1. Introduction

Fast physical random number generators (RNGs) play a crucial role in applications in information security and numerical simulations due to their non-reproducibility and non-periodicity. The speed of physical RNGs based on thermal noise in semiconductors has been limited to hundreds of megabits per second (Mb/s). Recently, optical physical RNGs based on chaotic lasers [1–23], amplified spontaneous noise [23–27], and quantum noise [28–33] have been extensively developed. One of the advantages of chaotic-laser-based RNGs is the speed of random number generation, from gigabits per second (Gb/s) to terabits per second (Tb/s), due to the fast dynamics of the chaotic laser output. Fast physical RNGs are key in new information-theoretic security systems [34–36] and large-scale parallel computation [37].

The generation rate of RNGs based on chaotic laser dynamics has been rapidly increasing in recent years. A generation rate of 1.7 Gb/s was reported in 2008 for the first demonstration of an optical-chaos-based RNG with two semiconductor lasers with optical feedback [1]. Since then, several techniques have been proposed for increasing the frequency bandwidth of chaotic temporal waveforms [3,5,13] and implementing complicated post-processing methods [2–5]. These efforts have resulted in generation rates on the order of Tb/s [12–15] in recent years. However, most studies on fast RNGs used a fast digital oscilloscope to detect chaotic temporal waveforms, and random numbers were generated by computer through offline post-processing. The generation rates reported in the literature are “equivalent” values if the methods used to obtain them can be implemented in real-time post-processing for RNGs using fast electronic devices.

Real-time hardware implementations of optical-chaos-based RNGs have been reported [1,7]. Random bit sequences have been generated at 4.5 Gb/s in real time with a fiber-optics-based semiconductor laser with optical feedback and an electronic circuit for post-processing [7]. The format of the generated random bit sequences is non-return-to-zero (NRZ), which has been used as a modulation signal in optical communications. However, random bits in NRZ format cannot be directly used in a computer as a secret key for cryptography or a random seed in numerical simulations. A real-time RNG with a computer-friendly format is needed for engineering applications involving RNGs. In addition, real-time quantum RNGs having generation rates of up to 3.2 Gb/s have been reported [31,33]. A Boolean chaos network has been implemented on a field programmable gate array (FPGA) electronic board to generate random numbers, and a real-time generation rate of 12.8 Gb/s was achieved [38]. However, the generation rate for each Boolean chaos system is limited to 100 Mb/s.

Photonic integrated circuits (PICs) are a promising way to minimize the size of RNGs for practical implementation [17–21] because many studies on chaos-based RNGs are based on commercially available semiconductor lasers and photodetectors placed on a large optical table. Photonic integrated circuits can provide stable operation of chaotic light sources and are robust against air turbulence and temperature fluctuations. Photonic integrated circuits consisting of a semiconductor laser with optical feedback have been fabricated and reported as chaotic laser sources for RNGs [17–21]. However, no studies on real-time RNG with PICs have been reported. Very recently, streaming of a real-time RNG using a PIC has been performed at a throughput up to 4 Gb/s with 1-GHz sampling rate and 8-bit resolution [39]. They used a PIC with a semiconductor laser with optical feedback, and random bit sequences are generated using post-processing implemented in a personal computer by software.

In this study, we demonstrate a real-time fast physical RNG with a PIC and an FPGA electronic board for a full hardware implementation with a generation rate of over 20 Gb/s. We produce a large amount of random bit sequences in binary format in a personal computer and evaluate their statistical randomness using NIST SP 800-22 [40] and TestU01 [41]. We also evaluate the real-time generation rate of the RNG.

