## Abstract

In this work, a method of generating all-optical random numbers based on optical Boolean chaotic entropy source is proposed. This all-optical random number generation system consists of a Boolean chaotic entropy source and an optical D flip-flop. The Boolean chaotic entropy source is composed of an optical XOR gate and two self-delayed feedback; meanwhile, the optical D flip-flop is composed of two optical AND gates and one SR latch. The optical Boolean chaotic signal possesses the dynamic characteristics of complexity and binarization, so random numbers would be generated only by extracted from chaotic signals with the optical D flip-flop. This all-optical random number generation system achieves the result of 5 Gb/s random numbers that is testable. The whole process of random number generation could be completed in the optical domain without photoelectric conversion, more importantly, the device could be integrated.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

The famous “one-time pad” theory proposed by Shannon can realize absolutely secure transmission of information [1]. The theory requires that the key applied for encryption is truly random and cannot be reused, meanwhile, the length of the key is not less than that of the plaintext. To achieve such a perfect cipher, it is necessary to find a method to generate high-speed physical random numbers in real-time.

Traditional physical random numbers are mainly extracted from entropy sources such as thermal noise, electronic device jitter and photon noise [2–6]. While, it is difficult to generate high-speed physical random numbers by these physical entropy sources since the low bandwidth and insufficient amplitude.

To improve the rate of random numbers, in recent years, chaotic laser entropy source has been widely studied for its unique advantages of wide bandwidth and high amplitude. In 2008, Uchida *et al*. proposed optical chaos generated by two optical feedback semiconductor lasers as an entropy source at the first time, meanwhile, real-time physical random numbers of 1.7 Gb/s were generated [7]. Since then, increasing the bandwidth of chaotic entropy source has become an effective way to improve the random number rate. Moreover, the rate of random number could be further enhanced by combining complex post-processing with high bandwidth entropy source. By these efforts, the rate of random numbers has been promoted to the order of Gb/s [8–16]. The study of Reidler *et al*. indicated that 12.5 Gb/s random bits were extracted from the first derivative between the digital chaotic signal and its time shifted version [9]. In addition, Kanter *et al*. demonstrated that the rate of the random number could be improved to 300 Gb/s by higher order derivative of digital chaotic signal [10]. Recently, the rate of random number generator shows the potential of increasing to Tb/s by similar multi-bit extraction [17–20]. Admittedly, this method combined with post-processing could greatly improve the rate of random number. Nevertheless, most of these ultra-fast random numbers are generated by offline computer processing after collecting data that is not real-time. Some methods generating the real-time random numbers by chaotic lasers are proposed [21–25]. Notably, sampling and quantization exist in the process of random number generation. Moreover, some studies revealed that the random numbers could be obtained by real-time differential XOR processing after electric comparator. While, the process of extraction is complicated.

To simplify the extraction, Boolean chaotic random number entropy source has been explored [26–28]. The output time series of Boolean chaos show the characteristic of binarization, so Boolean chaos could be used as entropy source to generate random numbers without complex quantization process [26], in which only D flip-flop trigger is necessary to complete the extraction of random numbers. Lately, 12.8 Gb/s random numbers could be generated by cascading Boolean chaotic entropy sources, which just needs trigger sampling without complex quantization [28]. Yet, the rate of single channel random number is only 100 Mb/s because of the limitation of electronic bottleneck.

Different from the conventional Boolean chaos, a method of generating Boolean chaos in the whole optical domain has been proposed in our previous work [29]. The optical Boolean chaos displayed high bandwidth and it was unrestricted to the electronic bottlenecks, so that it could be used to generate high-speed physical random numbers. Based on the previous work of optical Boolean chaos, the objective of this work is to extract high-speed physical random numbers in real-time by a simple method.

Herein, the optical Boolean chaos as entropy source combined with an optical D flip-flop has been applied to generate all-optical random numbers. Optical Boolean chaos is composed of an optical XNOR gate and two delay lines. The D flip-flop consists of two optical AND gates and an optical SR latch. The high and low power signals in the optical Boolean chaotic output sequence show Gaussian distribution, which ensures that unbiased random numbers could be extracted. Finally, 5 Gb/s random numbers that can pass the NIST test is produced by the all-optical random numbers system. There is no photoelectric conversion in the entire process of random number extraction and the final output could be adapted to optical communication networks.

