. Conversely, as plotted in Figs. 5(c)-(d), the temporal waveforms of the driving source and the chaotic output processed by SMMM appear in totally different profiles with a correlation value of 0.056, indicating that SMMM can effectively suppress the residual correlation, that is, the correlation between driving source and output. In consequence, the random bits extracted from the public transmission link cannot be used to generate the correct key.
Fig. 5. (a)-(b) Temporal waveforms and the corresponding correlation plots of the chaos output from SMMMA and SMMMB. (c)-(d) Temporal waveforms and the corresponding correlation plots of the chaos output from drive source and SMMMA.
In addition, hardware parameters in SMMM play a crucial role in enhancing key security. As shown in Fig. 6, the PM modulation coefficient serves as a key hardware parameter in SMMM and significantly impacts both residual correlation and effective bandwidth. The effective bandwidth of chaotic output gradually increases with the PM modulation coefficient, effectively suppressing and reducing the residual correlation to 0.05. According to previous studies, a cross-correlation higher than 0.9 is sufficient for chaotic key distribution [35]. However, there's a trade-off between cross-correlation and effective bandwidth. To optimize the effective bandwidth, the PM modulation coefficient should increase, which results in the decrease of cross-correlation between chaotic outputs and the deterioration of chaotic synchronization. In this scheme, we set the PM modulation coefficient to 3.3, resulting in a cross-correlation of approximately 0.92 between legitimate chaotic outputs and an effective chaotic bandwidth of around 24 GHz. Apart from this, the residual correlation remains incredibly low at only 0.06, ensuring the failure of eavesdroppers to acquire any information that can be used to generate keys.
4.2 Effects of mutual modulation module parameters mismatch
Figures 7(a)-(d) demonstrates the effect of hardware parameters mismatch on our key distribution system, including DLI’s delay time, PM modulation coefficient, delay time in mutual modulation module, and dispersion. When one of these parameters is adjusted on the Bob side while keeping Alice's constant, the cross-correlation decreases as a result of the hardware parameter mismatch. As shown in Figs. 7(a)-(d), the tolerance for mismatch is defined as a correlation value greater than 0.90. Based on the figures, the tolerances for DLI's delay time, PM modulation coefficient, the delay time in the mutual modulation module, and dispersion are 800 fs, 0.83, 11 ps, and 11 ps/nm, respectively. Notably, these tolerances fall within the controllable range of commercial hardware. Using these mismatch tolerances and the controllable range of commercial hardware, we calculate the key space by dividing the controllable range of hardware parameters by their corresponding mismatch tolerances. Multiplying all key spaces of hardware parameters yields the total key space. The introduction of the SMMM module in the key distribution system has achieved a remarkable total key space enhancement of 235 [36].
Fig. 7. (a)-(d) Cross-correlation versus mismatch of DLIs’ delay time, PM modulation coefficient, delay time, and dispersion.
4.3 Interference spectrum-shift keying with mutual modulation
Based on the above chaos synchronization results, we further investigate the switching characteristics of short-term cross-correlation in the case of interference spectrum shift keying. Figure 8(a) provides a typical result of interference spectrum shift keying based on chaotic synchronization with a time window of 100 ns. The upper and middle rows of Fig. 8(a) display the waveforms of random control codes for two legitimate users, while the short-term cross-correlation between their waveforms is exhibited at the bottom. The time window for the control code and the short-term cross-correlation are adopted as 5 ns and 0.25 ns, respectively. High synchronization, characterized by a cross-correlation exceeding 0.9, is evident when the control codes are identical between the two legitimate users. Conversely, a non-synchronization state, with a cross-correlation of approximately zero, is observed when the control codes are different. It should be noted that the introduction of SMMM enhances the security of the system, as it becomes challenging for an eavesdropper to acquire all the necessary hardware, replicate the system, and operate it under the same conditions as legitimate users to obtain the correct codes.
Fig. 8. (a) Randomly-shifted keying codes and the short-term cross-correlation of Alice and Bob’s chaotic output. (b) the enlarged view of chaotic waveform and difference of the temporal waveforms.