2. Experimental setup and chaotic temporal waveform

2.1 Experimental setup

Figure 1 shows a schematic diagram of the PIC used for the RNG [18,21]. The PIC consists of a photodetector (PD), a distributed-feedback (DFB) semiconductor laser, two optical semiconductor amplifiers (SOA 1 and SOA 2), a passive waveguide, and an external mirror for optical feedback. The lengths of the DFB laser, SOA 1, SOA 2, and PD are 0.3, 0.2, 0.1, and 0.05 mm, respectively. The structures of the DFB laser, SOAs, and PD are similar to those described in Ref [19], where ridge waveguide type of structure is used. The DFB laser, SOAs, and PD sections contain a strained InGaAs/InGaAsP multi-quantum well active layer grown by metal-organic vapor phase epitaxy on an n-InP substrate [19]. The DFB laser emits light around a wavelength of 1550 nm. The two SOA sections are introduced to control the strength of the optical feedback. The PD is a PIN photodiode and the detection bandwidth of the PD is estimated as 20 GHz. The PD section is electrically isolated from the rest of the device to avoid mixing between the photocurrent and the injection current. One of the edges of the PIC has a high-reflection (HR) coating that acts as an external mirror. The external cavity length is 10.3 mm, which corresponds to the distance between the right facet of the DFB laser and the external mirror, and the external cavity frequency is 3.8 GHz. The width of the waveguide is 2 μm, which ensures single-mode propagation.

 

Fig. 1 Schematic diagram of photonic integrated circuit (PIC). The PIC consists of a photodetector (PD), a distributed-feedback (DFB) semiconductor laser, two semiconductor optical amplifiers (SOA 1 and SOA 2), a passive waveguide, and an external mirror. The external cavity length is 10.3 mm.

Download Full Size | PDF

The optical output from the DFB laser is reflected by the mirror and re-injected into the laser to produce chaotic intensity fluctuations. The lasing power is controlled by the injection current of the DFB laser. The injection currents of the SOAs are adjusted to control the strength of the feedback light for which chaos can be generated. We observed different temporal dynamical regimes at various parameter values of the injection current and the feedback strength [42,43]. We selected a chaotic oscillation used for the RNG when the injection currents of the DFB laser, SOA1, and SOA2 were set to 39.00 (3.0 times the lasing threshold), 13.00, and 5.00 mA, respectively. We selected the operational parameters in order to generate chaotic oscillations with continuous RF spectrum (no sharp peak) [43]. On the contrary, we avoided low-frequency fluctuations (LFF), which degrades the quality of generated random bit sequences due to slow intensity dropouts. The choice of the operational parameters is not severe to generate good random bit sequences, as long as chaotic oscillations are obtained.

Figure 2 shows a photograph of the real-time RNG, the components of which are stored in a box having dimensions of 310 mm × 270 mm × 140 mm. The RNG consists of the PIC, two electric amplifiers (Newport, 1422-LF, 20-GHz bandwidth), an analog-to-digital converter (ADC) (Tokyo Electron Device, TD-BD-FMC3.6GADC, 3.6 GigaSamples/s (GS/s), 12-bit vertical resolution), an FPGA electronic board (Tokyo Electron Device, TB-7K325T-10GPCIE with Kintex7 XC7K325T FFG900-2 Speed Grade (Xilinx)), and a PCI Express Gen2 x8 (PCIE) output bus connected to a personal computer (OS: Windows 7 Professional 64 bit, CPU: Intel Core i7-4770 (3.4-GHz clock), random access memory: 32 GBytes). The electronic output from the photodetector in the PIC is amplified by the two amplifiers and is sent to the ADC at 3.6 GS/s with a 12-bit vertical resolution. The eight most significant bits (MSBs) are extracted from each 12-bit signal and the remaining four least significant bits (LSBs) are discarded in order to avoid the effect of ADC noise and the limitation of the transmission rate of the PIC Express Gen2 x8 (32 Gb/s = 4 Gb/s × 8 bits at maximum). The eight extracted MSBs are used for generating random bit sequences on the FPGA. The generated random numbers are sent directly to the random access memory on the computer in binary format via the PCI Express Gen2 x8. The maximum real-time generation rate for the RNG is 28.8 Gb/s ( = 3.6 GS/s × 8 MSBs), as determined by the ADC, in this system.

 

Fig. 2 Photograph of experimental setup for real-time random number generator with photonic integrated circuit and FPGA electronic board. The dimensions of the box are 310 mm × 270 mm × 140 mm.

Download Full Size | PDF

2.2 Chaotic temporal waveform and its characteristics

Figures 3(a) and 3(b) show the chaotic temporal waveform and the corresponding RF spectrum of the laser output generated from the PIC, respectively. The temporal waveforms are detected by the ADC in the RNG box. Chaotic irregular oscillation is observed, and a broadband RF spectrum is obtained. Figure 3(c) shows the histogram of the chaotic temporal waveform. A non-smooth distribution is obtained, resulting from the quantization errors of the ADC. In order to obtain a smoother histogram, we introduce a differential method [2,6],in which the difference between a chaotic signal and its time-delayed signal (1.94-ns delay) is used for random number generation. Figure 3(d) shows the histogram of this difference signal. The differential method generates a smoother and more symmetric distribution for the histogram, as compared to that for the original histogram in Fig. 3(c).