## 2. Principle

As shown in Fig. 1, all-optical random number generator is composed of an optical Boolean chaotic entropy source and an optical D flip-flop. The optical D flip-flop consists of two optical AND gates [30,31] and a SR latch [32]. The random numbers rate is controlled by an external optical clock.

The detailed device for generating optical random numbers based on the optical Boolean chaos is shown in Fig. 2. The entropy source of optical Boolean chaos has been reported before [29]. Specifically, it is mainly composed of an optical XOR gate realized by a semiconductor optical amplifier Mach-Zehnder interferometer (SOA-MZI), an optical NOT gate realized by SOA and two self-feedback delays. The degradation effect [33,34] in SOA is the physical mechanism of Boolean chaos generated in the system. When a narrow pulse is input into the SOA, an incomplete pulse could be output because of the incomplete response of SOA, subsequently, the optical Boolean chaos would be output.

The optical Boolean chaotic signal shows dynamic characteristics of complexity and binarization, so random numbers could be generated only by extracted from chaotic signals with the optical D flip-flop. The optical D flip-flop is composed of two optical AND gates and one SR latch. The two AND gates composed of SOA 4 and SOA 5 are used to generate “Set” and “Reset” signals for SR latch. Specifically, the output of the optical Boolean chaos is divided into two channels which are injected into the two optical AND gates respectively. In SOA 4, the four-wave mixing effect (FWM) occurs between the chaotic signal and the clock signal, which is equivalent to the AND operation between the clock signal the Boolean chaotic signal. Subsequently, the sideband frequency signal filtered by the band-pass filter 1 (BPF1) is taken as the “Set” signal. In SOA 5, the cross-gain modulation effect (XGM) occurs between the chaotic signal and the clock signal, which is equivalent to the AND operation between the clock signal and the signal that is complementary to the Boolean chaotic signal. The filtered signal by the BPF2 is used as the “Reset” signal.

Furthermore, the optical SR latch is mainly composed of SOA 6 and SOA 7. The SR latch possesses two output states (*Q* = 1 or *Q* = 0) which depend on the two control signals those are injected into the “Set” port and the “Reset” port. Notably, the “Set” signal and “Reset” signal are complementary, which ensures the high extinction ratio of the output signal. The carriers of SOA 7 are depleted when the “Set” signal arrives. At this moment, the SR latch outputs continuous wave amplified by SOA 6 (*Q* = 1). Similarly, the “Reset” signal depletes the carriers of SOA 6 and the continuous optical signal cannot be amplified. As a result, the SR latch outputs low power signal (*Q* = 0). The all-optical D flip-flop can be realized through the above process of generating the control signal and SR-latch, and then the trigger sampling of the optical Boolean chaos would be completed.

## 3. Simulation and result

To verify the feasibility of all-optical random number generation based on optical Boolean chaos. The system of optical Boolean chaos combined with optical D flip-flop is built in the simulation software VPItransmission Maker. The main parameters of the SOA are shown in Table 1 [29] and other parameters are as the standard parameters in VPI [35].

It should be pointed out that the “Reset” signal of the sideband signal filtered out by the four-wave mixing effect in SOA 4 is low. Therefore, in order to ensure the normal operation of the subsequent SR latch, the bias current setting of SOA 4 is set to 400 mA.

#### 3.1 Optical Boolean chaos and all-optical random numbers generation

To generate random numbers that could pass the randomness test, the spectrum and autocorrelation curve of optical Boolean chaotic entropy source should be analyzed firstly. The two delay times of Boolean chaos used in this manuscript are the same as that in the Ref. [29]. The shortest delay time and delay time difference are 1 and 0.15 ns, respectively. The spectrum characteristics and autocorrelation curve of the optical Boolean chaotic entropy source are analyzed, as shown in Fig. 3. The 10 dB bandwidth of the optical Boolean chaos reaches 14 GHz [26], which is greatly improved compared with that of the electric Boolean chaos. In addition, the autocorrelation time of the optical Boolean chaos falls off to close to zero within 0.07 ns and no time delay signatures appear. The aperiodicity of output random sequence is guaranteed by the autocorrelation curve without correlation peak. Accordingly, these characteristics of optical Boolean chaos allow the external optical clock with high frequency to extract random numbers.