In our experiment, we utilized a 2 × 2 dual-output modulator with a swift switching time of <100 ps. This significantly reduces the transition time from non-synchronization to synchronization, resulting in an improved key distribution rate. As plotted in Fig. 8(b), an enlarged view of the temporal waveforms of two legitimate users and the difference between them shows the shifting process from asynchronization to synchronization. The synchronization recovery time is precisely determined as the shortest duration between the start time with the same control code and the point where the differences between the two temporal waveforms converge within the range of ±10% [28–30]. In Fig. 8(b), a green dashed line on the left represents the start time with the same control code, and the other indicates the time of synchronization recovery. Notably, the recovery time is calculated to be <100 ps, which is extremely shorter than the recovery time in previous keying schemes. Throughout all state-shifting processes, the recovery time remains below 100 ps, effectively enhancing the retention rate of raw random bits and optimizing the key distribution rate.
4.4 Entropy rate estimation and key extraction
The common driving induced synchronization key distribution scheme is based on the key generation model proposed by Maurer, whereby when two users can sample related random sources, they can generate a shared key through the exchange of messages on a public channel [37]. Performance metrics commonly utilized to evaluate the effectiveness of the physical layer key distribution system include key generation rate, error rate, and randomness. The rate of the entropy source will have a great influence on the maximum key distribution rate. The maximum rate of the generated random bits, with confirmed randomness, is determined by the single-threshold quantization method described below. The chaotic temporal waveform is sampled at different sampling rates and quantized into a binary sequence using the median value of intensity as the single quantization threshold that is selected to ensure that the quantized bits of “0” and “1” have equal probability with guaranteed randomness. The final test sequence is generated by executing the exclusive-or operation between the original binary random bits and its delayed copy. The generated binary sequences with 1000 samples of 1Mbit data is then tested by the National Institute of Standards and Technology Special Publication 800-22 statistical tests (NIST SP800-22), consisting of 15 statistical test items, to confirm its level of randomness. If all of the test items are passed, the tested sequences are deemed to satisfy all the requirements of randomness. Figures 9(a)-(b) depicts the passed items at different sampling rates for the outputs from the response laser and the chaos after SMMM. The maximum entropy rate is highlighted with a red arrow. As can be seen from Fig. 9(a), the maximum entropy source rate of the response laser is examined to reach the rate of 10.0 Gbit/s. However, the entropy source rate of chaotic output processed by SMMM can reach up to 20 Gbit/s, indicating the effective optimization of the entropy source rate by SMMM and a substantial enhancement in the final key rate.
Fig. 9. (a) Entropy rate for the signal from DFB. (b) Entropy rate for the signal from the final chaotic output.
In the subsequent step, the chaotic temporal output of interference spectrum-shift keying is quantized into raw random bit sequences using a robust dual-threshold quantization method. These sequences are then stored in recorders alongside their respective keying codes. After the exchange and comparison of keying codes, Alice and Bob can derive the retained random bits as the final keys for achieving key distribution. In the dual-threshold quantization method, the two thresholds are described as follows [35]:
where ${T_{th,u}}$ and ${T_{th,l}}$ are the upper and lower threshold values normalized by the maximum intensity of the temporal outputs, respectively. m and $\sigma $ represent the mean value and standard deviation of the normalized temporal outputs, while ${C_ + }$ and ${C_ - }$ denotes the threshold coefficients used to determine the two normalized threshold values. Depending on the dual-threshold quantization method with upper and lower thresholds, samples falling between them are regarded as susceptible to noise and will be consequently discarded, so the bit error rate (BER) of the shared keys can be decreased to an acceptable extent. However, there exists a trade-off between BER and the shared key rate, as reducing BER inevitably decreases the retained bits, resulting in a decline in the shared key rate. The final key rate can be described as follows [30]:To determine the optimal retention rate for different threshold coefficients, we examine the probability of bit ‘0’ occurrence, BER, and retention rate in the final keys. Figure 10(a) and Fig. 10(d) illustrate the probability of bit ‘0’ occurrence as a function of C+ and C- in the case of same-frequency extraction and frequency multiplication extraction. The ideal probability for random bits (‘0’ or ‘1’) is 0.5. The dashed