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Using two mutually coupled semiconductor lasers (MC-SLs) outputs as chaotic entropy sources, a scheme for generating Tbits/s ultra-fast physical random bit (PRB) is demonstrated and analyzed experimentally. Firstly, two entropy sources originating from two chaotic outputs of MC-SLs are obtained in parallel. Secondly, by adopting multiple optimized post-processing methods, two PRB streams with the generation rate of 0.56 Tbits/s are extracted from the two entropy sources and their randomness are verified by using NIST Special Publication 800-22 statistical tests. Through merging the two sets of 0.56 Tbits/s PRB streams by an interleaving operation, a third set of 1.12 Tbits/s PRB stream, which meets all the quality criteria of NIST statistical tests, can be further acquired. Finally, after additionally taking into account the restriction of the min-entropy, the generation rate of two sets of PRB stream from the two entropy sources can still attain 0.48 Tbits/s, and then a third set of merging PRB stream is 0.96 Tbits/s. Moreover, for the sequence length of the order of 10 Gbits, the statistical bias and serial correlation coefficient of three sets of PRB streams are also analyzed.
© 2015 Optical Society of America
1. Introduction
Compared with pseudo-random number, nondeterministic physical random number is more promising for information security application due to its non-predictability, non-reproducibility and non-periodicity [1]. Usually, the entropy sources of physical random number can be obtained by some stochastic physical processes such as thermal noise in resistors [2], frequency jitter of oscillators [3], spatial resolution of single photon emission [4, 5 ], arrival time of single photon [6, 7 ], phase noise of lasers [8, 9 ], and amplified spontaneous emissions from super-luminescent diodes [10, 11 ].
In recent years, an intriguing possibility for acquiring high-speed physical random number has continued to receive special attention, where the chaotic outputs of semiconductor lasers (SLs) under external disturbances are used as the entropy sources [12–29 ]. In 2008, Uchida et al. demonstrated for the first time a fast physical random bit (PRB) stream based on two chaotic lasers, where the periodicity in the chaotic signal at the photon round trip frequency is eliminated by sampling the fluctuations of two independent lasers with 1-bit analog-to-digital converters (ADCs) [12] and the generation rate of PRB is 1.7 Gbits/s. Since then, based on the SLs chaotic system, some novel schemes for generating PRBs have been proposed successively, and the generation rate of PRBs has been increasing very rapidly. In 2009, Reidler et al. reported 12.5 Gbits/s PRBs generation with 5 least significant bits (LSBs) sampled at 2.5 GHz [13]. In 2010, based on the bandwidth-enhanced chaos in SLs, Hirano et al. demonstrated a scheme for generating ultrafast PRBs at 75 Gbits /s with 6 LSBs sampled at 12.5 GHz [14]. In the same year, through introducing high-order differential method, Kanter et al. reported 300 Gbits/s PRB generation [15]. In 2012, after adopting a novel post-processing method named as the bit-order reversal method, Akizawa et al. demonstrated the generation of 400 Gbits/s PRBs with 8 LSBs sampled at 50 GHz [16]. Some other schemes for PRBs generated at several hundreds of Gbits/s have also been proposed [17–20 ]. Except modifying the post-processing method, some technologies for improving the quality of the entropy source have also been proposed. For example, via optical injection, the bandwidth of chaotic entropy source can be enhanced for obtaining high-speed PRBs [14, 16, 28 ]. Through adopting photonic integrated semiconductor lasers as the entropy sources, the PRB generator possesses the advantages of small size, high performance and especially robustness [17, 23, 24 ]. By taking mutually coupled SLs (MC-SLs), we have preliminarily demonstrated an entropy source which has a potential to generate multi-channel PRBs with dozens Gbits/s [29].