Moreover, whether the optical Boolean chaotic entropy source would output biased random sequences should also be considered, which is very important for the random numbers to pass the randomness test. Based on the optical Boolean chaotic entropy source, only optical D flip-flop is needed to extract random numbers. The distribution of time differences displays great influence on the distribution of “0” and “1” in random numbers. Furthermore, the time differences could be obtained by the following: the “1” in the output sequence is represented by the time interval between the rising edge and the falling edge; the “0” is defined by the time interval between the falling edge and the rising edge; finally, the distribution of time differences is defined as difference between the above two-time intervals.

In order to analyze the uniformity of the output random number, we obtain a 2 μs optical Boolean chaotic output timing when the sampling rate is 100 Gb/s. The time interval between transition edges is obtained by the following method. The median in the time series is used as the comparison threshold, the value smaller than the threshold is set to 0, and the value larger than the threshold is set to 1. Through this method, the optical Boolean chaotic sequence is transformed into a sequence of Boolean variables, and the time interval between transition edges is obtained. As illustrated in Fig. 4, the time differences Δ*t* show highly symmetry, which guarantees that unbiased “0” and “1” could be uniformly extracted. This is beneficial to obtaining the testable random numbers from the optical Boolean chaos.

In this work, the carrier recovery time of SOA is about 60 ps, and the 10 dB bandwidth of the optical Boolean chaotic entropy source is 14 GHz. With these parameters, random numbers with a rate of 5 Gb/s that pass the test could be obtained. Qualitatively, the rate of random numbers depends on the bandwidth of optical Boolean chaos, and the bandwidth is related to the speed of carrier recovery time of SOA. Specifically, short carrier recovery time of SOA will lead to high bandwidth optical Boolean chaos, and then high-speed random number can be extracted.

The random numbers generated by the optical Boolean chaos is shown in Fig. 5. A 5 GHz optical clock is used to trigger Boolean chaotic signals. As Fig. 5 presented, the “Set” pulse would be generated only when the four-wave mixing effect (FWM) occurs between the optical clock and Boolean chaos with high power in SOA 4. The “Reset” pulse would be generated when the cross-gain modulation effect (XGM) occurs between the optical clock and Boolean chaos with low power in SOA 5. Conversely, carriers in SOA would be consumed by the high-power Boolean chaos, resulting in the fact that clock signal cannot be amplified.

In this work, the wavelength of clock signal injected into SOA 4 and SOA 5 is 1550 nm. The output signals of the two SOAs pass through a band-pass filter, respectively. Specifically, the center wavelength of BPF1 is 1546 nm and its filter width is 1 nm. Different from BPF 1, the center wavelength of BPF2 is 1550 nm. The power of two continuous waves (*P*_{IN3} and *P*_{IN4}) injected into SR latch is 1 mW. The continuous signal *P*_{IN3} could be obtained by the same light source as *P*_{IN1} and the wavelength of continuous signal *P*_{IN4} is 1546 nm. In the initial state, SOA 4 and SOA 5 are kept in balance when no external pulse passes through the SR latch. When the “Set” pulse signal enters into SOA 7, abundant carriers are consumed by the pulse signal, which greatly reduces the gain of the continuous wave. Meanwhile, the carriers in SOA 6 begin to recover, which allows SOA 6 to amplify the continuous optical signal, which finally leads to the output of a high-power signal. When SOA 6 is injected by the “Reset” pulse signal, large amounts of carriers are consumed. As a result, the gain of SOA 6 to continuous wave is greatly reduced and eventually a low-power signal is output. In summary, the high-power signal “1” would be output from the SR latch when a “Set” pulse signal is injected into the SR latch. Additionally, the low power signal “0” would be output when a “Reset” pulse signal enters the SR latch. The original output state would be maintained by the system if continuous pulses exist in the “Reset” or “Set” signal. Eventually, all optical random numbers based on the optical Boolean chaos would be achieved.

