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Abstract
The entropy of white chaos is evaluated to certify physical random number generators. White chaos is generated from the electric subtraction of two optical heterodyne signals of two chaotic outputs in semiconductor lasers with optical feedback. We use the statistical test suites of NIST Special Publication 800-90B for the evaluation of physical entropy sources of white chaos with an eight-bit resolution. The minimum value of entropy is 2.1 for eight most significant bits data. The entropy of white chaos is enhanced from that of the chaotic output of the semiconductor lasers. We evaluate the effect of detection noise and distinguish between the entropy that originates from the white chaos and the detection noise. It is found that the entropy of five most significant bits originates from white chaos. The minimum value of entropy is 1.1 for five most significant bits data, and it is considered that the entropy can be obtained at at least one bit per sample.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Random number generation is a key technology for information security and numerical simulations. It is necessary to generate fast and trusted random numbers for these applications. Photonic systems are useful for generating fast physical random numbers, such as chaotic semiconductor lasers [1–20], quantum noise [21–26], and spontaneous emission noise [27–31]. In particular, fast physical random number generation using chaotic lasers has been reported for a decade, over gigabits per second (Gb/s) [1,2]. To improve the generation rate, cascaded lasers are used, and bandwidth enhancement of chaos is achieved [3,5,14]. Using this technique, the rates of random number generation have been achieved over terabits per second (Tb/s) with post-processing [13–16]. Miniaturization of random number generators has also advanced with the use of photonic integrated circuits [32–38]. In addition, all-optical random number generators have been proposed [17–19].
The evaluation of physical random number generators is an important issue. In most studies, random bit sequences are generated using complicated post-processing, and the randomness of the random bit sequences is evaluated using de-facto standard statistical test suites, known as NIST Special Publication (SP) 800-22 [39]. However, the effect of post-processing is included in this statistical evaluation. It is difficult to differentiate between the bit sequences generated from physical random number generators and those from pseudo-random number generators. The evaluation of entropy for a physical source has been limited, due to little knowledge on the subject [40–42]. It is necessary to quantitatively evaluate entropy directly from the physical entropy sources. In addition, it is not clear as to how many bits of one sampled data can be guaranteed as physically non-reproducible bits. Here, entropy is defined as the uncertainty of sampled data obtained from physical entropy sources, with the progression of time.
Recently, white chaos has been used to generate a flat radio-frequency (RF) spectrum for random number generation [43]. White chaos is generated from a differential signal of two chaotic heterodyne signals of semiconductor laser outputs. Flat and broadband spectrum such as white noise can be obtained with no periodicity; the bandwidth of white chaos can be controlled by changing the optical frequency detuning between the two lasers. Random number generation using white chaos has been reported [44]. However, the evaluation of the entropy of white chaos has not been reported yet. In addition, it is not clear whether the entropy can be enhanced by generating white chaos from two original chaotic laser outputs. It is also important to distinguish between the entropy obtained from white chaos and that from detection noise to evaluate physical entropy sources.
In this study, we experimentally generate white chaos from two optical heterodyne signals of chaotic outputs in semiconductor lasers with optical feedback. We evaluate the entropy of white chaos by using statistical tests of randomness for the evaluation of the physical entropy sources. We compare the entropy between white chaos and the original chaotic signals. We also investigate the entropy of the detection noise and evaluate the entropy of white chaos, without the effect of the detection noise.
2. Experimental setup for white chaos generation
Figure 1 shows our experimental setup for the generation of white chaos. We use two distributed-feedback (DFB) semiconductor lasers (NTT Electronics, KELD1C5GAAA, wavelength of 1548 nm), referred to as laser 1 and laser 2. The injection currents of lasers 1 and 2 are set to 60.0 mA (6.2I1,th) and 60.0 mA (6.8I2,th), respectively, where I1,th = 9.7 mA and I2,th = 8.8 mA are their lasing thresholds. A fiber reflector is connected to each laser through a fiber coupler to introduce optical delayed feedback. Chaotic oscillations are generated in the presence of optical feedback. The external cavity lengths from the laser facet to the fiber reflector are mismatched between the two lasers to avoid the peaks of RF spectra, and are set to 4.55 m and 5.61 m, respectively (one-way). The corresponding round-trip feedback delay times (frequencies) are 43.9 ns and 54.1 ns (22.8 MHz and 18.5 MHz), respectively. Two chaotic signals are made to interfere with each other through a fiber coupler to generate two optical heterodyne signals. The two optical heterodyne signals are injected into a balanced receiver (Discovery Semiconductors, DSC-R410-89-FC/APC-G-3, bandwidth of 25 GHz), where they are converted into two separate electric signals by photodetectors, and the difference in the two electric signals is detected. This differential signal is referred to as white chaos. Optical delay lines (General Photonics, MDL-002-I-15-56-PP-FC/APC) are inserted before the balanced receiver to adjust the optical path lengths for the two optical heterodyne signals. The two optical paths need to be carefully adjusted to match the oscillation phase of the heterodyne signals. The output of the balanced receiver is amplified by an electric amplifier (Picosecond Pulse Labs, 5881-219, bandwidth of 43 GHz) and sent to a digital oscilloscope (Tektronix, DPO72304DX, 23 GHz bandwidth, 100 GigaSample/s) and a radio-frequency (RF) spectrum analyzer (Agilent, N9010A-544, 44 GHz bandwidth) to observe temporal waveforms and RF spectra, respectively. The optical spectra of lasers 1 and 2 are measured by an optical spectrum analyzer (Yokogawa, AQ6370C).
