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We experimentally demonstrate fast physical random bit generation from bandwidth-enhanced chaos by using three-cascaded semiconductor lasers. The bandwidth-enhanced chaos is obtained with the standard bandwidth of 35.2 GHz, the effective bandwidth of 26.0 GHz and the flatness of 5.6 dB, whose waveform is used for random bit generation. Two schemes of single-bit and multi-bit extraction methods for random bit generation are carried out to evaluate the entropy rate and the maximum random bit generation rate. For single-bit generation, the generation rate at 20 Gb/s is obtained for physical random bit sequences. For multi-bit generation, the maximum generation rate at 1.2 Tb/s ( = 100 GS/s × 6 bits × 2 data) is equivalently achieved for physical random bit sequences whose randomness is verified by using both NIST Special Publication 800-22 and TestU01.
© 2015 Optical Society of America
1. Introduction
Fast physical random number generators are promising for information security applications due to its non-reproducibility and non-periodicity, compared with pseudo-random number generators. Optical physical random number generators based on chaotic lasers [1–17], amplified spontaneous noise [18–20], and quantum noise [21–23] have been developed intensively in recent years. One of the advantages of chaotic-laser based random number generators is the speed of random bit generation over Gigabit per second (Gb/s) due to the fast dynamics of chaotic laser output. Fast physical random number generators are key technologies for the applications of a new type of security systems [24,25] and parallel computation.
One of the important applications with large amount of physical random numbers is the secure key distribution based on information theory [24,25]. In this scheme, two legitimate users randomly and independently select the parameter values of their laser systems and they can share secret keys by using common-signal induced synchronization of the laser systems. The two users discard some of the bits to generate common secret keys, and the generation ratio of the secret keys to the discarded bits is exponentially decreased as the number of cascaded laser systems is linearly increased to protect the sampling attack by an eavesdropper [25]. Similarly, a quantum cryptographic system requires large amount of random numbers to select the modulation parameter of the optical phase for single photons [26]. In addition, it has been reported that pseudo-random number generators starting from different initial conditions show some correlation that causes the degradation of the performance in parallel computing systems [27]. Fast physical random number generators are required for these modern applications.
The development of random number generators based on chaotic laser dynamics has been progressing recently in terms of generation rate. A number of new schemes for random number generation have been proposed since the first demonstration was carried out in 2008 [1], and the generation rate of random bits has been increasing very rapidly. The generation rate at 1.7 Gb/s has been reported with single-bit random number generation at 1.7 GHz in 2008 [1]. In 2009, the generation rate is 12.5 Gb/s was demonstrated with 5 least significant bits (LSBs) sampled at 2.5 GHz [2]. In 2010, the generation rate at 75 Gb/s was demonstrated with 6 LSBs sampled at 12.5 GHz [3]. At the same time, a fast post-processing method that brings up to 300 Gb/s was proposed with high-order differential method [4], where a complicated post-processing method was introduced. In 2012, the bit-order reversal method was demonstrated at 400 Gb/s with 8 LSBs sampled at 50 GHz [5]. Some other schemes for random numbers generated at several hundreds of Gb/s have been proposed [6–10]. Very recently, a method for “pseudo”-random bit generation at 2.2 Terabit per second (Tb/s) has been proposed [11], however, the generation rate is beyond the original generation rate of physical random bits, which is determined by the product of the sampling rate and the number of the sampled significant bits.
To increase the generation rate of random bits, both pre-processing and post-processing schemes have been improved. For pre-processing, the frequency bandwidth of the chaotic laser output needs to be enhanced for fast generation rate. A technique for bandwidth enhancement of chaos has been developed by using optical injection from one laser to another laser [28–33]. A method of dual optical injection was also demonstrated that results in the broad bandwidth up to 32 GHz in experiment [32]. A definition of the frequency bandwidth that is suitable to quantify chaotic broad spectra was also introduced, so-called the effective bandwidth [34]. For post-processing, several methods have been developed to improve the randomness of generated bit sequences, for example, exlusive-or (XOR) [1], LSB extraction [2], bit-order-reversal [5], differential comparison [2,12], and high-order differential methods [4]. In particular, the high-order differential method have been proposed [4,11,35], where the difference between two chaotic data is calculated and the original n bit sequence is converted to a n + m bit sequence for the m-th repetition of the differential operation. The obtained m bits could be more susceptible to the effects of physical noise in the analog-to-digital converter (ADC), which is separate from physical randomness of the chaotic laser dynamics. In addition, the randomness of the bit sequences generated from the post-processing is not dependent of the laser parameter values, which indicates that the characteristics of the obtained random bits do not originate from the laser dynamics. This post-processing method thus can result in “pseudo”-random bit sequences with fast generation rate over several orders of magnitude higher than that of the original physical generation rate. The generated bit sequence is more likely pseudo-random bits, rather than physical random bits [11].
In this study, we experimentally demonstrate a scheme for “physical” random bit generation over Tb/s by using bandwidth-enhanced chaos in three-cascaded semiconductor lasers. We measure the bandwidth and flatness of the chaotic radio-frequency (RF) spectra. We introduce two schemes of single-bit and multi-bit extraction methods for random bit generation. The generation rate of random bit sequences is evaluated within the limit of physical sampling rate and the number of significant bits.