 

Fig. 3 Experimental data for chaotic laser output of photonic integrated circuit. (a) Temporal waveform, (b) RF spectrum, (c) histogram of temporal waveform, and (d) histogram of difference between chaotic signal and its time-delayed signal (1.94-ns delay).

Download Full Size | PDF

Figure 4(a) shows the autocorrelation function for the chaotic temporal waveform obtained from the PIC. The absolute value of the autocorrelation function is plotted on a logarithmic scale on the vertical axis. The autocorrelation function decays with increasing delay time for the calculation of the autocorrelation. Figure 4(b) shows an enlarged view of the autocorrelation function. It is seen to decay rapidly and decrease from 10⁻¹ to 10⁻³ for delay times longer than 0.6 ns.

 

Fig. 4 (a) Experimental data for absolute value of autocorrelation function for chaotic laser output of photonic integrated circuit. (b) Enlarged view of (a).

Download Full Size | PDF

3. Experimental results for random number generation

3.1 Post-processing using the bit-order reversal method

We implement a post-processing method in the FPGA based on the combination of the differential method [2,6], the bit-order reversal method [5], the exclusive-OR (XOR) method [2,3], and the LSB extraction method [2,3]. We also use another post-processing method, i.e., the bit-shift and bit-rotation method [44], as described in the Appendix.

Figure 5 shows the post-processing scheme for the RNG. A chaotic temporal waveform generated from the PIC is converted into a 12-bit digital signal using the ADC at a sampling rate of 3.6 GS/s. Eight MSBs are extracted from the 12-bit digital signal and the remaining four LSBs are discarded in order to avoid the influence of quantization noise in the ADC. The extracted 8-bit digital signal is used for the post-processing in Fig. 5. The 8-bit signal is stored in memory and is used to generate three time-delayed signals (referred to as delayed signals 1, 2, and 3) with delay times of 1.94, 4.44, and 6.39 ns, respectively, on the FPGA board. The delay times are selected such that the autocorrelation function for the chaotic temporal waveform becomes small. For example, the autocorrelation values are 4.2 × 10⁻², 8.6 × 10⁻³, and 8.7 × 10⁻³ for delay times of 1.94, 4.44, and 6.39 ns, respectively, in Fig. 4(b). The difference is calculated between the original chaotic signal and delayed signal 1, and between delayed signals 2 and 3. The bit-order reversal method [5] is applied for the second difference signal, where the order of each significant bit of the 8-bit signal is reversed, i.e., the MSB becomes the LSB, the second MSB becomes the second LSB, and so on. Bit-wise XOR operation is then carried out between the first difference signal and the bit-order-reversal signal. Finally, some of the LSBs are extracted and used as random bit sequences. We change the number of extracted LSBs to evaluate the statistical randomness of the generated bit sequences.

 

Fig. 5 Schematic diagram of random number generation using bit-order reversal method. AD: analog-to-digital, LSB: least significant bit, PIC: photonic integrated circuit.

Download Full Size | PDF

These procedures for the RNG are implemented on the FPGA board, and the generated random bits are sent to the computer via PCI Express Gen2 x8 in real time. The random numbers are stored in binary format in computer memory, and can be directly used for applications in cryptography and numerical simulations without further offline post-processing. We experimentally confirmed stable generation of random bit sequences for several hours without changing the performance of RNG.

3.2 Evaluation of random number generation

We generate 1 Tbit of random bit sequences in real time, which are used for a simple statistical test of randomness referred to as the bias [22]. The bias b for the occurrence of bit 1 is defined as b = |P - 0.5|, where P is the probability of the occurrence of 1 for random bits 0 or 1. Here, b becomes 0 for ideal random bits of infinite length (i.e., P = 0.5). A smaller bias indicates better random bit sequences with a finite bit length.