Very recently, researchers indicate that Tbits/s PRBs can be obtained based on such chaotic laser entropy sources. In 2014, Li et al. reported a scheme of ultrafast random bit generation based on the high-order differential method. In this scheme, the initial data, which is originated from chaotic signal sampled at 40 GS/s in a 8-bit resolution oscilloscope, is transformed into a 64-bit integer type, and finally 55 LSBs are successfully extracted per sample to obtain a ultrafast random bit at 2.2 Tbits/s rate ( = 40 GS/s × 55 bits) [27]. However, due to the use of high-order differential method for post processing, the relevance between the generated bit streams and the chaotic entropy source is partly reduced, and the rate of obtained random bits violates the limit set by information theory [28]. After taking information-theoretic assessment into account, the actual generation rate of PRB is 160 Gbits/s. In 2015, Sakuraba reported the generation of 1.2 Tbits/s ( = 100 GS/s × 6 bits × 2 data) by using bandwidth-enhanced chaos in three-cascaded semiconductor lasers [28], where the bandwidth of the chaotic entropy source is enhanced up to 35.2 GHz via optical injection and the chaotic waveforms are sampled at a rate of 100 GS/s by a high-speed digital oscilloscope (33 GHz bandwidth, 100 GS/s, 8-bit vertical resolution).
In this work, by combining our previous research experience on MC-LDs system with recently reported post-processing schemes proposed by other scholars, after continuous exploration, trial and improvement, we experimentally demonstrate three sets of Tbits/s PRBs generation based on MC-LDs chaotic entropy source. The rates for the three sets of PRB streams are 0.56 Tbits/s, 0.56 Tbits/s, and 1.12 Tbits/s, respectively, and their randomness are verified by using NIST Special Publication 800-22 statistical tests. After additionally taking information-theoretic assessment into account, the actual generation rates for the three sets of PRB streams can still attain 0.48 Tbits/s, 0.48 Tbits/s, and 0.96 Tbits/s, respectively. Although the highest generation rate (0.96 Tbits/s) in this work is slightly lower than 1.2 Tbits/s reported in [28] due to a lower sample rate of the digital oscilloscope (80 GS/s) used in this experiment than that used in [28] (100 GS/s), this presented scheme has a unique potential to realize multi-set parallel generation of PRB streams if more than two SLs are used in such MC-SLs system.
2. Experimental setup
Figure 1 shows our experimental setup for multi-channel PRBs. The entropy source is originated from a MC-SLs chaotic system, in which two similar 1550 nm InGaAsP/InP distributed feedback Bragg SLs (SL 1 and SL 2) are mutually coupled through an optical fiber link including a tunable attenuator (TA) and a polarization controller (PC), and the coupling optical power is monitored by a power meter (PM, Thorlabs PM100D). Each SL is driven by a high stability and low noise laser diode controller (ILX-Lightwave, LDC-3724C). By selecting suitable bias current, operating temperature and coupling strength, two broadband chaotic signals can be simultaneously generated from SL 1 and SL 2 respectively. The chaotic signal output from each SL is divided into two parts by an optical coupler (OC) respectively. One part is fed into the test equipment module, which is composed of an optical spectrum analyzer (OSA, ANDO AQ6317C), an electrical spectrum analyzer (ESA, Agilent E4407B). The other part is fed into the sampling and post-processing module to generate PRB. Here, we take the chaotic signal output from SL 1 as an example to illustrate how to conduct sampling and post-processing. Firstly, via a 50/50 OC, the chaotic signal is divided into two parts. One part is converted to electric signal through a photo-detector (PD, New Focus 1544-B) and is sampled by a 8 bit analog-to-digital converter (ADC) with the sample rate of 80 GS/s in an oscilloscope (Agilent X91604A), and then the order of each 8 sampling bits is reversed [20]. The other part is time-delayed by a fiber delay line (FDL), converted into electric signal, sampled by another 8 bit ADC with 80 GS/s rate, and then combined together with the order-reversed binary stream by a bitwise exclusive-OR (XOR) operation. Furthermore, the m least significant bits (LSBs) of each 8 bits are extracted to form the final binary stream which is output from channel 1 (CH 1). The same process is implemented in parallel for the generation of the final binary stream in channel 2 (CH 2).Furthermore, a third set of bit stream with double rate can be acquired in channel 3 (CH 3).