2. Bandwidth enhancement of chaos in three-cascaded semiconductor lasers
2.1 Experimental setup
Figure 1 shows our experimental setup for bandwidth enhancement of chaos and physical random bit generation. We used three distributed-feedback (DFB) semiconductor lasers mounted in butterfly packages with optical fiber pigtails (NTT Electronics, NLK1C5GAAA, the optical wavelength of ~1548 nm). One laser (referred to as Laser 1) was used for the generation of chaotic intensity fluctuations induced by optical feedback. The other two laser (referred to as Laser 2 and Laser 3) were used for the bandwidth enhancement of chaotic waveforms. The injection current and the temperature of the semiconductor lasers were adjusted by a current-temperature controller (Newport, 8000-OPT-41-41-41-41). The lasing thresholds of the injection currents for Laser 1, 2, and 3 were 10.0, 8.8 and 10.0 mA, respectively. The injection currents of the Laser 1, 2, and 3 were set to 58.50, 59.00, and 59.00 mA, respectively. The relaxation oscillation frequencies of the solitary Laser 1, 2, and 3 were ~6.5 GHz. All the three lasers were prepared without standard optical isolators, to allow optical feedback and injection.
Fig. 1 Experimental setup for bandwidth enhancement of chaos and physical random bit generation using three-cascaded semiconductor lasers. FC, fiber coupler; ISO, optical fiber isolator; PD, photodetector.
Laser 1 was connected to a fiber coupler and a variable fiber reflector which reflects a fraction of the light back into the laser, inducing high-frequency chaotic oscillations of the optical intensity. The amount of the optical feedback light was adjusted by the variable fiber reflector. The fiber length between Laser 1 and the variable fiber reflector was 4.55 m, corresponding to the feedback delay time (round-trip) of 43.8 ns. On the other hand, there was no optical feedback for both Laser 2 and 3. Polarization maintaining fibers were used for all the optical fiber components.
A portion of the chaotic Laser 1 beam was injected into Laser 2. An optical fiber isolator was used to achieve one-way coupling from Laser 1 to Laser 2. A portion of the Laser 2 output was injected into Laser 3 unidirectionally through an optical fiber isolator as well. The injection power was adjusted by using optical attenuators, and the injection power from Laser 1 to 2 and from Laser 2 to 3 were set to 2.07 and 1.93 mW, respectively. The wavelengths of the three-cascaded lasers were precisely adjusted in order to generate bandwidth-enhanced chaotic output of Laser 3, as described below. The optical output of Laser 3 was converted into an electronic signal by using a high-speed AC-coupled photodetector (New Focus, 1474-A, 38 GHz bandwidth). The electronic signal was divided into two signals, one of which was time-delayed to avoid cross-correlation between the two signals for random bit generation. The electronic delay was implemented using a 1-meter-long coaxial cable with a 4.51-ns delay time. The two electronic signals were sent to a high-speed digital oscilloscope (Tektronix, DPO73304D, 33 GHz bandwidth, 100 GigaSamples/s, 8-bit vertical resolution) and a radio-frequency (RF) spectrum analyzer (Agilent, N9010A-544, 44 GHz bandwidth) to observe temporal waveforms and the corresponding RF spectra, respectively. The optical wavelengths of the lasers were measured using an optical spectrum analyzer (Yokogawa, AQ6370C-20).
The optical wavelengths of the three-cascaded semiconductor lasers were precisely controlled by changing the temperature of the lasers. For bandwidth enhancement of chaos, the optical wavelengths for coupled lasers need to be mismatched without using injection locking. In this experiment, the wavelength of the Laser 1 with optical feedback was set to 1547.782 nm (the temperature of the Laser 1 was 21.91 °C). The wavelength of the solitary Laser 2 was set to 1547.830 nm (17.90 °C), and the wavelength was shifted to 1547.914 nm in the presence of the optical injection from Laser 1 to 2. The wavelength of the Laser 2 was larger than that of the Laser 1, where injection locking does not occur [16,17]. The detuning of the optical wavelengths between the Laser 1 and 2 was 0.132 nm (16.5 GHz in frequency, where 0.01 nm corresponds to 1.25 GHz for this wavelength). In addition, the wavelength of the solitary Laser 3 was set to 1548.151 nm (20.09 °C), and the wavelength was shifted to 1548.166 nm due to the optical injection from Laser 2 to 3. The detuning of the optical wavelengths between the Laser 2 and 3 was 0.252 nm (31.5 GHz in frequency). These frequency components corresponding to the optical wavelength detunings appear on the RF spectrum of Laser 3, and the existence of the wavelength detuning is crucial to achieve bandwidth enhancement of chaos.