Figure 6 shows the bias b for the original chaotic waveform and the random bit sequences generated using the post-processing in Fig. 5 with the eight LSBs (i.e., no bits are discarded). Five sequences are used to calculate the average bias. The dotted lines indicate the 3σ criterion for the bias, where $\sigma =1/\left(2\sqrt{N}\right)$, and N is the number of random bits [7,24]. The bias exceeds the 3σ criterion for the original chaotic waveforms for N > 10⁸, as shown in Fig. 6(a), which cannot be directly used as random bits. However, the bias is always lower than the 3σ criterion for the generated random bits for all N up to 10¹² (1 Terabit), as shown in Fig. 6(b). This result indicates that the bias for the generated random bit sequences is small enough to satisfy the 3σ criterion.

 

Fig. 6 Bias (blue curves) of generated random bit sequences as function of number of generated bits N (a) without and (b) with the post-processing in Fig. 5. The black dotted-dashed line indicates the 3σ criterion for the bias of random bits, where $\sigma =1/\left(2\sqrt{N}\right)$.

Download Full Size | PDF

The generated random bit sequences are evaluated using statistical tests for randomness. We used two de-facto standards for statistical tests of randomness: NIST Special Publication 800-22 (NIST SP 800-22) [22,40] and TestU01 [41]. NIST SP 800-22 requires a 1-Gbit sequence for evaluation and has been commonly used to evaluate the randomness of RNGs in the literature. TestU01 has been developed in order to evaluate a larger amount of random bit sequences and consists of five packages: Rabbit, Alphabit, SmallCrush, Crush, and BigCrush [41]. The required amounts of random bit sequences for SmallCrush, Crush, and BigCrush are 1.64, 41.0, and 410 Gbits, respectively, which require larger amounts of random bits than those for NIST SP 800-22. Therefore, more reliable detection of non-randomness for a large amount of random bits can be realized by using TestU01. In contrast, Rabbit and Alphabit require only 1.05 and 33.6 Mbits, respectively. We used all five of the packages of TestU01, including BigCrush with 410-Gbit random bit sequences.

We first evaluate random bit sequences using NIST SP 800-22. We change the number of extracted LSBs and generate random bit sequences. Figure 7 shows the number of passed NIST tests as a function of the number of extracted LSBs. Nine 1-Gbit sequences of random bits are used for each NIST test and the median of the nine test results is plotted in Fig. 7, with error bars indicating the maximum and minimum values. On the vertical axis, 15 indicates that all of the NIST tests were passed. As can be seen, we succeeded in generating random bits that can pass all 15 statistical tests of randomness from one to eight LSBs, except for the case of three LSBs. In the case of three LSBs, all the 15 tests are passed twice and the rest of seven trials are failed. However, the median of the nine trials is 14 and we consider that the test results are close to be passed.

 

Fig. 7 Number of passed NIST SP 800-22 tests as function of extracted LSBs in post-processing for random number generation. On the vertical axis, 15 indicates that all of the NIST tests were passed. Nine 1-Gbit sequences of random bits are used for each NIST test, and the median of the nine test results is plotted with error bars indicating the maximum and minimum values.

Download Full Size | PDF

We also use TestU01 tests for the evaluation of the generated random bit sequences. Figure 8 shows the results of the five packages of TestU01 (Rabbit, Alphabit, SmallCrush, Crush, and BigCrush) while the number of the extracted LSBs is changed. Nine and five sequences of random bits were used for each test in Figs. 8(a) and 8(b), respectively, and the median of the test results is plotted with error bars indicating the maximum and minimum values. In Fig. 8(a), on the vertical axis, 38, 17, and 15 indicate that all of the Rabbit, Alphabit, and SmallCrush tests, respectively, were passed. The random bit sequences generated from all of the LSBs passed the Rabbit, Alphabit, and SmallCrush tests. In Fig. 8(b), on the vertical axis, 144 and 160 indicate that all the Crush and BigCrush tests, respectively, were passed. The random bit sequences generated from five to eight LSBs also passed both Crush and BigCrush tests. Therefore, we confirm that the randomness of the large amount of generated bit sequences (up to 410 Gbits for BigCrush) is statistically certified by TestU01 tests.

