1. Introduction

Random bits are widely applied in various fields including cryptographic systems, information security, scientific simulation, stochastic modeling, and even lotteries [1–4]. Compared with pseudo-random bits, physical random bits are more suitable for areas with higher security requirement such as “one-time-pad” encryption because of the unique advantage of true unpredictability. Moreover, the bits rate of physical random bit generators (RBGs) also determines the data transmission rate of security communication and cryptography. Thus, it is increasingly important to develop fast physical RBGs for adapting to future applications in large-capacity security communication.

Over the past few decades, photonic broadband entropy sources such as spontaneous emission noise [5–7], laser phase fluctuations [8–10], and laser chaos [11–23] have been widely explored for increasing the physical random bits generation rate. In particular, laser-chaos-based RBGs have attracted extensive attention because the achieved bit rates can readily exceed tens of Gbps enabled by fast laser dynamics. The pioneering work was done by Uchida et al. in 2008 with combined binary digitization of two independent chaotic semiconductor lasers (SLs) [11]. Since then, for faster bit rates, considerable efforts have been dedicated [12–23]. However, due to laser’s relaxation oscillation, generated chaotic signals usually characterize with uneven power spectrum and narrow bandwidth, which would limit the ultimate generation rate by the roots. In addition, chaotic lasers exhibit strong termed time-delay signature (TDS) (typically introduced by external cavity feedback). The existence of TDS not only hinders the passing of randomness tests in RBG, but also seriously threatens the security of communication.

To overcome above issues, large efforts have been devoted to enhancing the bandwidth [13–17], suppressing the TDS [24–26], or simultaneously eliminating these defects to generate white chaos with the optical heterodyne technique [27]. However, these above-mentioned schemes have complicated structures and are difficult to generate a broadband random entropy source signal with a 3-dB bandwidth of more than 30 GHz. Besides, we notice that most of them focus on single-mode SLs and only few of them study the use of multi-mode lasers such as FP-LDs to generate random entropy sources. For example, Chen et al. [28] implemented a wavelength-tunable chaotic signal generation based on an integrated chip consisting of an FP-LD and a photodiode (PD) subjected to filtered optical feedback. Zhang et al. [29] proposed the chaos bandwidth could be enhanced by dual-wavelength injection into the external-cavity feedback FP-LD. Recently, based on an FP-LD, Li et al. [30] reported a new proposal for flat chaos generation in low-frequency components by enabling single-mode chaotic signal output with a tunable optical filter. Hu et al. [31] demonstrated multi-channel chaotic signals generation could be realized with the use of two unidirectionally coupled weak-resonant-cavity FP-LDs. However, while the newly developed methods based on the FP-LD have exhibited superior performance in chaotic signal generation, none of them has efficiently exploited the multi-longitudinal modes of the FP-LD (i.e., it can be equivalent to multiple independent external cavity feedback chaotic SLs).

Considering such advantage of FP-LDs, its application for the generation of broadband chaotic signals and fast random bits should be reasonable, but no relevant studies have investigated it yet. Here, we proposed and demonstrated, for the first time to our knowledge, the optically beating of two multimode FP-LDs subject to optical feedback with different mode intervals via a fast photodetector can produce a broadband flat MMW chaos signal with perfect characteristics of the white noise. In our proof-of-concept experiment, an MMW white noise signal covering 50 GHz (limited by the measurement bandwidth) with a flatness of 2.9 dB has been actually achieved. Then, utilizing the generated broadband and flat MMW noise signal as the entropy source, 640 Gb/s physical random bits with verified randomness are successfully obtained by directly retaining 4 LSBs from 8-bit digitization at 160 GS/s sampling rate without any other post-processing.

2. Experiment Setup

As shown in Fig. 1, the whole experimental setup includes two parts: generation and quantization of MMW white noise. The generation setup of MMW white noise consists of two FP-LDs subject to optical feedback, which are referred to as two multi-mode chaotic light sources, and an optical heterodyne detection unit, which can be implemented by virtue of a high-bandwidth PD after a 3-dB fiber coupler (FC). For the generation of two multi-mode chaotic light, two external-cavity lasers (ECLs) are constructed, each of which contains a FP-LD_1,2 subject to optical feedback from a fiber mirror (M_1,2). Note that the laser facet and the mirror comprise a feedback external cavity. In the feedback cavity, a polarization controller (PC_1,2) is used to match the polarization state of the feedback light injected to the FP-LD_1,2, and a variable attenuator (VOA_1,2) is utilized to adjust the feedback strength to change the dynamic state of the FP-LD_1,2. Under suitable feedback conditions, two multi-longitudinal co-existing FP-LDs are driven into chaotic states. Meanwhile, their optical spectra can be significantly broadened. After passing through the optical isolators (ISO_1,2), two multi-mode chaotic light are coupled through the 3-dB FC, and then further amplified by an erbium-doped fiber amplifier (EDFA). Finally, the optically beating MMW white noise signal after a high-speed PD is inputted to and digitized by an 8-bit ADC: each sampling point is converted into 8-bit binary digits. For 8-bit ADC, D₀ and D₇ are considered as the least significant bit (LSB) and the most significant bit (MSB), respectively. Here, it is worth mentioning that the 8-bit ADC used in our experiments is provided by a 59 GHz real-time digital oscilloscope (OSC, Lecroy, LabMaster10-59Zi, 160 GS/s sampling rate, 8-bit vertical resolution).