Fig. 1. Experimental setup for the generation of white chaos. FC, fiber coupler; ISO, optical isolator.
The optical heterodyne signal can be generated in this experimental setup. The optical wavelength detuning between lasers 1 and 2, Δλ = λ2 − λ1 (or the optical frequency detuning Δf = f1 − f2, which is defined such that the signs of Δλ and Δf are matched), is a crucial parameter for the optical heterodyne signal (λ1,2 are the optical wavelengths of lasers 1 and 2 in the presence of optical feedback). Without optical feedback in both lasers 1 and 2, a sinusoidal temporal waveform with the oscillation frequency Δf can be obtained from the balanced receiver. In the presence of optical feedback, the frequency component Δf is responsible for the modulation of chaotic signals, and the RF spectrum of the optical heterodyne signal can be broadened if Δf is set to a larger value than the relaxation oscillation frequency of the chaotic signals (∼ 7 GHz).
3. Experimental results of white chaos generation
We measure temporal waveforms, RF spectra, and the histograms of the amplitudes of the temporal waveforms. Figure 2 shows the temporal waveforms and the RF spectra of the chaotic laser outputs of laser 1 and the white chaos signal generated from the optical heterodyne between the outputs of lasers 1 and 2. The temporal waveform of the chaotic laser output shows irregular oscillations (Fig. 2(a)), and the peak frequency of ∼7 GHz is observed in the RF spectrum (Fig. 2(c)), corresponding to the relaxation oscillation frequency. For the white chaos, faster oscillation of the temporal waveform is observed in Fig. 2(b). A flatter and broader RF spectrum is also observed in Fig. 2(d).
Fig. 2. (a), (b) Temporal waveforms and (c), (d) RF spectra of the chaotic laser output of the laser 1 and the white chaos signal generated from the optical heterodyne between the outputs of lasers 1 and 2. (a), (c) laser 1 and (b), (d) white chaos. The optical wavelength detuning is set to Δλ = 0.15 nm (Δf = 18.8 GHz) for white chaos.
Figure 3 shows the temporal waveforms and the RF spectra when the optical wavelength detuning Δλ is changed between lasers 1 and 2. For Δλ = 0.00 nm (Δf = 0 GHz), the maximum spectral power is found at 0 GHz and the spectral components decrease monotonically. For Δλ = 0.15 nm (Δf = 18.8 GHz), flat spectral components are observed from 0 to Δf (where, Δλ = 0.01 nm corresponds to Δf = 1.25 GHz at λ = 1548 nm). However, for larger Δλ = 0.40 nm (Δf = 50.0 GHz), the peak spectrum appears at ∼7 GHz, corresponding to the relaxation oscillation frequency of the semiconductor lasers. The RF spectrum resembles those of lasers 1 and 2. Therefore, detuning is not affected for larger values of Δλ.
Fig. 3. (a), (b), (c) Temporal waveforms and (d), (e), (f) RF spectra when the optical wavelength detuning Δλ between lasers 1 and 2 is changed. (a), (d) Δλ = 0.00 nm (Δf = 0 GHz), (b), (e) Δλ = 0.15 nm (Δf = 18.8 GHz), and (c), (f) Δλ = 0.40 nm (Δf = 50.0 GHz).
Figures 4(a)–4(c) show the histogram of the intensity of the temporal waveforms of white chaos, corresponding to Figs. 3(a)–3(c). The values of the intensity are normalized to 8-bit integers between -128 and 128 to generate the histogram, as shown in Fig. 4. The width of the histogram decreases as Δλ increases. In addition, the shape of the histogram becomes symmetric as Δλ increases. The shape of the histogram affects the entropy of white chaos (see Section 4.1).
Fig. 4. Histogram of the intensity of temporal waveforms of white chaos when the optical wavelength detuning Δλ between lasers 1 and 2 is changed. (a) Δλ = 0.00 nm (Δf = 0 GHz), (b) Δλ = 0.15 nm (Δf = 18.8 GHz), and (c) Δλ = 0.40 nm (Δf = 50.0 GHz). The horizontal axis indicates the normalized intensity of the temporal waveform of white chaos from -128 to 128 (8-bit integers).