 

Fig. 8 Number of passed TestU01 tests as function of extracted LSBs in post-processing for random number generation. (a) Rabbit (black solid line with circles), Alphabit (blue dashed line with squares), and SmallCrush (red dotted line with diamonds) tests. On the vertical axis, 38, 17, and 15 indicate that all of the Rabbit, Alphabit, and SmallCrush tests, respectively, were passed. Nine sequences of 1.05-Mbit, 33.6-Mbit, and 1.64-Gbit data were used for the Rabbit, Alphabit, and SmallCrush tests, respectively, and the median of the nine test results is plotted with error bars indicating the maximum and minimum values. (b) Crush (black solid line with circles) and BigCrush (blue dashed line with squares) tests. On the vertical axis, 144 and 160 indicate that all of the Crush and BigCrush tests, respectively, were passed. Five sequences of 41-Gbit and 410-Gbit data were used for the Crush and BigCrush tests, respectively, and the median of the five test results is plotted with error bars indicating the maximum and minimum values.

Download Full Size | PDF

In order to investigate noise effects on the generated random bits, we generate random bit sequences without chaotic signals from the PIC, but with the same post-processing as in Fig. 5. We evaluate the randomness of the generated bit sequences using NIST SP 800-22. Although the bit sequences generated from one LSB can pass all of the statistical tests, the bit sequences generated from two to eight LSBs cannot pass the tests. This result indicates that the noise effect is minimal, and it is necessary to use a chaotic signal of the PIC as an entropy source to obtain trusted random bits.

4. Speed of real-time random number generation

We measure the real-time generation rate for the RNG from the FPGA board to the memory in the computer. We use ten 1-Gbit random bit sequences for the measurement. Figure 9 shows the average real-time generation rate for the RNG while the number of extracted LSBs is changed. The real-time generation rate increases linearly with increasing number of LSBs, with the exception of eight LSBs. The generation rate saturates at seven LSBs, as shown in Fig. 9. A maximum real-time generation rate of 21.1 Gb/s is achieved in this experiment.

 

Fig. 9 Real-time generation rate for RNG as function of number of extracted LSBs in post-processing using bit-order-reversal method.

Download Full Size | PDF

We consider that the generation speed is mainly limited by the sampling rate of the ADC. This rate is estimated by the sampling rate of the ADC (3.6 GS/s) multiplied by the number of LSBs. However, for 8 LSBs, the transmission rate of PCI Express Gen2 x8 could be the main limitation of the generation rate, which is seen as clear saturation of the generation rate in Fig. 9. On the contrary, the bandwidth of the chaotic signals from the PIC is not a bottleneck for the generation rate. The real-time generation rate could be further improved by using a faster ADC, FPGA, and PCI Express bus for post-processing of the RNG and for data transmission to the computer.

Figure 9 shows slight mismatch between the observed generation rate and the rate estimated from the sampling rate of the ADC and the number of LSBs. We found that a few percent of the generated bits is lost for each LSB. This fact may result from the limitation of direct memory access (DMA) transfer to memory buffers on the FPGA board. We consider that this slight loss of bits does not strongly affect the quality of generated random bits.

We consider the theoretical upper limit of the physical entropy rate [45,46]. The information-theoretical upper limit of the entropy rate is described as R = S h_min, where S is the sampling rate of the ADC (3.6 GS/s in our RNG) and h_min is the min-entropy defined as h_min = -log₂(P_max). P_max is the maximum value of the probability density function of the histogram of the original chaotic signal used for RNG. P_max = 0.01096 is obtained from the histogram in Fig. 3(c), and the min-entropy of h_min = 6.51 is obtained. The theoretical upper limit of the entropy rate is thus R = 23.4 Gb/s ( = 3.6 GS/s × 6.51 bit). We confirmed that the maximum generation rate of 21.1 Gb/s in our RNG (see Fig. 9) is below the theoretical upper limit of the entropy rate R.

5. Conclusions

We demonstrated a hardware implementation of a real-time fast physical random number generator with a photonic integrated circuit and a field programmable gate array electronic board. We generated 1-Tbit random bit sequences and evaluated their statistical randomness using NIST SP 800-22 and TestU01. All of the BigCrush tests in TestU01 were passed using 410-Gbit sequences of random bits. We achieved a maximum real-time generation rate of 21.1 Gb/s in binary format, and this data can be directly used in applications involving secret keys in cryptography and random seeds in large-scale numerical simulations.