2.2 Definition of bandwidth and flatness
We measure frequency bandwidth and flatness of RF spectra obtained from the laser output. We introduce two types of frequency bandwidths. One is called the standard bandwidth, where the bandwidth is defined as the maximum frequency including 80% of the total spectral power from the DC component, as shown in Fig. 2(a) [28]. The other definition is called the effective bandwidth, which sums up large discrete spectral segments of the chaotic power spectrum accounting for 80% of the total power, as shown in Fig. 2(b) [34]. The effective bandwidth measures only the bandwidths that possess significant amounts of power in the chaos spectra, and the broadband chaotic states can be clearly distinguished from the narrowband periodic oscillation states [34], which is suitable for the qualification of broadband chaotic spectra. The effective bandwidth represents the regions of the dominant frequency components in chaotic broadband spectra, whereas the standard bandwidth shows the maximum oscillation components in the spectra. The effective bandwidth is thus smaller than the standard bandwidth in most cases. We also introduce the flatness as the difference in decibel between the maximum and minimum spectral values within the effective bandwidth, as shown in Fig. 2(b). Smaller flatness indicates flatter distribution of the broadband spectra. It is expected that good-quality random bit generation could be achieved from the laser output with large frequency bandwidth and small flatness.
Fig. 2 Examples of the measurement of (a) standard bandwidth, (b) effective bandwidth and flatness of RF broadband spectra.
The flatness of the RF spectrum is related to the auto-correlation function of the temporal waveform. A flatter spectrum corresponds to a smaller peak of the auto-correlation function at the delay time (known as the time-delay signature). The maximum Lyapunov exponent is also related to the time-delay signature and the flatness of RF spectrum [36]. A large Lyapunov exponent corresponds to a flat RF spectrum and a small auto-correlation value at the time delay. The estimation of the maximum Lyapunov exponent from experiment data is not accurate, on the contrary, the flatness of the RF spectra is experimentally accessible and the flatness can be an alternative measure of the maximum Lyapnov exponent.
2.3 Experimental results of bandwidth enhancement of chaos
We set the optical frequency detuning between Laser 1 and 2 Δf12 = f1 – f2 = 16.5 GHz, and that between Laser 2 and 3 Δf23 = f2 – f3 = 31.5 GHz for bandwidth enhancement of chaos, where fi is the optical carrier frequency of Laser i, estimated from the optical wavelength detuning as described in the previous section. The condition of positive frequency detuning (f1 > f2 and f2 > f3) needs to be satisfied to avoid injection locking among the three unidirectionally-coupled lasers, and the frequency detuning components remain in the RF spectra. We observe temporal waveforms and RF spectra for the three-cascaded semiconductor lasers. Figure 3 shows the temporal waveforms and the corresponding RF spectra for Laser 1, 2, and 3. Chaotic oscillations are obtained for the temporal waveforms of all the laser outputs. The temporal waveform of Laser 2 output oscillates faster than that of Laser 1. The oscillation of Laser 3 output is also faster than that of Laser 2. In the RF spectra, the relaxation oscillation frequency is found at fr1 ~6.5 GHz for the Laser 1 output. A higher peak appears at 17.5 GHz in the RF spectrum of Laser 2, which roughly corresponds to the optical frequency detuning between Laser 1 and 2 (Δf12 = 16.5 GHz). Flatter and wider spectral components are observed in the RF spectra of Laser 3, compared with that of Laser 2. A peak appears at 31.8 GHz, which roughly corresponds to the optical frequency detuning between Laser 2 and 3 (Δf23 = 31.5 GHz). Therefore, the existence of the optical frequency detuning plays an important role for bandwidth enhancement of chaos.
Fig. 3 (a), (c), (e) Chaotic temporal waveforms and (b), (d), (f) their corresponding RF spectra of the three unidirectionally-coupled laser outputs. (a), (b) Laser 1, (c) (d) Laser 2, and (e), (f) Laser 3. The relaxation oscillation frequency of Laser 1 is fr1 ~6.5 GHz, the optical frequency detuning between Laser 1 and 2 is Δf12 = 16.5 GHz, and the optical frequency detuning between Laser 2 and 3 is Δf23 = 31.5 GHz.
The measurement of the standard bandwidth, the effective bandwidth and the flatness for the three laser outputs are summarized in Table 1. Both the standard and effective bandwidths increases as i is increased for Laser i (i = 1, 2, and 3). We measured the standard bandwidth of 35.2 GHz and the effective bandwidth of 26.0 GHz for Laser 3, which are comparable with the values in the literature (32.3 GHz [32]). Large flatness is obtained for Laser 1 and 2 due to the existence of the peak components near fr1 ~6.5 GHz and Δf12 = 16.5 GHz, respectively. However, smaller flatness of 5.6 dB is obtained for Laser 3, which is also comparable with the values in the literature (within ± 3.5 dB [32]). Careful control of both the optical frequency detunings and the injection strengths among the three-coupled lasers is required in order to obtain these flat spectra.
Table 1. Standard bandwidth, effective bandwidth, and flatness of the RF spectra for the Laser 1, 2, and 3 outputs, shown in Fig. 3.