 

Fig. 1. Schematic of fast physical random bit generation based on a broadband flat MMW white noise signal. FP-LD_1,2, Fabry-Perot laser diode; PC_1,2, polarization controller; FC_1,2,3, fiber coupler; VOA_1,2, variable optical attenuator; M_1,2, fiber mirror; ISO_1,2, optical isolator; OSA, optical spectrum analyzer; EDFA, erbium-doped fiber amplifier; PD: photodetector; ESA: electrical spectrum analyzer; ADC: analog-to-digital converter; MSB: most significant bit; LSB, least significant bit.

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3. Experiment Results

3.1 Generation of MMW white noise by optical beating of two FP-LDs

The basic properties of the two free-running FP-LDs without external optical feedback are firstly depicted here. We adjust currents of two FP-LDs (Eblana Photonics, EP1550-FP-B) which are biased at 18.82 mA (1.94 I_th) and 21.97 mA (1.91I_th), and use the laser diode temperature controllers (ILX Lightwave LDC-3706) to adjust the temperature to 20.3 °C and 22.6 °C, respectively, so that the center wavelengths of the two lasers are precisely aligned at 1550 nm. The feedback intensity is about 20% and the distance from the FP-LDs to the fiber mirrors are about 9.1 m. Experimental optical spectra results are recorded using an optical spectrum analyzer (OSA, APEX, AP2081-A) with a resolution of 0.01 nm. According to the obtained experimental data, it is calculated that the mode intervals are about 1.26 nm and 1.36 nm, respectively. Similarly, through further analysis of the optical spectra of the two free-running FP-LDs, it is known that the 3-dB linewidths of the multi-longitudinal modes are 0.011 nm and 0.012 nm, respectively.

To facilitate the comparison of the optical spectra evolution, the optical spectra of the two multi-mode lasers before and after optical feedback for FP-LD₁ and FP-LD₂ are depicted in Figs. 2(a1) and 2(b1) respectively. By contrast, one can clearly observe that the optical spectra for each longitudinal mode are significantly broadened after optical feedback, as shown in the inset of Figs. 2(a1) and 2(b1). It is confirmed that 3-dB optical spectral linewidths of the two multi-mode chaotic lasers have reached up to 0.052 nm and 0.054 nm, respectively. This means that the 3-dB optical spectral linewidths of both FP-LDs are broadened nearly five times after the additional optical feedback. It is also an important manifestation of the evolution of free-running lasers into chaotic oscillations. Figures 2(a2) and 2(b2) show the obtained frequency spectra output from optical feedback FP-LD₁ and FP-LD₂ after photoelectric conversion by a PD under the same operation parameters (i.e., center wavelengths, bias currents, feedback ratios, etc). Obviously, the relaxation oscillation frequencies of FP-LD₁ and FP-LD₂ are about 9.2 GHz and 10.5 GHz, respectively. In this case, most of the energies of the obtained multi-mode chaotic signals are concentrated around the relaxation oscillation frequencies. Consequently, their 3-dB bandwidths are only 4.4 GHz and 4.3 GHz, respectively. Moreover, one can also notice that when the frequency reaches 20.1 GHz, the power spectra substantially coincide with the noise floor.

 

Fig. 2. Comparison of characteristics of multi-mode external-cavity optical feedback chaotic signals and the optically beating MMW white noise signal. (a1), (b1) and (c1) optical spectra; (a2), (b2) and (c2) radio-frequency (RF) spectra; (a3), (b3) and (c3) time series; (a4), (b4) and (c4) amplitude probability distributions; (a5) (b5) and (c5) autocorrelation function traces.