Figure 5(a) shows the bandwidth and flatness of the RF spectra when the optical wavelength detuning Δλ is changed. We used two definitions of frequency bandwidths, i.e., the standard bandwidth [45] and the effective bandwidth [46]. The flatness is defined using the effective bandwidth [14]. It has been reported that a larger bandwidth and smaller flatness are suitable for fast random number generation [14]. The maximum values of the standard and effective bandwidths are obtained (21.6 and 21.1 GHz, respectively) at a detuning value of Δλ = 0.25 nm (Δf = 31.3 GHz). On the contrary, a minimum flatness of 2.9 dB is obtained at a detuning value of Δλ = 0.15 nm (Δf = 18.8 GHz). The conditions for the largest bandwidth and the smallest flatness are slightly different. We select a detuning value of Δλ = 0.15 nm for the smallest flatness to evaluate white chaos in the following section.
Fig. 5. (a) Standard bandwidth, effective bandwidth, and flatness of the RF spectra. BW: bandwidth. (b) Standard deviation of the histogram when the optical wavelength detuning Δλ is changed.
Figure 5(b) shows the standard deviation of the histogram of the intensity of the temporal waveforms, when Δλ is changed. The standard deviation of the histogram decreases monotonically as Δλ increases. The optical injection reduces the amplitude of the temporal waveforms as shown in Figs. 3 and 4, and the standard deviation decreases.
4. Evaluation of entropy from experimental data
4.1 Entropy of white chaos with 8-bit resolution
We evaluate the entropy of white chaos using NIST SP 800-90B statistical tests of randomness [47]. This version of these tests aims to evaluate the entropy generated from physical random number generators. The NIST SP 800-90B tests differ from the NIST SP 800-22 statistical tests [39] that are used for evaluating the statistical randomness of the digital bits (“0” and “1”) generated from pseudorandom number generators. In contrast, the signals with n-bit vertical resolutions (n-bit signals) are used for the NIST SP 800-90B tests. The minimum entropy (min-entropy) of the n-bit signals generated from physical entropy sources is evaluated. The statistical tests consist of ten different entropy measures, which are listed as follows: the most common value (MCV), collision, Markov, compression, t-Tuple, LRS, MultiMCW, Lag, MultiMMC, and LZ78Y [47]. The last four tests are categorized into prediction tests that evaluate the unpredictability of the n-bit signals. We selected n-bit signals from the first to the n-th most significant bits (MSBs) of the detected 8-bit data (denoted as n MSBs) and evaluated the entropy of the n MSBs data. The maximum value of the entropy is n for n MSBs data. We used 1 Mega point data to evaluate the entropy using NIST SP 800-90B.
Figure 6 shows the entropy of the NIST SP 800-90B statistical tests of randomness for the 8-bit signals of white chaos generated at Δλ = 0.15 nm, when the number of MSBs of the 8-bit signals n is changed. The Markov test only provides the entropy up to 6 MSBs data because it requires a larger number of data for larger values of n. In Fig. 6, the entropy increases with an increase in the number of MSBs for all the NIST SP 800-90B tests. For 8-bit data (8 MSBs), the maximum value of entropy obtained from the MCV test is 5.9. Entropy values of 4.7–6.0 are obtained from the prediction tests. In contrast, the entropy is relatively small for the collision (3.1) and compression (2.1) tests for 8 MSBs, respectively. We estimate that the minimum value of the entropy is 2.1 (compression test) for the 8 MSBs data.
Fig. 6. Entropy of the NIST SP 800-90B statistical tests of randomness for the 8-bit signals of white chaos generated at Δλ = 0.15 nm when the number of most significant bits (MSBs) of the 8-bit signals n is changed. (a) MCV, Collision, Markov, Compression, (b) t-Tuple, LRS, (c) MultiMCW, Lag, MultiMMC, and LZ78Y tests.
From these results, smaller entropy values are obtained from the collision and compression tests, compared to other tests. It should be noted that a similar characteristic is observed for the 8-bit data of white Gaussian noise generated from a pseudorandom number generator (see the Appendix for details). The histogram of the 8-bit signal of white chaos is similar to the Gaussian distribution. We speculate that the shape of the histogram presents smaller entropy values from the compression and collision tests. This is because the collision (the appearance of the same value) occurs more often when the peak value of the Gaussian distribution is larger. For the compression tests, a larger peak of the histogram results in lesser compression of the 8-bit data. Therefore, the shape of the histogram strongly affects the entropy evaluation. A flatter histogram of the 8 MSBs data is required to increase the entropy of the collision and the compression tests.
We evaluate the entropy of white chaos for different optical wavelength detuning values. Figure 7 shows the entropy of the NIST SP 800-90B statistical tests when Δλ is changed. The entropy decreases monotonically as Δλ is increased. For example, the entropy of the MCV test decreases from 6.6 to 4.4. Other tests show similar curves for a change in entropy between 7 and 3. However, the entropy of collision and compression is relatively small, between 1.3 and 3.2. We speculate that the decrease in entropy is related to the change in the standard deviation of the histogram of the 8-bit signals, as shown in Fig. 5(b). This is because a larger number of similar values of the 8-bit signals are obtained for the histogram with a smaller standard deviation.