2.4 Parameter dependence of bandwidth and flatness on optical frequency detuning
We investigate the dependence of the bandwidths and the flatness of Laser 3 on the optical frequency detuning. Figure 4 shows the standard bandwidth, the effective bandwidth, and the flatness of the bandwidth-enhanced chaos of the Laser 3 output as the optical frequency detuning between Laser 2 and 3 Δf23 is varied. Δf23 is continuously changed from –50 to 50 GHz. The optical frequency detuning between Laser 1 and 2 is fixed at Δf12 = 16.5 GHz. For Fig. 4(a), the standard bandwidth increases as the absolute value of Δf23 is increased up to ~40 GHz, where the measurement is limited by the bandwidth of the photodetector (~38 GHz). The increase of the bandwidth is significant for positive values of Δf23 than negative values, because the injection locking range is asymmetrically located in the negative detuning region due to the linewidth enhancement factor [16,17]. It seems that the standard bandwidth is increased monotonically as |Δf23| is increased, except Δf23 > 40 GHz (i.e., bandwidth limitation of the photodetector). However, the peak height of the RF spectral component corresponding to Δf23 becomes higher for large |Δf23|, which is not suitable for random bit generation.
Fig. 4 (a) Standard bandwidth, (b) effective bandwidth and flatness as a function of the optical frequency detuning between Laser 2 and 3 Δf23. The optical frequency detuning between Laser 1 and 2 is fixed at Δf12 = 16.5 GHz.
For Fig. 4(b), the effective bandwidth shows a similar curve to that for the standard bandwidth in Fig. 4(a), however, the value of the effective bandwidth is smaller than that of the standard bandwidth. The effective bandwidth increases as |Δf23| is increased, and saturated at ~26 GHz for positive Δf23. This result indicates that too large |Δf23| cannot result in large effective bandwidth due to the existence of large peak component at |Δf23|. On the contrary, the flatness is changed for different values of Δf23, and smaller flatness is obtained for positive Δf23. The minimum flatness (5.0 dB) can be obtained at Δf23 = 20.0 GHz, however, the standard bandwidth (28.7 GHz) and effective bandwidth (24.5 GHz) are not so large in this condition. We thus selected Δf23 = 28.0 GHz to obtain both large standard bandwidth (35.2 GHz) and large effective bandwidth (26.0 GHz) for random bit generation, even though the flatness is slightly degraded (5.6 dB), as shown in Table 1.
There are three important parameters to determine the maximum rate of single-bit extraction method: the standard bandwidth (the maximum oscillation frequency), the effective bandwidth (the effective frequency region in the broadband spectrum), and the flatness (no periodicity and no time-delay signature). Large bandwidths and small flatness result in high rate of random number generation, which will be described in the next section.
3. Single-bit extraction method for random bit generation
3.1 Two schemes of single-bit and multi-bit extraction methods
We introduce two extraction methods for physical random bit generation. The first scheme is a single-bit extraction method to evaluate the entropy rate of bandwidth-enhanced chaos, which is the rate for the generation of non-deterministic bits [15,37,38]. The entropy rate needs to be independent of complicated post-processing techniques for generating random bit sequences, and the entropy rate can be quantitatively evaluated by a simple single-bit generation from chaotic temporal waveforms. We use the single-bit extraction method in Section 3. The second scheme is a multi-bit extraction method to evaluate the maximum generation rate of random bit sequences that can pass the statistical tests of randomness for the purpose of engineering applications. Both the bandwidth enhancement of chaos and the multi-bit extraction method are crucial to maximize the random-bit generation rate in this scheme. The result of the multi-bit extraction method will be described in Section 4.
The purposes of the single-bit and multi-bit extraction schemes are different as follows: For the single-bit scheme, the entropy rate is used for the evaluation of physical entropy source, and it is important to guarantee theoretical limit of the rate of uncertainty from the physics point of view. On the contrary, for multi-bit scheme, statistical tests can be used for the evaluation of the statistical properties of randomness for generated bits with finite bit lengths, and this evaluation may be satisfactory from the practical point of view for engineering applications.
3.2 Single-bit extraction method
The single-bit extraction method used for the estimation of entropy rate is shown in Fig. 5 [1,6,15]. A chaotic waveform and its time-delayed signal are sampled by using a 1-bit ADC with a periodic clock, and converted into a binary signal (bit 0 or 1), based on whether the sampled data is below or above a threshold value of the 1-bit ADC, respectively. The threshold value for the 1-bit ADC is set to the mean value of the chaotic waveform in this experiment. Bitwise XOR operation is carried out for the two bits to generate a new random bit. One bit is obtained at one sampling point of the two chaotic waveforms, and the sampling rate of the ADC directly corresponds to the generation speed of random bit sequences.
3.3 Result of single-bit extraction method for random bit generation
We generated random bit sequences by using the single-bit extraction method with the chaotic temporal waveforms of all the three lasers to evaluate the entropy rate. The generated random bits are evaluated using National Institute of Standards and Technology Special Publication 800-22 (NIST SP 800-22) [39]. This is a de-fact standard statistical test of randomness, consisting of 15 tests. Random numbers that can pass all the 15 tests indicate that their randomness is statistically equivalent to ideal random numbers. (see also Table 2 in Sec. 4.3.1 for details.)
Table 2. Result of NIST SP 800-22 for random bit sequences generated from the multi-bit extraction method with 7 LSBs. Significance level is set to α = 0.01. To pass the tests, the P-value of the uniformity of p-values should be larger than 0.0001, and the proportion of sequences satisfying p-value > α for 1000 samples of 1 Mbit data should be in the range of 0.99 ± 0.0094392 [39]. For tests which produce multiple P-values and proportions, the worst case is shown.