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To overcome above issues, we propose to use optical beating technique to produce a broadband flat MMW white noise signal. Figure 2(c1) shows the tuned optical spectra of FP-LD₁ and FP-LD₂ in the range of 1543.2 nm to 1555.4 nm, where the main modes coincide at 1550 nm. For the convenience of subsequent analysis, the longitudinal mode with the highest energy (center wavelength of 1550 nm) is defined as 0 mode (m = 0). At the same time, the two longitudinal modes close to 0 are defined as +1 mode (m = +1) and −1 mode (m = −1), respectively. The other patterns are sequentially defined as +2, −2, +3, −3, +4, −4 mode etc. Then, the upper and lower FP lasers would experience optical heterodyne in the 50 GHz PD (Finisar, XPDV2120R) after optical feedback. In our experiment, the generated noise temporal waveform and radio-frequency (RF) spectra are measured by the 59 GHz digital OSC (Lecroy, LabMaster10-59Zi) and a 50 GHz electronic spectrum analyzer (ESA, Rohde & Schwarz, FSW50). The polarization controllers and optical attenuators of the upper and lower channels are used to broaden their spectrum of the FP-LDs, and then a wide flat MMW white noise signal can be generated, as shown in Fig. 2(c2), and the achieved flatness is 2.9 dB. We can observe that the flatness deteriorates while approaching 50 GHz, the reason is that the noise floor of the electrical spectrum analyzer increases and exceeds the MMW power a little bit around this frequency, as shown in Fig. 2(c2).

The characteristics of the achieved MMW white noise signal including the temporal intensities and the autocorrelation function (ACF) are further investigated. Figures 2(c3) and 2(c4) show the temporal waveforms and the corresponding probability density function (PDF) of the heterodyne beating result, which behaves dramatically noise intensity oscillation and symmetrical amplitude distribution. Quantitatively, it has a small skewness of 0.0261. Figures 2(a3) and 2(b3) display the temporal waveforms of two multi-mode chaos outputs from external-cavity optical feedback FP-LD₁ and FP-LD₂, which exhibit typical chaotic fluctuation. The corresponding PDFs are depicted in Figs. 2(a4) and 2(b4). The asymmetric PDFs can also be identified by comparing it with the Gaussian fitted curves (plotted as the black dotted lines). After calculation, their skewness are both about 0.926, which are significantly larger than the skewness of the obtained MMW white noise. Figures 2(a5), 2(b5), and 2(c5) show the autocorrelation function of two single channel chaotic signals and the generated MMW signal after heterodyning, respectively. From the Figs. 2(a5) and 2(b5), one can clearly observe that the two feedback delays of optical feedback FP-LD₁ and FP-LD₂ are τ₁= 91.3 ns and τ₂ = 93.7 ns, corresponding to the external-cavity round trip time of the two FP-LDs. In contrast, as depicted in Fig. 2(c5), no more peaks appear at the delay in the autocorrelation function of the heterodyne signal. Thus, the generated signals exhibit typical white noise characteristics.

In order to clarify the physical origination reason why the beat frequency of FP lasers can produce broadband flat MMW white noise, the principle for the broadband flat MMW white noise generated by the optically mixing of the two FP-LDs is also briefly explained here. Since the 0 mode of FP-LD₁ and FP-LD₂ are aligned, that is, they have the same central wavelength, and the mode interval of FP-LD₁ and FP-LD₂ are 1.26 and 1.36 nm, respectively. The +/−1, +/−2, +/−3, and +/−4 mode difference of FP-LD₁ and FP-LD₂ are 0.1 nm, 0.2 nm, 0.3 nm and 0.4 nm, respectively. The center frequency of the beat term will appear in 12.5 GHz, 25 GHz, 37.5 GHz, and 50 GHz in the frequency spectrum. In fact, the noise bandwidth generated by the beat frequency between the single mode of the laser is relatively narrow, and the linear superposition of the beat frequencies of multiple modes can generate broadband flat noise. This is also the reason why we use multi-mode lasers, the modes of FP-LDs are fully utilized here. Correspondingly, after the superposition of these beating terms, a flat frequency spectrum can be easily achieved.

3.2 Generation of random bits by an 8-ADC

In this section, we successfully realized fast random bit generation through a typical combination of multi-bit quantization (an 8-bit ADC embedded in the 59 GHz real-time digital OSC is employed to quantize the MMW noise signal) and m-LSBs retention. The selection of a few LSBs from the multi-bit digitized outputs has been widely recognized as a critical procedure for efficiently improving the uniformity and eliminating the residual correlations of generated random bits [32,21]. In practice, to ensure the randomness, particular attention should be given to the following two key aspects (i) the equilibrium of the probability distribution of digitization levels, which can equivalently reflect the deviation of logical 0/1 values; (ii) the autocorrelation function of the digitization values obtained from the 8-bit ADC, which can be used for judging whether the correlation degree of the statistical bits sequence is significant.