Fig. 1 Experimental setup for generating multi-channel high-speed PRNs. SL: semiconductor laser, CC: current controller; TC: temperature controller; TA: tunable attenuator; PC: polarization controller; OC: optical coupler; OI: optical isolator; PD: photo-detector; FDL: fiber delay line; OSA: optical spectrum analyzer; ESA: electrical spectrum analyzer; PM: power meter; ADC: analog-to-digital converter; XOR: logical exclusive-OR; CH: channel; m-LSBs: the m least significant bits.
3. Results and discussion
3.1 Dynamical property of MC-SLs chaotic system
The quality of the generated bit stream is generally depended on the characteristics of entropy source, and then it is necessary to analyze the dynamical property of the chaotic entropy source. During the whole experiment, the currents of SL 1 (SL 2) is fixed at 12.1 mA (13.2 mA), which is about 1.48 (1.55) times of its threshold current. The temperatures are stabilized at 19.3 °C for SL 1 and 20.6 °C for SL 2, respectively. Under these circumstances, the output power for free-running SL 1 is the same as that for free-running SL 2. Via a fiber link, a mutually coupled system can be established. In this work, the coupling delay time τ is fixed at about 29.45 ns and the coupling strength η can be varied via a tunable attenuator (TA). The coupling strength η is defined as follows:
where Pout is the output power of free-running SL 1 (or SL 2), and Pc is the coupling optical power which is estimated from the measured power by PM in Fig. 1. Through varying the coupling strength η, multiple nonlinear dynamical states can be observed. For η is varied within the region of (−28 dB, −10 dB), the outputs of SL 1 and SL 2 are both chaotic signals, and the optical spectra, radio-frequency (RF) spectra, and chaotic bandwidth evolution with coupling strength η are shown in Fig. 2 . As shown in Fig. 2(a), with the increase of η, the optical spectra of SL 1 and SL 2 are obviously broadened, and the RF spectra possess similar evolution tendency (as shown in Fig. 2(b)). Furthermore, the bandwidth evolutions of the two chaotic outputs are given in Fig. 2(c). With the increase of η, the bandwidths of chaotic outputs from SL 1 and SL 2 increase firstly and then decrease. For η = −12 dB, the bandwidths of chaotic output from SL 1 and SL2 reach their maximum, which are 10.96 GHz for SL 1 and 10.40 GHz for SL 2, respectively.Fig. 2 Mappings of optical spectra (first row), RF spectra (second row), and chaotic bandwidth (third row) evolution with coupling strength η varied from −28 dB to −10 dB. Left column corresponds to SL 1, and right column is corresponds to SL 2. Different colors represent different amplitude values of optical spectra (a1, a2) and RF spectra (b1, b2).
The time delay (TD) signature of chaotic signal is harmful for generating PRBs [12, 13, 15 ]. Hence, it is meaningful to analyze the TD signature of chaotic signal generated by the MC–SLs system. In this work, self-correlation (SF) method is used to analyze TD signature of MC–SLs system. For a delay-differential system, the SF function can be defined as [13]:
where P(t) and Δt represent chaotic time series and the time shift, respectively. In Fig. 3 , the self-correlation curve evolutions of chaotic outputs from SL 1 (a) and SL 2 (b) and the corresponding variation curve of amplitude ρ (ρ is the maximum of the SF peak in the time shift window 58 ns < Δt < 60 ns) are drawn to show the impact of different coupling strength on the TD signature. As shown in Fig. 3 (a1 and b1), peaks of the TD signature appear at Δt ≈2τ = 58.9 ns. In the range of −23 dB > η > −28 dB, TD signature seems inconspicuous. As shown in Fig. 3 (a2 and b2), the minimal TD signature of SL 1 is 0.072 under η = −24 dB meanwhile the minimal TD signature of SL 2 is 0.074 under η = −24 dB. On the other hand, the maximal TD signature of SL 1 is 0.53 under η = −10 dB, and the maximal TD signature of SL 2 is 0.47 under η = −10 dB. These results indicate that the original chaos signals of MC-SLs always contain TD signatures within the investigated range of η. Therefore, the post-processing operation for acquiring PRBs is necessary to reduce the TD signature effect.Fig. 3 The Maps (a1, b1) of self-correlation curves evolution and the variation curve (a2, b2) of amplitude ρ with coupling strength η varied from −28 dB to −10 dB. Different colors represent different SF function values. The amplitude ρ is the maximum of the SF peak in the time shift window 58 ns < Δt < 60 ns. Left column is for SL 1, and right column is for SL 2.