line in Fig. 10(a) denotes pairs of threshold coefficients where the probability of ‘0’ bit occurrence in the final keys is 0.500. As shown in Fig. 10(b), by increasing the quantization threshold, the sampling points affected by the noise can be remarkably eliminated, resulting in the decrease of BER and the generation rate. For information reconciliation, a (255, 247) Bose-Chaudhuri Hocquenghem (BCH) error-correction code is employed, exhibiting an error-correction capability of 4.0 × 10−3 and a code rate of 96.9%. Consequently, identical, error-free binary key bits are shared among legitimate parties. However, the HD-FEC hard decision threshold utilized in this study stands at 3.8 × 10−3, representing the limit of uncorrectable errors that can be effectively handled by existing production hardware. The dashed line in Fig. 10(b) and Fig. 10(e) represents the threshold for the hard-decision forward error correction (HD-FEC), which is the maximum tolerable BER in our proposed scheme [38]. To achieve a ‘0’ bit occurrence probability around 0.500 and a BER below HD-FEC, we set the threshold coefficients to C+ = 0.2 and C- = 0.6. Figure 10(c) shows that the optimal retention rate of 0.686 is achieved with these threshold coefficients in the same-frequency extraction scenario and Fig. 10(e) depicts the optimal retention rate of 0.652 in the frequency multiplication extraction scenario. And then, it can be obtained that the bit error rate of communication parties is 3.74 × 10−3, while the bit error rate of the eavesdropper is 0.49746 in the case of frequency multiplication extraction. Actually, in the case of same-frequency extraction method, the key extraction rate is the same as the keying sequence rate, and only one bit is extracted during one keying duration. Therefore, the one-bit key extraction can be realized as long as chaos synchronization is achieved. The synchronization recovery time accounts for a negligible contribution to the final key rate, which is mainly determined by the keying extraction rate and the retention ratio in this scenario. The final key rate is calculated as 200 × 0.652 / 2 = 65.2 Mbit/s in the case of same-frequency extraction, while, with frequency multiplication extraction, multiple bit keys can be extracted at the sampling rate of 20.0 Gbit/s during each keying code duration, greatly optimizing the final key rate with confirmed randomness. Therefore, the final key rate is calculated to be 0.5 × 0.686 × 20 Gbit/s × (5 ns - 0.09 ns) / 5 ns = 6.7 Gbit/s with a BER lower than the HD-FEC threshold [29,30]. Figure 11 shows the randomness test results for the final keys. It’s noteworthy that all P-values exceed 0.0001, and the proportions fall within the confidence interval of 0.99 ± 0.0094392 [39]. This demonstrates that the generated key with the rate of 6.7 Gbit/s successfully passes all the 15 NIST tests, meeting the required level of randomness.
Fig. 10. (a)-(c) Probability of the occurrence of bit ‘0’, bit error rate and retention rate as functions of threshold coefficients ${C_ + }$ and ${C_ - }$ in the same-frequency extraction scenario. (d)-(e) Probability of the occurrence of bit ‘0’, bit error rate and retention rate as functions of threshold coefficients ${C_ + }$ and ${C_ - }$ in the frequency multiplication extraction scenario.
5. Conclusions
In conclusion, we propose and experimentally demonstrate a high-speed physical key distribution scheme based on interference spectrum shift keying chaos synchronization with chaos mutual modulation in commonly-driven chaotic synchronization. The results highlight the effectiveness of DLIs in dividing chaotic input into two outputs with low correlation. These DLIs, with their same arms achieving chaotic synchronization and different arms remaining asynchronous, eliminate the need for two identical sets of hardware to generate two signals with low correlation. The utilization of 2 × 2 optical switches significantly reduce the synchronization recovery time to below 100 ps. The introduction of SMMM enhances the entropy source rate to an impressive 20 Gbit/s, eliminating residual correlation between the driving source and the final chaotic output. Thanks to interference spectrum-shift keying chaotic synchronization and the privacy of the hardware modules in the proposed scheme, the security performance of key distribution can be considerably guaranteed. It is worth noting that the security of the proposed scheme can be further enhanced by cascading SMMMs. In our experiments, we achieved a high key distribution rate of up to 6.7 Gbit/s over a 40 km fiber transmission, underscoring the promise and potential of our proposed scheme for high-speed and secure key distribution.
Funding
National Key Research and Development Program of China (2023YFB2906000); National Natural Science Foundation of China (62004047, 62375055, U2001601, U22A2087); Guangdong Basic and Applied Basic Research Foundation (2023B1515020088); Guangdong Introducing Innovative and Enterpreneurial Teams of “The Pearl River Talent Recruitment Program” (2019ZT08X340).
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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