6 Appendix

6.1 Post-processing using the bit-shift and bit-rotation method

We use another post-processing method, called the bit-shift and bit-rotation method, proposed by Dichtl [44]. Figure 10 shows the post-processing scheme with the bit-shift and bit-rotation method [22,44]. The procedure for generating two difference signals is the same as that shown in Fig. 5. Bit-shift and bit-rotation signals are generated from the first difference signal. For example, a one-bit-shifted signal is generated as follows. Each significant bit is shifted to the left by one bit, i.e., the second MSB becomes the first MSB, the third MSB becomes the second MSB, and so on, and the first MSB is rotated to become the LSB. A two-bit-shifted signal and four-bit-shifted signal are also generated by changing the number of bit shifts. Bit-wise exclusive-OR (XOR) operation is carried out among the first difference signal, one-bit-shifted signal, two-bit-shifted signal, four-bit-shifted signal, and the second difference signal. Some of the LSBs are then extracted and used as random bit sequences.

 

Fig. 10 Schematic diagram of random number generation using bit-shift and bit-rotation method. AD: analog-to-digital, LSB: least significant bit, PIC: photonic integrated circuit.

Download Full Size | PDF

We implement two minor methods by changing the direction of the bit shift and rotation: to the left shift and to the right shift. We compare the performance of these two methods. Note that the original method proposed by Dichtl is based on the left bit-shift and bit-rotation [44]. However, right bit-shift and bit-rotation could be more effective to avoid the correlation between the first MSB and the second MSB, which exists in chaotic laser signals [5].

Figure 11 shows the bias b for the original chaotic waveform and the random bit sequences generated using the bit-shift and bit-rotation method (both left and right bit-shift) with eight LSBs. The bias for the generated random bit sequences is small enough compared with the 3σ criterion for both cases. However, the bias becomes close to the 3σ criterion for a large N for the left bit-shift method, as shown in Fig. 11(a).

 

Fig. 11 Bias (blue curves) for random bit sequences generated by bit-shift and bit-rotation method as function of number of generated bits N. (a) Left bit-shift and bit-rotation method. (b) Right bit-shift and bit-rotation method. The dotted-dashed line indicates the 3σ criterion for the bias of random bits, where $\sigma =1/\left(2\sqrt{N}\right)$.

Download Full Size | PDF

Figure 12 shows the number of passed NIST SP 800-22 tests as a function of the number of extracted LSBs. On the vertical axis of Fig. 12, 15 indicates that all of the NIST tests were passed. For Fig. 12(a), the bit sequences generated from two to seven LSBs can pass all 15 statistical tests of randomness, which is not the case for bit sequences generated from one or eight LSBs. In contrast, for Fig. 12(b), the bit sequences generated from one to eight LSBs can pass all 15 statistical tests. This result indicates that the right bit-shift method is better than the left bit-shift method. We speculate that this fact results from the existence of a correlation among the several MSBs in the chaotic laser signals.

 

Fig. 12 Number of passed NIST SP 800-22 tests as function of extracted LSBs for (a) left bit-shift and bit-rotation method and (b) right bit-shift and bit-rotation method. On the vertical axis, 15 indicates that all of the NIST tests were passed. Nine 1-Gbit sequences of random bits were used for each NIST test, and the median of the nine test results is plotted with error bars indicating the maximum and minimum values.

Download Full Size | PDF

Funding

Grants-in-Aid for Scientific Research from the Japan Society for the Promotion of Science (JSPS KAKENHI Grant Number JP16H03878) and Management Expenses Grants from the Ministry of Education, Culture, Sports, Science and Technology in Japan.

Acknowledgments

The authors thank Morihiro Ohtsuki and Fumihiko Sugawara for technical support for hardware implementation of RNG. The authors thank Susumu Shinohara, Kenichi Arai, and Peter Davis for helpful discussions.

References and links

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]   [PubMed]  

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]   [PubMed]  

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 Photon. 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]   [PubMed]  

7. 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]   [PubMed]  

8. 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]  

9. 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]  

10. 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. Cir. Syst. I 61(3), 888–901 (2014).

11. 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]   [PubMed]  

12. 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]   [PubMed]  

13. 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]   [PubMed]  

14. 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]   [PubMed]  

15. 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]   [PubMed]  

16. 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]   [PubMed]  

17. 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]   [PubMed]  

18. 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]  

19. 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]   [PubMed]  

20. 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]   [PubMed]  

21. 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]   [PubMed]  

22. A. Uchida, Optical Communication with Chaotic Lasers, Applications of Nonlinear Dynamics and Synchronization (Wiley-VCH, 2012).