Fig. 7. Entropy of the NIST SP 800-90B statistical tests when Δλ is changed. (a) MCV, Collision, Markov, Compression, (b) t-Tuple, LRS, (c) MultiMCW, Lag, MultiMMC, and LZ78Y tests.
4.2 Entropy of chaotic laser output
In this subsection, we measure the entropy of the chaotic output of a single semiconductor laser with optical feedback (laser 1 in Fig. 1), before generating optical heterodyne signals. We compare the entropy of the chaotic output of laser 1 with that of white chaos to investigate the amount of entropy that can be increased by optical heterodyne.
Figure 8 shows the entropy of the chaotic output of laser 1 in the semiconductor laser with optical feedback, whose temporal waveforms and RF spectra are shown in Figs. 2(a) and 2(c), respectively. Compared to Fig. 8 and Fig. 6, the entropy of the chaotic output of laser 1 is smaller than that of white chaos for all tests. The entropy of the MCV tests for the chaotic output of laser 1 is similar to that of white chaos because it is determined by the shape of the histogram of the temporal waveforms. On the contrary, the entropy of the prediction tests for the chaotic output of laser 1 decreases (e.g., the entropy is ∼3 for 8 MSBs), while that for white chaos is around 5–6 for 8 MSBs. This result indicates that the entropy is enhanced by generating white chaos from the chaotic output of laser 1. This is performed to eliminate the correlation of the temporal waveforms (i.e., the improvement of the flatness in the RF spectrum, as seen in Fig. 2(d)). Therefore, heterodyning of two chaotic signals results in an increase in complexity. We speculate that the optical phase dynamics is converted to optical intensity dynamics by heterodyning the two chaotic signals, thereby enhancing complexity. We also speculate that entropy of the heterodyne signal is enhanced, because fast phase oscillations are extracted by heterodyning the two chaotic waveforms.
Fig. 8. Entropy of the NIST SP 800-90B statistical tests of randomness for the 8-bit signals of the chaotic output of the laser 1 (Figs. 2(a) and 2(c)) when the number of MSBs of the 8-bit signals n is changed. (a) MCV, Collision, Markov, Compression, (b) t-Tuple, LRS, (c) MultiMCW, Lag, MultiMMC, and LZ78Y tests.
4.3 Elimination of noise
The 8-bit signals of white chaos may include noise in the detection and measurement equipment, such as the balanced receiver and the digital oscilloscope. It is important to distinguish between the entropy generated from white chaos and that from the detection noise [41,47]. Thus, we evaluate the entropy of the detection equipment without injecting the chaotic signal. Figure 9 shows the entropy of the NIST SP 800-90B for the noise signals generated from the balanced receiver without injecting the two chaotic laser outputs. The noise is detected with the same vertical resolution as in Fig. 6. The entropy is almost zero from 1 to 5 MSBs and slightly increases beyond 5 MSBs. This result indicates that entropy is generated from the noise for the three least significant bits (LSBs) of the 8-bit signals. We consider that 3 LSBs should be excluded to estimate the entropy originated from the 8-bit white chaos signals. Therefore, we use 5 MSBs (from the first to the fifth MSB) for the evaluation of the entropy of white chaos in the following section. In addition, it should be noted that the effective number of bits (ENOB) of the fast-digital oscilloscope has been reported as ∼5 bits, and for entropy evaluation, it is reasonable to use 5 MSBs to avoid the influence of noise in the detection equipment.
Fig. 9. Entropy of the NIST SP 800-90B for the noise signals generated from the balanced receiver without injecting the two chaotic laser outputs when the number of MSBs are changed. (a) MCV, Collision, Markov, Compression, (b) t-Tuple, LRS, (c) MultiMCW, Lag, MultiMMC, and LZ78Y tests.
Figure 10 shows the entropy value of the 5 MSBs of white chaos excluding the effect of noise when the optical wavelength detuning Δλ is changed. The entropy for some NIST tests decreases monotonically as Δλ increases, which is similar to the results shown in Fig. 7. However, the entropy of the Markov, t-Tuple, and other prediction tests (MultiMCW, Lag, and LZ78Y) show a peak at Δλ = ∼0.15 nm. This peak indicates that the entropy is maximized at the optimal value of Δλ, where the flatness of the RF spectrum is minimized, as shown in Fig. 5(a). Therefore, the entropy is related to the flatness of the RF spectrum (i.e., the autocorrelation in time domain), when 5 MSBs are used for entropy evaluation. It is found that the relationship between entropy and flatness of the RF spectrum is clearly seen, after excluding the detection noise from 8-bit signals in Fig. 10. We found that the minimum flatness of the RF spectrum is more crucial than the maximum bandwidth to obtain large entropy, according to the comparison between Figs. 5(a) and 10. This result is consistent with the result shown in Ref. [48].
Fig. 10. Entropy of the 5 MSBs of the white chaos as the optical wavelength detuning Δλ is changed. 3 LSBs are excluded. (a) MCV, Collision, Markov, Compression, (b) t-Tuple, LRS, (c) MultiMCW, Lag, MultiMMC, and LZ78Y tests.