We investigated how the single-bit extraction rates are changed for the three laser outputs. We changed the sampling rate of the single-bit extraction method and generated random bit sequences at each sampling rate. The randomness of the bit sequences generated at different sampling rates is evaluated by using NIST SP 800-22 to estimate the maximum rate of random bit generation.
Figure 6 shows the number of passed NIST SP 800-22 tests for random bits generated from the chaotic temporal waveforms of Laser 1, 2, and 3 when the sampling rate is changed. We used the chaotic temporal waveforms of Laser 1, 2, and 3 shown in Fig. 3. The lower horizontal axis indicates the sampling time, and the upper horizontal axis indicates the sampling rate (the inverse of the sampling time), which corresponds to the random-bit generation rate. The vertical axis indicates the number of the NIST tests that pass the statistical tests of randomness, where “15” corresponds to the results that all the NIST tests are passed. Five 1-Gbit sequences of random bits are used for each NIST test and the median of the five test results is plotted with error bars of the maximum and minimum values in Fig. 6. For Laser 1, the sampling rates that can pass all the NIST tests are distributed discretely, as shown in Fig. 6(a). The maximum rate of random bit generation with certified randomness is 6.67 Gb/s for Laser 1. For Laser 2, the maximum generation rate 4.76 Gb/s is obtained in Fig. 6(b), which is smaller than the case of Laser 1. We speculate that this result originates from the existence of the large peak corresponding to Δf12 in the RF spectra in Fig. 3(d), and the worse flatness of Laser 2 results in slower maximum generation rate than that for Laser 1. For Laser 3, the generation rate is improved from both Laser 1 and 2, and the maximum generation rate 20.0 Gb/s is obtained in Fig. 6(c). Both large bandwidth and small flatness for the Laser 3 output results in such a high generation rate. The single-bit generation rate 20.0 Gb/s is reasonable since this value is 0.57 times the standard bandwidth (35.2 GHz) of the bandwidth-enhanced chaos [6]. The sampling rate can be larger than half of the standard bandwidth, however, it is smaller than the standard bandwidth.
Fig. 6 Number of passed NIST tests for random bit sequences generated from chaotic temporal waveforms of (a) Laser 1, (b) Laser 2, and (c) Laser 3 as a function of the sampling time. The lower horizontal axis indicates the sampling time, and the upper horizontal axis indicates the sampling rate (the inverse of the sampling time), which corresponds to the random-bit generation rate. The vertical axis indicates the number of the NIST tests that pass the statistical tests of randomness, where “15” corresponds to the results that all the NIST tests are passed. Five 1-Gbit sequences of random bits are used for each NIST test and the median of the five test results is plotted with error bars of the maximum and minimum values. The red numbers on the upper horizontal axis indicate that the random bits generated at this sampling rate can pass all the NIST tests.
Figure 7 shows the relationship between the autocorrelation function of the chaotic temporal waveforms of Laser 3 and the result of the NIST tests for random bit sequences generated from the Laser 3 output. The horizontal axis of Fig. 7 indicates the delay time for the calculation of the autocorrelation function, where the delay time is treated as the sampling time for random bit generation. The red circles indicate that the random bits generated at the sampling rate can pass all the 15 NIST tests, whereas the blue triangles indicate that the random bits generated at the sampling rate fail some of the NIST tests. In some cases, random bits generated at the sampling rate with small autocorrelation can pass all the NIST tests. The autocorrelation value less than 7.7 × 10−2 is a necessary condition to pass all the 15 NIST tests of randomness. However, low autocorrelation values do not mean that all the NIST tests are passed in Fig. 6(c) for chaotic data [6].
Fig. 7 Autocorrelation function (absolute value) of the chaotic temporal waveform of Laser 3 as a function of the delay time and the result of NIST tests for generated random bit sequences. The results of NIST tests correspond to Fig. 6(c). Red circles: all the NIST tests are passed. Blue triangles: one or more NIST tests are failed. The dotted line indicates the autocorrelation value 7.7 × 10−2. The red numbers on the upper horizontal axis indicate that the random bits generated at this sampling rate can pass all the NIST tests.
4. Multi-bit extraction method for random bit generation
4.1 Multi-bit extraction method
In this section, we used a multi-bit extraction method for random bit generation to evaluate the maximum generation rate of physical random bit sequences that can pass the statistical test of randomness. We combined several post-processing methods to maximize the random-bit generation rate. It is worth noting that we evaluate the generation rate within the information limit L of the physical random bit generation, determined by the following equation.