#### 3.2 Randomness verification

The industry standard statistical test suite of the National Institute of Standards Technology Special Publication 800-22 (NIST SP 800-22) is usually used to test the statistical characteristics of random numbers [36]. This standard that is widely accepted in the field of secure communications consists of 15 test items and the names of the items are shown in Fig. 6. In this work, 100 groups of samples are set in the test process, and each group contains one million bits when the significance level *α* is 0.01. Notably, each bit is obtained by encoding the output waveform. Two indicators are used as signs that random numbers pass the test, specifically, the *P*-value of each test item is greater than 0.0001 and the all proportions are in the range of 0.99 ± 0.0094392. The test result shown in Fig. 6 indicates that all test items pass the standard.

Additionally, it is necessary to evaluate the entropy of physical entropy source quantitatively. In this work, the entropy of the final output random number was evaluated by randomness statistical test NIST SP 800-90B [37].

Different from NIST SP 800-22 which focuses on the statistical randomness of digit bits (“0” and “1”) in random sequences under the “null hypothesis testing”, NIST SP 800-90B is aimed at using a new method to calculate the minimum entropy of test sequences. The minimum entropy of *n*-bit signal generated from physical entropy source could be calculated by the test method.

The statistical tests are composed of ten test items listed as follows: the most common value (MCV), LRS, Markov, t-Tuple, compression, collision, MultiMCW, MultiMMC, Lag, and LZ78Y. The last four items are prediction tests, which are used to analyze the unpredictability of the test sequences. In this work, one million data bits extracted from the optical Boolean chaos are applied to calculate the entropy by NIST SP 90-B. Table 2 shows that the minimum entropy of the ten items is 0.8630, which indicates that the entropy source is random.

## 4. Conclusion and discussion

With the decrease of the volume of communication equipment and the increase of optical communication rate, higher requirements are put forward for all-optical random number generator. The integration of SOA has been reported and the technology is mature. Moreover, the study revealed that the integrated delay line technology is feasible. Generally, the system generating random numbers based on optical Boolean chaos can be integrated theoretically. The rate of random numbers generation depends on the bandwidth of optical Boolean chaos. The system of optical Boolean chaos is mainly composed of optical logic gates. It could be concluded that improving the processing speed of logic gate can fundamentally solve the problem of low rate of random number generation. For example, additional probe light injected into the SOA could be used to reduce the carrier recovery time of the SOA. In addition, maximizing the rate of random numbers generation based on the optical Boolean chaos is also the future work.

The current methods for generating high-speed random numbers based on laser chaos have the disadvantages that they cannot be generated in real time or the extraction process is too complicated. In addition, due to the existence of electronic bottleneck, it is difficult to generate high-speed random numbers based on electrical Boolean chaotic entropy source. Therefore, it is a promising choice to use optical Boolean chaos, which is not limited to the electronic bottleneck, as the entropy source of random numbers, and it has not been reported yet. Although there are some non-ideal characteristics in SOA in practical applications, such as nonlinear polarization rotation and power fluctuation, the influence on the output can be reduced by changing external parameters and adding polarization controller. Therefore, the actual non-ideal factors would not affect the significance of this work for generating real-time all-optical random numbers based on optical Boolean chaos.

In this work, the optical Boolean chaos as entropy source combined with an optical D flip-flop has been applied to generate all-optical random numbers. Optical Boolean chaos is composed of an optical XNOR gate and two delay lines. The optical Boolean chaos shows the advantages of short correlation time and Gaussian distribution of consecutive Boolean transitions time difference in the output signal. Therefore, it can be used to generate high-speed random numbers. The D flip-flop consists of two optical AND gates and an optical SR latch. The Boolean chaotic signal and external clock signal produce FWM and XGM effects in the two AND gates, respectively. The output signals of these two AND gates are applied as “Set” and “Reset” signals of SR latch. Finally, 5 Gb/s random numbers that can pass the NIST test is produced by the all-optical random numbers system. This work provides a strategy to realize random number in all-optical domain.

## Funding

National Natural Science Foundation of China (61731014, 61805171, 61927811, 61961136002); Natural Science Foundation of Shanxi Province (201801D121145); Guangdong Province Introduction of Innovative R&D Team; Shanxi Scholarship Council of China (2017-key-2).

## Disclosures

The authors declare no conflicts of interest.

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