For Δλ = 0.15 nm, the entropy of the MCV and Markov tests are 2.9 and 2.4, respectively. For the four prediction tests, the entropy ranges from 1.6 to 2.6. In contrast, smaller entropy values are obtained from the collision (1.4) and compression (1.1) tests. Therefore, it is found that the minimum value of entropy is 1.1 at Δλ = 0.15 nm, for the compression test. It is considered that the entropy can be obtained at least in one bit per sample. The entropy generation rate is estimated to be 110 Gb/s ( = 1.1 bit × 100 GS/s).
4.4 Effect of sampling time
In this subsection, we investigate the effect of the autocorrelation on the entropy by changing the sampling time of 5 MSBs data. Figure 11 shows the entropy of the 5 MSBs data when the sampling time is changed. The entropy is almost constant for large sampling times over 20 ps (50 GS/s in frequency). The entropy decreases for a sampling time of 10 ps (100 GS/s), except in the MCV. In particular, the decrease in the entropy is clearly seen for the L-Tuple, LRS (Fig. 11(b)), and the prediction tests (Fig. 11(c)), at a sampling time of 10 ps.
Fig. 11. Entropy of the 5 MSBs of the white chaos as the sampling time is changed. The detection noise of 3 LSBs is excluded. (a) MCV, Collision, Markov, Compression, (b) t-Tuple, LRS, (c) MultiMCW, Lag, MultiMMC, and LZ78Y tests.
This characteristic can be explained by the autocorrelation function, as shown in Fig. 12. In this figure, a large correlation value of 0.72 is obtained for a delay time (i.e., the sampling time) of 10 ps, because the sampling time is small, in comparison with the average period of the white chaos signals (∼ 50 ps). The correlation value decreases to less than ∼0.2 for a delay time of 20 ps or larger values in Fig. 12. We thus consider that the degradation of entropy for small sampling times in Fig. 11 is a result of the large autocorrelation of the 5 MSBs data.
Fig. 12. Autocorrelation of the temporal waveform of white chaos used in Fig. 11. The delay time corresponds to the sampling time in Fig. 11.
5. Discussion
We used NIST SP 800-90B tests to evaluate the entropy of the n-bit signals that were generated from white chaos. The entropy ranges from 2.1 to 6.0 for 8 MSBs data. This entropy is larger than that for the chaotic output of the laser with optical feedback. We exclude the detection noise and evaluate the 5 MSBs data, obtaining the range of entropy from 1.1 to 2.9. Therefore, it is found that the entropy can be obtained at a rate of at least one bit per sample for the 5 MSBs data. This result is a good certification of the physical entropy sources when white chaos is used as a single-bit random number generator. This is because the randomness of one bit per sample is guaranteed. In fact, it is difficult to determine which bits in 5 MSBs carry the uncertainty. However, the statistical tests show that one can extract at least one bit of entropy from the 5 MSBs. For example, a certified physical random bit can be generated by extracting the fifth bit of the 5 MSBs to avoid autocorrelation. In contrast, the entropy for the multi-bit random number generation cannot be certified for the white chaos generated in this experiment, as the minimum value of the entropy is less than 2.
From Figs. 10 and 11, the curves of the entropy for collision and compression are smaller than the remaining curves. It should be considered whether collision and compression are suitable measures for physical entropy sources to generate n-bit signals with a Gaussian-shaped distribution. As described in the Appendix, the entropy of the collision and compression tests are comparatively smaller than the other tests, even for the white Gaussian noise generated from a pseudorandom number generator without correlation. We consider that it might be better to exclude the collision and compression tests during physical entropy evaluation. Thus, we can obtain a minimum entropy value of 2.8 for a sampling time of 20 ps (50 GS/s), and 1.6 for a sampling time of 10 ps (100 GS/s). These values show the feasibility of the multi-bit random number generation, which requires the entropy to be more than one bit per sample. The entropy generation rates are estimated to be 140 Gb/s ( = 2.8 bit × 50 GS/s) and 160 Gb/s ( = 1.6 bit × 100 GS/s), respectively.
For the generation of physical random bits, one must extract the number of bits that are lesser than the entropy values, to certify physical random number generation. Some LSBs include detection noise (3 LSBs in our case), and some MSBs have large autocorrelation. Therefore, we recommend that the middle bits between LSBs and MSBs are extracted to certify physical random number generation.
6. Conclusions
We experimentally generated white chaos from optical heterodyne signals of two chaotic outputs in semiconductor lasers with optical feedback. We estimated the entropy of white chaos by using NIST SP 800-90B for the evaluation of physical entropy sources. The minimum entropy is 2.1 for 8 MSBs data. This entropy is larger than that of the chaotic output of the laser. We also evaluated the entropy of detection noise, finding that the detection noise has an entropy of 3 LSBs. Therefore, the entropy of 5 MSBs is originated from the white chaos. The minimum entropy is 1.1 for 5 MSBs data, without detection noise. The entropy is maximized for white chaos with minimum flatness of RF spectrum and large sampling time and the entropy generation rate is estimated to be 110 Gb/s ( = 1.1 bit × 100 GS/s).