Where S is the sampling rate of the ADC used for detecting chaotic waveforms, V is the vertical resolution (the number of significant bits) of the ADC, and D is the number of physically detected chaotic data. In our experiment, the information limit is considered as 1.6 Tb/s = 100 GigaSamples/s (GS/s) × 8-bit vertical resolution × 2 data. In this sense, we evaluate the generation rate of “physical” random bits, rather than physical-based “pseudo” random bits [11]. The generation rate of physical random bit sequences cannot exceed the amount of the original physical information determined by the information limit. The Eq. (1) is similar to the formula proposed in [8] when the sampling rate is set to twice the chaotic analog bandwidth and one chaotic signal is used. We note that the sampling rate is not limited by the bandwidth of chaotic analog signals, and the sampling rate can be larger than twice the bandwidth when some of the most significant bits (MSBs) are discarded, as seen in the following results.The multi-bit extraction scheme is different from the single-bit extraction scheme. The entropy rate of 20 Gb/s is obtained for the single-bit extraction scheme, where the MSB becomes unpredictable when sampled at this rate. The entropy rate is related to only the MSB, but not for the rest of the significant bits. For example, the entropy rate for the second MSB can be faster than that for the first MSB, since the width of the partitions for the ADC becomes exponentially smaller for lower significant bits (e.g. 2n partitions for the n-th MSB). The sampling rate can be increased by discarding some MSBs for the multi-bit extraction scheme.
Figure 8 shows the scheme for the multi-bit extraction method for random bit generation. A chaotic temporal waveform (referred to as A in Fig. 8) and its physically time-delayed waveform (referred to as B) are detected by the digital oscilloscope at the sampling rate 100 GS/s with 8-bit vertical resolution. The physical time delay is introduced by using an extra electric coaxial cable whose length determines the delay time of 4.51 ns. The delay time is selected to avoid the correlation between the original and the time-delayed signals. Note that the original chaotic signal and its physically time-delayed signal do not match completely because the sampling points are different on the chaotic analog signals. In addition, two additional time-delayed signals (referred to as A’ and B’) are generated from A and B by using software in a computer, whose delay times are determined to avoid the correlation between the original and delayed signals (i.e., 1.59-ns delay between A and A’, and 2.91-ns delay between B and B’). The order of each significant bit of the 8-bit time-delayed signals A’ and B’ is reversed independently, i.e., the MSB becomes the LSB, the second MSB becomes the second LSB, and so on [5]. The resultant bits after the bit-order-reversal procedure are referred to as A’R and B’R, respectively. Bit-wise XOR operation is carried out between A and A’R, and between B and B’R. The generated 8-bit signals are referred to as X and Y, respectively. n LSBs (1 ≤n ≤8) are extracted from 8-bit X and Y, and the extracted n bits (referred to as Xn~1 and Yn~1) are combined as a random bit sequence [3]. This procedure is repeated for each chaotic data sampled at 100 GS/s in off-line post-processing.
4.2 Selection of amplitude and off-set of temporal waveform
First, we optimize the condition for the multi-bit extraction method for random bit generation. We properly set both the amplitude and the off-set (mean value) of chaotic temporal waveforms when the 8-bit analog-to-digital conversion is applied. The temporal waveform is quantized in the digital oscilloscope with 8-bit vertical resolution, and converted to the values ranging from −127 to + 127 in integer. The amplitude of the temporal waveform is adjusted continuously by using the variable fiber attenuator in front of the photodetector. The standard deviation σ of the chaotic temporal waveform is measured. The detection window size of the digital oscilloscope is denoted as ± mσ, where m is a positive number [6]. The amplitude of the chaotic waveform is controlled by changing the detection window size ± mσ, where larger m indicates smaller amplitude of a chaotic temporal waveform. We also changed the center position (the mean value) of the detected temporal waveform by adjusting the off-set function in the digital oscilloscope.
Figure 9 shows the histograms (probability density functions) of the chaotic temporal waveforms of Laser 3 when the detection window size or the off-set of the temporal waveform is changed. For Fig. 9(a), the detection window size is changed from ± 2.5σ to ± 6.0σ. A broad distribution with high peak at −127 is observed for ± 2.5σ. The center peak of the distribution increases as m is increased. A sharp distribution with small amplitude is obtained for ± 6.0σ. For Fig. 9(b), the center peak moves from the negative to the positive values as the off-set is increased. From these results, the probability distribution is controlled by changing the detection window size and the off-set.
Fig. 9 Histograms (probability density functions) of the temporal waveforms when (a) the detection window size or (b) the off-set of the temporal waveforms is changed. (a) The off-set is fixed at 0. (b) The detection window size is fixed at ± 4.0σ, corresponding to ± 50 mV.
To optimize the amplitude and the off-set, we investigate the bias of the occurrence of bit ‘1’ for each significant bit of the 8-bit data for the historgrams shown in Fig. 9. The bias b of the occurrence of bit 1 is defined as follows.
Where P is the probability of the occurrence of 1. b becomes 0 for ideal random bits with infinite bit length (P = 0.5). Smaller bias indicates better random bit sequences with finite bit length.Figure 10 shows the bias b for each significant bit of the 8-bit chaotic waveform for different amplitude and off-set for 1-Gbit chaotic data. For Fig. 10(a), the bias for LSB 1~4 (LSB 1 corresponds to the least significant bit) decreases and saturates at ~10−5 as m is increased, because the peak height of the probability distribution at −127 decreases. On the contrary, the bias for LSB 8 (i.e., the most significant bit) and LSB 7 is almost constant for different m, because the off-set of the probability distribution is fixed. The bias for LSB 5 and 6 is changed slightly for different m, and the bias for LSB 6 increases for large m. These results indicate that larger m results in better bias for LSB 1~4. For too large m, however, the distribution becomes too narrow and no probability is found in many bins of the histogram. We thus selected ± 4.0σ as the optimal detection window size (amplitude) of the temporal waveform from Fig. 10(a).