3.2 PRBs generation and performance analysis
For a multi-bit extraction method, the detection window size and the off-set of the temporal waveforms obviously affect the bias of the random bit streams. In this work, after referring to [18] and combining with our experience, we fixed the detection window size by setting the vertical-resolution of OSA at 37.5 mv/div, and the off-set value is set at −30 mv. Under these cases, the number of off-scale points is effectively restrained and the histograms of samples are shown as a Gaussian-like distribution. The evolution mappings of the amplitude probability density distributions (APDDs) with coupling strength η for the chaotic signals output from SL 1 and SL2 are displayed in Fig. 4 . For the chaotic signal output from SL 1 (shown as in Fig. 4(a)), under −22 dB < η < −13 dB, the distributions of APDDs are relatively flat and the peaks of APDDs are around 1.2 × 10−2, and such distributions are beneficial for PRB generation [18]. For the chaotic signal generated by SL 2 (as shown in Fig. 4(b)), a similar evolution tendency is appeared but the range of η for obtaining favorable APDD for PRB generation is from −23 dB to −16 dB approximately.
Fig. 4 Evolution mappings of the amplitude probability density distributions (APPDs) of sampled temporal waveforms for η varied from −28 dB to −10 dB, where (a) and (b) are for SL 1 and SL 2, respectively. Different colors represent different probability values.
Next, we will introduce the post-processing scheme used in this work. Figure 5 illustrates the detail of post-processing for multi-channel PRB generation. Here, the post-processing implemented on 8 bits sampled binary stream in CH 1 is taken as an example. As shown in Fig. 5, the original sampled binary stream of SL 1 is firstly bitwise reversed, and then the bitwise XOR operation is used between the reversed bit stream and the delayed bit stream to generate a set of XOR data. Next, the m least significant bits (m LSBs) of 8 bits are selected to form an m-bit Boolean stream, which is examined by the NIST Special Publication 800-22. The same process has been implemented to 8 bits sampled binary stream in CH 2. For the case that m-bit Boolean stream in both CH 1 and CH 2 can pass through all 15 terms of the NIST tests, the result of random bit streams output from CH 1 and CH 2 can reach m × 80 Gbits/s rate. Furthermore, by using the bitwise interleaving operation of binary streams between CH 1 and CH 2, a bit stream with a rate as high as 2 × m × 80 Gbits/s can be obtained in CH 3. And the bit streams are also judged by the NIST tests.
Fig. 5 Post-processing for PRB generation. The dotted box is for the post-processing in CH 1 which is taken as an example, and the post-processing in CH 2 is the same as that in CH 1.
Here, we use the NIST Special Publication 800-22 tests to evaluate the randomness of the bit streams generated by using the multi-bit extraction method in three channels. Figure 6 gives the number of failed NIST tests varying with the coupling strength under m LSBs selections (m = 8, 7, and 6). For the 8 LSBs (as shown in Fig. 6(a)), none of bit stream in CH 1, CH 2 or CH 3 can pass all the NIST standard tests. The reason may be related to the high sample rate of ADCs, which is set at 80 GHz and is about eight times of the maximum signal bandwidth (see Fig. 2). Under this case, the adjacent sampling points are not independent and have noticeable correlations between each other, which cannot be eliminated completely via the bit-order-reversal and XOR operations. Therefore, a relatively lower order m LSBs extraction method may be helpful to improve the randomness of bit stream. As shown in Fig. 6(b), for 7 LSBs, the random bit streams in three channels can pass all 15 terms of the NIST tests for −15 dB ≥ η ≥ −22 dB. For η < −22 dB or η > −15 dB, the results of tests become worse and the reason may be mainly associated with the deterioration of chaotic entropy source. For η < −22 dB, although the TD signature is not clear (as shown in Fig. 3), the bandwidth of the chaotic signal of SL 1 (or SL 2) are limited and located at a range of 6 GHz to 8 GHz, which is too small relative to the 80 GHz sample frequency of ADC. For η > −15 dB, the bandwidths of chaotic signals of SL 1 and SL 2 are enough large (as shown in Fig. 2(c)), but the TD signature is obvious, which means that the periodic feature of chaos signal is strong and the randomness property of chaos signal are degraded seriously. Interestingly, we have noted that the results of tests are roughly in accordance with the evolution behaviors in Fig. 4, i. e. the region of the coupling strength required for generating PRBs can be roughly estimated by the region of the coupling strength required for obtaining 8-bit streams with favorable APPDs. Furthermore, for 6 LSBs extraction (shown in Fig. 6(c)), the range of η within which the bit streams can pass the NIST tests is wider.