We can certify the physical entropy source of white chaos when it is used for single-bit random number generators, because a randomness of at least one bit per sample is guaranteed. On the contrary, the use of white chaos for multi-bit physical random number generators with entropy certification is challenging with the current data, since the entropy is not sufficiently large to extract multiple bits per sample.
The use of entropy evaluation is particularly important to certify physical entropy sources for physical random number generators, and further investigation is strongly encouraged.
Appendix
We investigate the entropy of white Gaussian noise (8-bit signals) from a pseudorandom number generator using the Mersenne Twister algorithm [49]. Figure 13 shows the results of NIST SP 800-90B tests as the standard deviation of the white Gaussian noise is changed. An entropy over 6 is obtained for MCV, t-Tuple, LRS, and four prediction tests. However, the entropy is lower for Markov (only used for 6-bit signals), collision, and compression, compared with other tests. The maximum entropy is ∼3 for compression and ∼4 for collision test, respectively, for the white Gaussian noise in Fig. 13. The entropy of collision and compression tests may be limited by the shape of the Gaussian distribution. Therefore, one may need to reconsider whether collision and compression tests should be used to evaluate physical random bits with a Gaussian-shaped distribution.
Funding
Japan Society for the Promotion of Science (JP16H03878); Core Research for Evolutional Science and Technology (JPMJCR17N2).
Disclosures
The authors declare no conflicts of interest.
References
1. A. Uchida, K. Amano, M. Inoue, K. Hirano, S. Naito, H. Someya, I. Oowada, T. Kurashige, M. Shiki, S. Yoshimori, K. Yoshimura, and P. Davis, “Fast physical random bit generation with chaotic semiconductor lasers,” Nat. Photonics 2(12), 728–732 (2008). [CrossRef]
2. I. Reidler, Y. Aviad, M. Rosenbluh, and I. Kanter, “Ultrahigh-speed random number generation based on a chaotic semiconductor laser,” Phys. Rev. Lett. 103(2), 024102 (2009). [CrossRef]
3. K. Hirano, T. Yamazaki, S. Morikatsu, H. Okumura, H. Aida, A. Uchida, S. Yoshimori, K. Yoshimura, T. Harayama, and P. Davis, “Fast random bit generation with bandwidth-enhanced chaos in semiconductor lasers,” Opt. Express 18(6), 5512–5524 (2010). [CrossRef]
4. I. Kanter, Y. Aviad, I. Reidler, E. Cohen, and M. Rosenbluh, “An optical ultrafast random bit generator,” Nat. Photonics 4(1), 58–61 (2010). [CrossRef]
5. Y. Akizawa, T. Yamazaki, A. Uchida, T. Harayama, S. Sunada, K. Arai, K. Yoshimura, and P. Davis, “Fast random number generation with bandwidth-enhanced chaotic semiconductor lasers at 8 × 50 Gb/s,” IEEE Photonics Technol. Lett. 24(12), 1042–1044 (2012). [CrossRef]
6. J. Zhang, Y. Wang, M. Liu, L. Xue, P. Li, A. Wang, and M. Zhang, “A robust random number generator based on differential comparison of chaotic laser signals,” Opt. Express 20(7), 7496–7506 (2012). [CrossRef]
7. R. M. Nguimdo, G. Verschaffelt, J. Danckaert, X. Leijtens, J. Bolk, and G. Van der Sande, “Fast random bits generation based on a single chaotic semiconductor ring laser,” Opt. Express 20(27), 28603–28613 (2012). [CrossRef]
8. A. Wang, P. Li, J. Zhang, J. Zhang, L. Li, and Y. Wang, “4.5 Gbps high-speed real-time physical random bit generator,” Opt. Express 21(17), 20452–20462 (2013). [CrossRef]
9. X.-Z. Li and S.-C. Chan, “Heterodyne random bit generation using an optically injected semiconductor laser in chaos,” IEEE J. Quantum Electron. 49(10), 829–838 (2013). [CrossRef]
10. N. Oliver, M. C. Soriano, D. W. Sukow, and I. Fischer, “Fast random bit generation using a chaotic laser: approaching the information theoretic limit,” IEEE J. Quantum Electron. 49(11), 910–918 (2013). [CrossRef]
11. X. Fang, B. Wetzel, J.-M. Merolla, J. M. Dudley, L. Larger, C. Guyeux, and J. M. Bahi, “Noise and chaos contributions in fast random bit sequence generated from broadband optoelectronic entropy sources,” IEEE Trans. Circuits Syst. I 61(3), 888–901 (2014). [CrossRef]