Figures 3(a)-7(h) show the probability distribution of the decimal digitization levels when the number of extracted LSBs gradually increases. Specifically, while 7-LSBs are retained, the probability distribution as depicted in Fig. 3(g), shows an obvious non-uniform distribution. Through further discarding several of the MSBs, as shown in Fig. 3(f) and Fig. 3(e) where 6-LSBs and 5-LSBs are retained respectively, the uniformity is significantly improved. When 4-LSBs are retained, a flat histogram can be achieved as shown in Fig. 3(d). In this case, its probability distribution is well approximated to a perfect uniform distribution, which is one of the most important prerequisites for producing un-biased and reliable random bits. To quantitatively evaluate the uniformity, we introduce the cumulative probability deviation $\Delta \textrm{ = }\sum\limits_{i = 1}^{{2^N} - 1} {|{{P_i} - {2^{ - N}}} |}$, where P_i is the probability of an N-bit binary number corresponding to a decimal integer between 0 and 2^N−1. For a uniform distribution, P_i= 2^-N, the cumulative probability deviation Δ should be equivalent to be 0. After calculation, we obtain the cumulative probability deviation with 4-LSB extraction Δ = 0.0015, greatly lower than that of the allowable probability deviation of 0.0050 in the same situation [18].

 

Fig. 3. Probability distribution histogram of the decimal digitization levels with 1 million data points generated by retaining a number of LSBs. Standard probability values of theoretical statistics (red dotted lines).

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Meanwhile, we also observe that there is no obvious correlation peak in the autocorrelation curves of the quantization values by retaining different significant bits as shown in Figs. 4(a)–5(h). Moreover, we notice that the main peak width of obtained correlation traces as depicted in illustrations is extremely narrow without broadening [the corresponding cases are that D₀, D₁, D₂, D₃ as shown in Figs. 4(a)–4(d)], which can further help us to determine and select the optimal quantized output bit sequences with good randomness.

 

Fig. 4. Autocorrelation functions (ACFs) of the quantization values generated by retaining different significant bits for cases: (a) D₀; (b) D₁; (c) D₂; (d) D₃; (e) D₄; (f) D₅; (g) D₆; and (h) D₇. The insets in (a)-(h) plot autocorrelation curves in a scale of 0.05 ns.

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Fig. 5. (a) Bias |e[N]| versus the sample size of the generated 40 Gb/s random bit stream; The red line in (a) is its three-standard-deviation line, 3σ_e = (3N^−1/2)/2 where n = 1, 2, 3, …, 16Mbits. (b) Autocorrelation coefficient C[K] as a function of the delay bit K for a 16 × 10⁶ bit.

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3.3 Randomness evaluation

As known, a physical random bit sequences should be unbiased and independent. Figures 5(a) and 5(b) depict the statistical bias level and the autocorrelation (AC) coefficients of the generated 640 Gb/s binary random bit stream, estimated utilizing the normalized Gaussian distribution estimation N (0, σ²). It can be confirmed from Fig. 5 that both the bias and the serial AC coefficients are below their three-standard-deviation written as 3σ_e= (3N^−1/2)/2 and 3σ_c = (3N^−1/2). Further, we use the state-of-the-art National Institute of Standards and Technology (NIST SP800-22) test suite with 15 statistical test items to examine the obtained random bits [33]. Each test item is performed using 1000 samples of 1-Mbit sequence, and the statistical significance level is set as α = 0.01. Figure. 6 depicts the test result. In general, the test criterion or success is that each P-value should be larger than 0.0001, and the proportion should be within the range of 0.99 ± 0.0094392. For tests that return multiple P-values and proportions, the worst case is given. All the results suggest that our generated random bits can be regarded to be unbiased and independent statistically.

 

Fig. 6. NIST test results: P-value (left column) and proportion (right column). Note, the 15 test items are shown along the horizontal axis.

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4. Conclusions

In conclusion, we experimentally demonstrated a method for fast physical random bit generation using a broadband flat MMW white noise signal generated by optical heterodyning of two FP-LDs subject to optical feedback. As a proof-of-principle demonstration, a MMW white noise signal with a 3-dB bandwidth beyond 50 GHz and the flatness less than 2.9 dB is successfully achieved. Utilizing the generated MMW white noise signal as the entropy source, 640 Gb/s physical random bits can be further obtained by directly selecting 4-LSB from each 8-bit digital signal at 160 GS/s sampling rate without additional complex post-processing. In fact, considering the multi-longitudinal beating approach has full potential to generate broadband flat noise signal in MMW region or even terahertz region, our proposed RBG also has great promise to reach a total bit rate at the order of Tb/s if the utilized devices bandwidth such as the PDs and ADCs are sufficiently large.