Fig. 6 Mappings of number of failed NIST tests for CH 1, CH 2 and CH 3 under η varied within (−28 dB, −10 dB). (a) 8 LSBs, (b) 7 LSBs, and (c) 6 LSBs.
Above results demonstrate that, for 7 LSBs, the random bit streams in three channels can pass all 15 terms of the NIST tests under −15 dB ≥ η ≥ −22 dB, and the highest generated rate of random bit stream whose randomness are verified by using NIST Special Publication 800-22 statistical tests can attain 1.12 Tbits/s. However, from security requirements for cryptographic applications, the generation rate of PRB should be restricted by the min-entropy of the chaotic entropy. The min-entropy, which is a measure of the maximal unpredictable information in the worst-case scenario, provides a conservative and strict assessment for the maximum value of LSB extraction from an entropy resource. As a result, the highest generated rate of PRB for one channel cannot exceed the bounds determined by the min-entropy. Here, the min-entropy is defined as , where pi is the probability of the ith outcome and is the probability of the most likely event [27]. Figure 7 depicts the calculated min-entropy for chaotic entropy output from SL 1 and SL 2 under different η. As shown in this diagram, the maximum value of min-entropy is 6.19 for SL 1 and 6.18 for SL 2, which means that no more than 6 out of 8 bits in each sample can be extracted for PRB generation theoretically. For −15 dB ≥ η ≥ −22 dB, the number of LSBs selection can be set at the maximum value 6 in CH 1 and CH 2. Therefore, based on information-theoretic determination, the highest rate of random bit streams output from CH 1 (or CH 2) will be 0.48 Tbits/s (6 bit × 80 GS/s), and the highest rate of random bit streams output from CH 3 is 0.96 Tbits/s.
Generally, for a hardware random bits generator, two basic characteristics of bit stream should be considered. One is the statistical bias b (defined as , where p(1) stands for probability of “1”), and the other is the serial autocorrelation coefficients Ck for generated bits streams. Ck is defined as [27]
where ai takes the value of “1” or “0”, k is a lag, and the averaging (denoted as ) is performed over index i.Figure 8 shows the variation of the statistical bias b with the sequence length N (a) and the serial autocorrelation coefficient Ck for N = 10 Gbits (b) under η = 20 dB. Here, standard deviations σB and σC are determined by for bias b and (N = 10 Gbits) for Ck, respectively. From Fig. 8 (a1) and (a2), it can be seen that, with the increase of N, the bias deviates from the criteria of 3σB. This may be due to the non-ideal distribution of temporal waveforms, which is originated from the statistical property of entropy sources and the nonlinearity of ADC sampling process. In addition, the correlations between the samples, which are not completely inhibited during the post-processing shown in Fig. 5, may also affect the bias level. From Fig. 8 (b1) and (b2), it can be seen that a large short-timescale correction above 3σC emergences within k0 = 3 for CH 1 and at k0 = 6 for CH 2, which is introduced by an over-sampling scheme.
Fig. 8 (a) Variation of statistical bias b with sequence length N, and (b) first 200 serial autocorrelation coefficients for the binary sequence of 10 Gbits under η = 20 dB. Left column is for CH 1, and right column is for CH 2.