12. M. Virte, E. Mercier, H. Thienpont, K. Panajotov, and M. Sciamanna, “Physical random bit generation from chaotic solitary laser diode,” Opt. Express 22(14), 17271–17280 (2014). [CrossRef]
13. N. Li, B. Kim, V. N. Chizhevsky, A. Locquet, M. Bloch, D. S. Citrin, and W. Pan, “Two approaches for ultrafast random bit generation based on the chaotic dynamics of a semiconductor laser,” Opt. Express 22(6), 6634–6646 (2014). [CrossRef]
14. R. Sakuraba, K. Iwakawa, K. Kanno, and A. Uchida, “Tb/s physical random bit generation with bandwidth-enhanced chaos in three-cascaded semiconductor lasers,” Opt. Express 23(2), 1470–1490 (2015). [CrossRef]
15. X. Tang, Z.-M. Wu, J.-G. Wu, T. Deng, J.-J. Chen, L. Fan, Z.-Q. Zhong, and G.-Q. Xia, “Tbits/s physical random bit generation based on mutually coupled semiconductor laser chaotic entropy source,” Opt. Express 23(26), 33130–33141 (2015). [CrossRef]
16. T. Butler, C. Durkan, D. Goulding, S. Slepneva, B. Kelleher, S. P. Hegarty, and G. Huyet, “Optical ultrafast random number generation at 1 Tb/s using a turbulent semiconductor ring cavity laser,” Opt. Lett. 41(2), 388–391 (2016). [CrossRef]
17. P. Li, Y. Sun, X. Liu, X. Yi, J. Zhang, X. Guo, Y. Guo, and Y. Wang, “Fully photonics-based physical random bit generator,” Opt. Lett. 41(14), 3347–3350 (2016). [CrossRef]
18. P. Li, J. Zhang, L. Sang, X. Liu, Y. Guo, X. Guo, A. Wang, K. A. Shore, and Y. Wang, “Real-time online photonic random number generation,” Opt. Lett. 42(14), 2699–2702 (2017). [CrossRef]
19. P. Li, Y. Guo, Y. Guo, Y. Fan, X. Guo, X. Liu, K. Li, K. A. Shore, Y. Wang, and A. Wang, “Ultrafast fully photonic random bit generator,” J. Lightwave Technol. 36(12), 2531–2540 (2018). [CrossRef]
20. A. Uchida, Optical Communication with Chaotic Lasers, Applications of Nonlinear Dynamics and Synchronization (Wiley-VCH, 2012).
21. M. Stipčević and B. Medved Rogina, “Quantum random number generator based on photonic emission in semiconductors,” Rev. Sci. Instrum. 78(4), 045104 (2007). [CrossRef]
22. J. F. Dynes, Z. L. Yuan, A. W. Sharpe, and A. J. Shields, “A high speed, post-processing free, quantum random number generator,” Appl. Phys. Lett. 93(3), 031109 (2008). [CrossRef]
23. C. Gabriel, C. Wittmann, D. Sych, R. Dong, W. Mauerer, U. L. Andersen, C. Marquardt, and G. Leuchs, “A generator for unique quantum random numbers based on vacuum states,” Nat. Photonics 4(10), 711–715 (2010). [CrossRef]
24. T. Symul, S. M. Assad, and P. K. Lam, “Real time demonstration of high bitrate quantum random number generation with coherent laser light,” Appl. Phys. Lett. 98(23), 231103 (2011). [CrossRef]
25. Y.-Q. Nie, L. Huang, Y. Liu, F. Payne, J. Zhang, and J.-W. Pan, “The generation of 68 Gbps quantum random number by measuring laser phase fluctuations,” Rev. Sci. Instrum. 86(6), 063105 (2015). [CrossRef]
26. X.-G. Zhang, Y.-Q. Nie, H. Zhou, H. Liang, X. Ma, J. Zhang, and J.-W. Pan, “Fully integrated 3.2 Gbps quantum random number generator with real-time extraction,” Rev. Sci. Instrum. 87(7), 076102 (2016). [CrossRef]
27. C. R. S. Williams, J. C. Salevan, X. Li, R. Roy, and T. E. Murphy, “Fast physical random number generator using amplified spontaneous emission,” Opt. Express 18(23), 23584–23597 (2010). [CrossRef]
28. X. Li, A. B. Cohen, T. E. Murphy, and R. Roy, “Scalable parallel physical random number generator based on a superluminescent LED,” Opt. Lett. 36(6), 1020–1022 (2011). [CrossRef]
29. A. Argyris, E. Pikasis, S. Deligiannidis, and D. Syvridis, “Sub-Tb/s physical random bit generators based on direct detection of amplified spontaneous emission signals,” J. Lightwave Technol. 30(9), 1329–1334 (2012). [CrossRef]
30. T. Yamazaki and A. Uchida, “Performance of random number generators using noise-based superluminescent diode and chaos-based semiconductor lasers,” IEEE J. Sel. Top. Quantum Electron. 19(4), 0600309 (2013). [CrossRef]