Fig. 10 Bias b for each significant bit of the 8-bit chaotic waveform as a function of (a) the window step size and (b) the off-set for 1-Gbit chaotic data. (a) The off-set is fixed at 0. (b) The detection window size is fixed at ± 4.0σ, corresponding to ± 50 mV. The light yellow region indicates that the peak value is larger than 10−4 at the edge −127 in the histogram of Fig. 9.
Figure 10(b) shows the bias of each significant bit as the off-set is changed. The bias for LSB 1~4 stays at ~10−5 for the off-set between −15 and 2 mV. In this region, the probability distribution has a peak at −127 whose peak height is less than 10−4. Therefore, it is important to select the distribution with small peak at the edge −127 in order to obtain small bias. The bias for LSB 8 and 7 shows the minimum value at the off-set of 2 and 3 mV, respectively. The bias for LSB 5 and 6 shows some local minima that are shifted from the minimum values of LSB 8 and 7. Therefore, the off-set cannot be determined to optimize the bias for all the significant bits. We selected 2 mV for the off-set in our experiment to minimize the off-set of LSB 8.
4.3 Result of multi-bit extraction method for random bit generation
We generated random bit sequences by using the multi-bit extraction method with the parameter settings of the detection window size of ± 4.0σ and the off-set of 2 mV of the temporal waveforms. The generated bit sequences are evaluated by using the statistical tests of randomness. We used two types of statistical tests of randomness, NIST SP 800-22 [39] and TestU01 [40]. TestU01 has been developed to evaluate large number of random bit sequences. There are five packages for TestU01, namely Rabbit, Alphabit, SmallCrush, Crush, and BigCrush, depending on the lengths of random bit sequences used for statistical tests of randomness.
4.3.1 NIST Special Publication 800-22
First we used NIST SP 800-22 for the evaluation of random bit sequences generated by using the multi-bit extraction method. Table 2 shows the results of NIST SP 800-22 for random bit sequences generated by using the multi-bit extraction method with 7 LSBs. The equivalent generation rate of the random bit sequences is 1.4 Tb/s ( = 100 GS/s × 7 LSBs × 2 data). We found that all the statistical tests are passed as shown in Table 2.
We changed the number of extracted n LSBs to evaluate the maximum rate of random bit generation. Figure 11 shows the number of passed NIST tests as a function of the number of extracted LSBs for the multi-bit extraction method. Five 1-Gbit sequences of random bits are used for each NIST test and the median of the five test results is plotted with error bars of the maximum and minimum values in Fig. 11. “15” on the vertical axis of Fig. 11 indicates that all the NIST tests are passed. For Fig. 11, we succeed in generating random bits that can pass all the 15 statistical tests of randomness from 1 to 7 LSBs. Only the runs test cannot be passed for the case of 8 LSBs in Fig. 11, indicating that short-term correlation still remains in the generated random bits. The existence of short-term correlation is one of the weaknesses for physical random bit generation. The failure of the statistical tests indicates that the random bit sequences obtained from some post-processing still possess the original physical feature of the chaotic dynamics. From Fig. 11, we found that the maximum random bit generation rate is equivalently achieved at 1.4 Tb/s by avoiding only the most significant bit and by using the rest of 7 LSBs (i.e., 1.4 Tb/s = 100 GS/s × 7 LSBs × 2 data).
Fig. 11 Number of passed NIST tests as a function of the extracted LSBs for the multi-bit extraction method for random bit generation. “15” on the vertical axis indicates that all the NIST tests are passed. Five 1-Gbit sequences of random bits are used for each NIST test and the median of the five test results is plotted with error bars of the maximum and minimum values.
4.3.2 TestU01
Next we used TestU01 for the evaluation of random bit sequences generated by using the multi-bit extraction method. TestU01 has been proposed by L’Ecuyer and Simard [40] for empirical testing of random number generators. There are five packages: Rabbit, Alphabit, SmallCrush, Crush, and BigCrush. The required amount of random bit sequences for Rabbit and Alphabit are small, where 1.05 and 33.6 Mbits are used in our experiment. On the contrary, the three packages of SmallCrush, Crush, and BigCrush require large number of random bit sequences for the statistical tests of randomness, i.e., 1.64, 41.0, and 410 Gbits are required, respectively. In this study, we used four packages of TestU01 except BigCrush, since too many bits are required for BigCrush. SmallCrush and Crush require larger number of bit sequences than that for NIST SP 800-22 (> 1 Gbit), and more precise detection of non-randomness can be executed for physical random numbers.