In order to decrease such short-timescale correlation, we adopt an optimized post-processing method proposed in [11], which is given in Fig. 9 . The last 6 bits of the 8-bit XOR sequence in Fig. 5 form the raw random sequence of {bi}. Sequence {Bi} is produced by applying the von Neumann unbiasing method to the 6 bits of the 8-bit raw sequence, and satisfies the result that no correlation exists between Bi. Bi + 1, …, Bi + m0-1, where m0>k0. Blocks in {Bi} are repeated s times. The final random bit sequence {bi’} is obtained by the XOR process between {bi} and {Bi}. Here, we set m0 = 10 for CH 1 and m0 = 16 for CH 2 after considering k0 = 3 for CH 1 and k0 = 6 for CH 2. Figure 10 shows the results of statistical bias b and serial autocorrelation coefficient Ck for final random bit streams in three channels under the case of η = 20 dB. Obviously, the statistical bias b for each channel always stays below the 3σB-criterion and converges to zero asymptotically with N increasing to large numbers, and the first 200 autocorrelation coefficients for a 10 Gbits bit sequence per channel are within the range bounded by ± 3σC-criterion. Therefore, final random bit streams in three channels with a sequence length up to 10 Gbits well satisfy the criteria of three-standard-deviations for bias and serial autocorrelation coefficients. Furthermore, the randomness of final random bit streams are evaluated by the NIST statistical tests, and the results show that three bit streams can pass all of the tests. Table 1 gives the results of NIST tests for CH 3 under η = 20 dB. Hence, a set of 0.96 Tbits/s PRB stream can be acquired under strict criteria of information-theoretic assessment.
Fig. 9 Further optimization process for obtaining final random bit stream. {bi} is the 6 LSBs random sequence, {Bi} is extracted from raw LSBs sequence by using the von Neumann unbiasing method.
Fig. 10 After adopting short-timescale correlations eliminated scheme, under the case for η = 20 dB, (a) the bias b as a function of the number of generated bits N; (b) the first 200 serial autocorrelation coefficients for the binary sequence of 10 Gbits length.
Table 1. Results of NIST tests for final random bit streams of CH 3 under the case for η = 20 dB. 1000 samples of 1 Mbit data are used for evaluation and the significance level a = 0.01. The P value (uniformity of p values) should be larger than 0.0001 and the proportion should be in the range of 0.99 ± 0.0094392. For tests which produce multiple P values and proportions, the worst case is shown.
4. Conclusions
In summary, we experimentally demonstrated a scheme of Tbits/s ultra-fast PRBs generation based on mutually coupled semiconductor laser (MC-SL) chaotic entropy source. Firstly, the dynamical property of this chaotic system is investigated. By selecting suitable coupling strength, both the two MC-SLs can be driven into chaotic state. The optical spectra, RF spectra and chaotic bandwidth evolution are experimentally examined, and the TD signatures of chaotic signals are analyzed by using SF method for η varied within the region of (−28 dB, −10 dB). Next, the chaotic outputs from the two MC-SLs are used as two physical entropy sources and then are transformed into binary streams by 8-bit ADCs with 80 GS/s rate. Time delay operation, bit-order reversal method, XOR operation, m LSBs extraction method are used in combination to eliminate the correlation among sampling points and optimize the statistical distribution of the generated bit streams. Two sets of 0.56 Tbits/s PRB streams and a third set of 1.12 Tbits/s PRB stream, which can pass through all the NIST tests, are obtained for η varied within the region of (−22 dB, −15 dB). Finally, based on strictly information-theoretic assessment, the min-entropy, the statistical bias and the serial correlation coefficients are further used to estimate the performances of the random bit sequences, and the results demonstrate that, two sets of 0.48 Tbits/s PRB streams and a third set of 0.96 Tbits/s PRB stream, which can pass through all the NIST tests and satisfy the strict criteria of mainstream information-theoretic assessment, can be generated under suitable operating parameters. We hope that this work can provide a helpful technical attempt for the generation of Tbits/s PRBs.
Acknowledgments
This work was supported by the National Natural Science Foundation of China under Grant 61178011, Grant 61275116, Grant 61475127, Grant 61575163, Grant 11474233, and Grant 11204248, and the Fundamental Research Funds for the Central Universities under Grant XDJK2014C079, Grant XDJK2014C120, Grant XDJK2013B037 and Grant XDJK2014C168.