31. M. Huang, A. Wang, P. Li, H. Xu, and Y. Wang, “Real-time 3 Gbit/s true random bit generator based on a super-luminescent diode,” Opt. Commun. 325, 165–169 (2014). [CrossRef]
32. A. Argyris, S. Deligiannidis, E. Pikasis, A. Bogris, and D. Syvridis, “Implementation of 140 Gb/s true random bit generator based on a chaotic photonic integrated circuit,” Opt. Express 18(18), 18763–18768 (2010). [CrossRef]
33. T. Harayama, S. Sunada, K. Yoshimura, P. Davis, K. Tsuzuki, and A. Uchida, “Fast nondeterministic random-bit generation using on-chip chaos lasers,” Phys. Rev. A 83(3), 031803 (2011). [CrossRef]
34. S. Sunada, T. Harayama, K. Arai, K. Yoshimura, P. Davis, K. Tsuzuki, and A. Uchida, “Chaos laser chips with delayed optical feedback using a passive ring waveguide,” Opt. Express 19(7), 5713–5724 (2011). [CrossRef]
35. S. Sunada, T. Harayama, K. Arai, K. Yoshimura, K. Tsuzuki, A. Uchida, and P. Davis, “Random optical pulse generation with bistable semiconductor ring lasers,” Opt. Express 19(8), 7439–7450 (2011). [CrossRef]
36. R. Takahashi, Y. Akizawa, A. Uchida, T. Harayama, K. Tsuzuki, S. Sunada, K. Arai, K. Yoshimura, and P. Davis, “Fast physical random bit generation with photonic integrated circuits with different external cavity lengths for chaos generation,” Opt. Express 22(10), 11727–11740 (2014). [CrossRef]
37. A. Karsaklian Dal Bosco, N. Sato, Y. Terashima, S. Ohara, A. Uchida, T. Harayama, and M. Inubushi, “Random number generation from intermittent optical chaos,” IEEE J. Sel. Top. Quantum Electron. 23(6), 1801208 (2017). [CrossRef]
38. K. Ugajin, Y. Terashima, K. Iwakawa, A. Uchida, T. Harayama, K. Yoshimura, and M. Inubushi, “Real-time fast physical random number generator with a photonic integrated circuit,” Opt. Express 25(6), 6511–6523 (2017). [CrossRef]
39. A. Rukhin, J. Soto, J. Nechvatal, M. Smid, E. Barker, S. Leigh, M. Levenson, M. Vangel, D. Banks, A. Heckert, J. Dray, S. Vo, and L. E. Bassham III, National Institute of Standards and Technology (NIST), Special Publication 800-22, Revision 1a (2010).
40. A. M. Hagerstrom, T. E. Murphy, and R. Roy, “Harvesting entropy and quantifying the transition from noise to chaos in a photon-counting feedback loop,” Proc. Natl. Acad. Sci. U. S. A. 112(30), 9258–9263 (2015). [CrossRef]
41. J. D. Hart, Y. Terashima, A. Uchida, G. B. Baumgartner, T. E. Murphy, and R. Roy, “Recommendations and illustrations for the evaluation of photonic random number generators,” APL Photonics 2(9), 090901 (2017). [CrossRef]
42. X.-Z. Li, J.-P. Zhuang, S.-S. Li, J.-B. Gao, and S.-C. Chan, “Randomness evaluation for an optically injected chaotic semiconductor laser by attractor reconstruction,” Phys. Rev. E 94, 042214 (2016). [CrossRef]
43. A. Wang, B. Wang, L. Li, Y. Wang, and K. A. Shore, “Optical heterodyne generation of high-dimensional and broadband white chaos,” IEEE J. Sel. Top. Quantum Electron. 21(6), 1800710 (2015). [CrossRef]
44. A. Wang, L. Wang, P. Li, and Y. Wang, “Minimal-post-processing 320-Gbps true random bit generation using physical white chaos,” Opt. Express 25(4), 3153–3164 (2017). [CrossRef]
45. F. Y. Lin and J. M. Liu, “Nonlinear dynamical characteristics of an optically injected semiconductor laser subject to optoelectronic feedback,” Opt. Commun. 221(1-3), 173–180 (2003). [CrossRef]
46. F.-Y. Lin, Y.-K. Chao, and T.-C. Wu, “Effective bandwidths of broadband chaotic signals,” IEEE J. Quantum Electron. 48(8), 1010–1014 (2012). [CrossRef]
47. E. Barker and J. Kelsey, “Recommendation for the entropy sources used for random bit generation,” NIST Draft Special Publication 800-90B, second draft (2016).
48. K. Kanno, A. Uchida, and M. Bunsen, “Complexity and bandwidth enhancement in unidirectionally coupled semiconductor lasers with time-delayed optical feedback,” Phys. Rev. E 93(3), 032206 (2016). [CrossRef]
49. M. Matsumoto and T. Nishimura, “Mersenne Twister: A 623-dimensionally equidistributed uniform pseudorandom number generator,” ACM Trans. Model. Comput. Simul. 8(1), 3–30 (1998). [CrossRef]