Table 3 shows the results of Rabbit, Alphabit, and SmallCrush for random bit sequences generated from 8 LSBs (i.e., all the LSBs are used). Rabbit, Alphabit, and SmallCrush consist of 38, 17, and 15 tests, respectively. The p-values for all the tests need to be greater than 0.001 or smaller than 0.999 to pass the statistical tests of randomness. We found that all the test results satisfy the criteria for the p-values in Table 3. Table 4 shows the results of Crush for 41-Gbit random bit sequences generated from 6 LSBs (i.e., the significant bits from LSB 1 to 6 are used). We also found that all the 144 tests are passed for the Crush tests in Table 4.
Table 3. Results of (a) Rabbit, (b) Alphabit, and (c) SmallCrush in TestU01 for random bit sequences generated from the multi-bit extraction method with 8 LSBs. Rabbit, Alphabit, and SmallCrush consist of 38, 17, and 15 tests, respectively. The p-values for all the tests need to be greater than 0.001 or smaller than 0.999 to pass the statistical tests of randomness [40].
Table 4. Results of Crush in TestU01 for 41-Gbit random bit sequences generated from the multi-bit extraction method with 6 LSBs. Crush consists of 144 tests. R indicates the result (S: Success, F: Fail) in Table 4. The p-values for all the tests need to be greater than 0.001 or smaller than 0.999 to pass the statistical tests of randomness [40].
Figure 12 shows the results of Rabbit, Alphabit, SmallCrush, and Crush when the number of extracted LSBs is changed. Five sequences of random bits are used for each test and the median of the five test results is plotted with error bars of the maximum and minimum values in Fig. 12(a). Random bit sequences generated from all the LSBs can pass Rabbit, Alphabit, and SmallCrush, as shown in Fig. 12(a). The maximum generation rate is equivalently 1.6 Tb/s ( = 100 GS/s × 8 LSBs × 2 data) for random bit sequences that can pass Rabbit, Alphabit, and SmallCrush. For Fig. 12(b), one sequence of 41-Gbit data is used for Crush tests and the test results are plotted. We found that random bit sequences generated from 6 LSBs can pass all the Crush tests. This result indicates that the maximum generation rate is equivalently 1.2 Tb/s ( = 100 GS/s × 6 LSBs × 2 data) for random bit sequences that can pass all the Crush tests.
Fig. 12 Number of passed TestU01 tests as a function of the extracted LSBs for the multi-bit extraction method for random bit generation. (a) Rabbit, Alphabit, SmallCrush, and (b) Crush. (a) “38,” “17,” and “15” on the vertical axis indicate that all the Rabbit, Alphabit, and SmallCrush tests are passed, respectively. Five sequences of random bits are used for each test and the median of the five test results is plotted with error bars of the maximum and minimum values. (b) “144” on the vertical axis indicate that all the Crush tests are passed. One sequence of 41-Gbit data is used for Crush tests.
4.3.3 Summary of the statistical evaluation of randomness
For multi-bit generation, we used two batteries of the statistical tests of randomness, NIST SP 800-22 and TestU01. We obtained different maximum rates of random bit generation that are verified by different statistical tests. The maximum generation rates are 1.6 Tb/s (verified by using Rabbit, Alphabit, and SmallCrush in Test U01), 1.4 Tb/s (NIST SP 800-22), and 1.2 Tb/s (Crush in TestU01), respectively, where the maximum generation rate is equivalently estimated from the formula: 100 GS/s × n LSBs × 2 data (n = 1, 2, …, 8). We found that it is more difficult to pass the statistical tests that require larger amount of random bit sequences, such as Crush test. For all the cases, the random-bit generation rate up to 1.2 Tb/s can be achieved with verified randomness by using both NIST SP 800-22 and TestU01 for the multi-bit extraction method.
5. Conclusions
We experimentally demonstrated fast physical random bit generation from bandwidth-enhanced chaos by using three-cascaded semiconductor lasers. The three semiconductor lasers are unidirectionally coupled from Laser 1 to 2 and from Laser 2 to 3 with positive optical frequency detuning. We obtained the bandwidth-enhanced chaos with the standard bandwidth of 35.2 GHz, the effective bandwidth of 26.0 GHz, and the flatness of 5.6 dB.
We used the bandwidth-enhanced chaos for physical random bit generation. We introduced single-bit and multi-bit extraction methods for random bit generation to evaluate the entropy rate and the maximum random bit generation rate, respectively. For single-bit generation, we obtained the generation rate at 20 Gb/s for physical random bit sequences. For multi-bit generation, we achieved the maximum generation rate at 1.2 Tb/s ( = 100 GS/s × 6 LSBs × 2 data) for physical random bit sequences whose randomness is verified by using both NIST Special Publication 800-22 and TestU01.
Our approach with cascaded semiconductor lasers results in large bandwidth and small flatness of the RF spectrum as a fast entropy source, and Tb/s random number generation can be realized. The miniaturization of the entropy source on a photonic integrated circuit and the real-time implementation with fast electronic devices for post-processing will be our further works.
The high-speed random bit generators over Tb/s are promising for engineering applications in information security, such as secure key distribution based on information theory and quantum cryptography, as well as high-performance parallel computation.
Acknowledgments
We acknowledge support from Grants-in-Aid for Scientific Research from Japan Society for the Promotion of Science, and Management Expenses Grants from the Ministry of Education, Culture, Sports, Science and Technology in Japan.
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