References and links
1. D. Eastlake, J. Schiller, and S. Crocker, “Randomness requirements for security,” RFC4086, [Online]. Available: http://tools.ietf.org/html/rfc4086 (2005).
2. W. T. Holman, J. A. Connelly, and A. B. Dowlatabadi, “An integrated analog/digital random noise source,” IEEE Trans. Circuits Syst. I 44(6), 521–528 (1997). [CrossRef]
3. M. Bucci, L. Germani, R. Luzzi, A. Trifiletti, and M. Varanonuovo, “A high-speed oscillator-based truly random number source for cryptographic applications on a smart card IC,” IEEE Trans. Comput. 52(4), 403–409 (2003). [CrossRef]
4. A. Stefanov, N. Gisin, O. Guinnard, L. Guinnard, and H. Zbinden, “Optical quantum random number generator,” J. Mod. Opt. 47, 595–598 (2000).
5. K. Svozil, “Three criteria for quantum random-number generators based on beam splitters,” Phys. Rev. A 79(5), 054306 (2009). [CrossRef]
6. H. Q. Ma, Y. Xie, and L. A. Wu, “Random number generation based on the time of arrival of single photons,” Appl. Opt. 44(36), 7760–7763 (2005). [CrossRef] [PubMed]
7. M. Wahl, M. Leifgen, M. Berlin, T. Röhlicke, H.-J. Rahn, and O. Benson, “An ultrafast quantum random number generator with provably bounded output bias based on photon arrival time measurements,” Appl. Phys. Lett. 98(17), 171105 (2011). [CrossRef]
8. H. Guo, W. Tang, Y. Liu, and W. Wei, “Truly random number generation based on measurement of phase noise of a laser,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 81(5), 051137 (2010). [CrossRef] [PubMed]
9. B. Qi, Y. M. Chi, H. K. Lo, and L. Qian, “High-speed quantum random number generation by measuring phase noise of a single-mode laser,” Opt. Lett. 35(3), 312–314 (2010). [CrossRef] [PubMed]
10. 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]
11. Y. Liu, M. Y. Zhu, B. Luo, J. W. Zhang, and H. Guo, “Implementation of 1.6 Tb s-1 truly random number generation based on a super-luminescent emitting diode,” Laser Phys. Lett. 10(4), 045001 (2013). [CrossRef]
12. 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]
13. 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]
14. 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]
15. 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]
16. 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]
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. 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]
19. 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]
20. 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]
21. N. Oliver, M. C. Soriano, D. W. Sukow, and I. Fischer, “Dynamics of a semiconductor laser with polarization-rotated feedback and its utilization for random bit generation,” Opt. Lett. 36(23), 4632–4634 (2011). [CrossRef] [PubMed]
22. X. Z. Li and S. C. Chan, “Random bit generation using an optically injected semiconductor laser in chaos with oversampling,” Opt. Lett. 37(11), 2163–2165 (2012). [CrossRef] [PubMed]
23. S. Sunada, T. Harayama, P. Davis, K. Tsuzuki, K. Arai, K. Yoshimura, and A. Uchida, “Noise amplification by chaotic dynamics in a delayed feedback laser system and its application to nondeterministic random bit generation,” Chaos 22(4), 047513 (2012). [CrossRef] [PubMed]
24. 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]
25. P. Li, Y. C. Wang, and J. Z. Zhang, “All-optical fast random number generator,” Opt. Express 18(19), 20360–20369 (2010). [CrossRef] [PubMed]
26. J. Wang, J. Liang, P. Li, L. Yang, and Y. Wang, “All-optical random number generation using highly nonlinear fibers by numerical simulation,” Opt. Commun. 321, 1–5 (2014). [CrossRef]
27. 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]
28. 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]
29. X. Tang, Z. M. Wu, J. G. Wu, T. Deng, L. Fan, Z. Q. Zhong, J. J. Chen, and G. Q. Xia, “Generation of multi-channel high-speed physical random numbers originated from two chaotic signals of mutually coupled semiconductor lasers,” Laser Phys. Lett. 12(1), 015003 (2015). [CrossRef]