23. 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]  

24. 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]   [PubMed]  

25. 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]   [PubMed]  

26. 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]  

27. 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]  

28. M. Stipčević and B. M. Rogina, “Quantum random number generator based on photonic emission in semiconductors,” Rev. Sci. Instrum. 78(4), 045104 (2007). [CrossRef]   [PubMed]  

29. 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]  

30. 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]  

31. 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]  

32. 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]   [PubMed]  

33. X.-G. Zhang, Y.-Q. Nie, H. Zhou, H. Liang, X. Ma, J. Zhang, and J.-W. Pan, “Note: Fully integrated 3.2 Gbps quantum random number generator with real-time extraction,” Rev. Sci. Instrum. 87(7), 076102 (2016). [CrossRef]   [PubMed]  

34. K. Yoshimura, J. Muramatsu, P. Davis, T. Harayama, H. Okumura, S. Morikatsu, H. Aida, and A. Uchida, “Secure key distribution using correlated randomness in lasers driven by common random light,” Phys. Rev. Lett. 108(7), 070602 (2012). [CrossRef]   [PubMed]  

35. H. Koizumi, S. Morikatsu, H. Aida, T. Nozawa, I. Kakesu, A. Uchida, K. Yoshimura, J. Muramatsu, and P. Davis, “Information-theoretic secure key distribution based on common random-signal induced synchronization in unidirectionally-coupled cascades of semiconductor lasers,” Opt. Express 21(15), 17869–17893 (2013). [CrossRef]   [PubMed]  

36. T. Honjo, A. Uchida, K. Amano, K. Hirano, H. Someya, H. Okumura, K. Yoshimura, P. Davis, and Y. Tokura, “Differential-phase-shift quantum key distribution experiment using fast physical random bit generator with chaotic semiconductor lasers,” Opt. Express 17(11), 9053–9061 (2009). [CrossRef]   [PubMed]  

37. H. Miyazawa and M. Fushimi, “An implementation of a 5-term GFSR random number generator for parallel computations,” in Proceedings of the International Symposium on Operations Research and Its Applications (ISORA, 2009) 448–452 (2009).

38. D. P. Rosin, D. Rontani, and D. J. Gauthier, “Ultrafast physical generation of random numbers using hybrid Boolean networks,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 87(4), 040902 (2013). [CrossRef]   [PubMed]  

39. S. Shinohara, K. Arai, P. Davis, S. Sunada, and T. Harayama are preparing a manuscript to be called “Chaotic laser based physical random bit streaming system with a computer application interface.”

40. 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, Special Publication 800–22, Revision 1a (2010).

41. P. L’Ecuyer and R. Simard, “TestU01: A C library for empirical testing of random number generators,” ACM Trans. Math. Softw. 33(4), 22 (2007). [CrossRef]  

42. A. Karsaklian Dal Bosco, Y. Akizawa, K. Kanno, A. Uchida, T. Harayama, and K. Yoshimura, “Photonic integrated circuits unveil crisis-induced intermittency,” Opt. Express 24(19), 22198–22209 (2016). [CrossRef]   [PubMed]  

43. A. Karsaklian Dal Bosco, S. Ohara, N. Sato, Y. Akizawa, A. Uchida, T. Harayama, and M. Inubushi, “Dynamics versus feedback delay time in photonic integrated circuits: Mapping the short cavity regime,” IEEE Photonics J.in press.

44. M. Dichtl, “Bad and good ways of post-processing biased physical random numbers,” Lect. Notes Comput. Sci. 4593, 137–152 (2007). [CrossRef]  

45. J. D. Hart, Y. Terashima, A. Uchida, G. B. Baumgartner, T. E. Murphy, and R. Roy, “Commentary: Evaluating photonic random number generators,” arXiv:1612.04415v2 (2016).

46. J. Y. Haw, S. M. Assad, A. M. Lance, N. H. Y. Ng, V. Sharma, P. K. Lam, and T. Symul, “Maximization of extractable randomness in a quantum random-number generator,” Phys. Rev. Appl. 3(5), 054004 (2015). [CrossRef]