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We propose and theoretically demonstrate an all-optical method for directly generating all-optical random numbers from pulse amplitude chaos produced by a mode-locked fiber ring laser. Under an appropriate pump intensity, the mode-locked laser can experience a quasi-periodic route to chaos. Such a chaos consists of a stream of pulses with a fixed repetition frequency but random intensities. In this method, we do not require sampling procedure and external triggered clocks but directly quantize the chaotic pulses stream into random number sequence via an all-optical flip-flop. Moreover, our simulation results show that the pulse amplitude chaos has no periodicity and possesses a highly symmetric distribution of amplitude. Thus, in theory, the obtained random number sequence without post-processing has a high-quality randomness verified by industry-standard statistical tests.
©2012 Optical Society of America
1. Introduction
Random number generators (RNGs) are essential components with a variety of applications from commerce to science, such as lottery games, cryptography and Monte-Carlo calculations. For the majority of these applications, random numbers are mostly generated by means of classical computer algorithms. Although this kind of methods possess the advantage of being fast and easy to implement, they are fully deterministic and exhibit a finite period, thus called as pseudorandom number generators (PRNGs). In a security system, the adoption of PRNGs can lead to catastrophic results [1]. John von Neumann once famously said, “Anyone who considers arithmetical methods of producing random digits is, of course, in a state of sin.” [2]
The other type of generators, called as physical random number generators or true random number generators (TRNGs), generate nondeterministic random numbers from stochastic physical phenomena, including stochastic noise [3,4], radioactive decay [5], frequency jitter of electronic oscillator [6] chaotic circuits [7–10] and chaotic lasers [11–18]. Among them, chaotic laser is the most attractive entropy source for high-speed true random number generation in recent years, due to its high bandwidth and large amplitude. Since Uchida et al. firstly realized random number generation by using chaotic laser in the experiment, there have been a large number of TRNGs based on chaotic lasers reported which utilize optoelectronic [11–16] or all-optical techniques [17,18]. All of them employ the semiconductor laser with optical feedback to produce chaotic signal. This kind of chaos is a continuous chaotic intensity signal and has an inherent periodicity associated with the external cavity. To achieve high-quality random bit sequence generation in these TRNGs, they must perform the following procedures:
However, in the transformation from a continuous analog signal to a discrete digital signal, the sampling aperture jitter may greatly deteriorate the conversion accuracy and signal to noise ratio (SNR) [19]. Moreover, the existence of the external clock and the post-process procedure will significantly increase the complexity of RNG.
In this paper, we theoretically present a direct method for all-optical random number generation using the discrete pulse amplitude chaos in a mode-locked fiber ring laser (MLFRL) rather than analog chaotic intensity signals from semiconductor lasers. The principal advantages of this method are listed as bellows. 1) The pulse amplitude chaos is a kind of discrete signal consisting of a train of chaotic pulses with a fixed repetition frequency but random intensities. Therefore, we can directly quantize them into random bit sequences without requiring the sampling procedure and the external clock. 2) The pulse amplitude chaos has no periodicity and possesses a highly symmetric distribution of amplitude. Thus, in theory, we do not require post-processing such as XOR operation and high-order derivatives algorithm and the generated random bit sequences therefore can successfully pass the standard statistical tests for randomness [20,21]. 3) The proposed RNG does all signal processing in the optical domain and thus can be compatible with the optical communication networks directly with no need of any external modulators.
2. Principle and Simulation
Figure 1 is the schematic diagram of the proposed all-optical RNG which consists of a mode-locked fiber ring laser (MLFRL) and a distributed-feedback laser diode (DFB). The pulse amplitude chaos generated by the MLFRL is split into two identical chaotic pulse trains by a 3-dB coupler (3 dB). The power of the pulse trains can be adjusted by the erbium-doped fiber amplifiers (EDFA) and attenuators (Att.). One of them is injected into the right-hand side of the DFB via a length of fiber delay line (FDL), while the other is injected into the left-hand side of the DFB, combined with a continuous-wave light (CW) by a wavelength-division-multiplexing coupler (WDM). Here, the DFB acts as an all-optical ñip-ñop (AOFF), which plays the role of quantizing the pulse amplitude chaos in the whole system. Finally, with a circulator and an optical band-pass filter (BPF), we can separate the light of the DFB laser from the injected light and visualize the random bit sequence on the oscilloscope (OSC). Specific simulation procedures are described in the followings.
Fig. 1 Schematic diagram of the all-optical RNG based on the pulse amplitude chaos in a mode-locked fiber ring laser (MLFRL). EDF, erbium-doped ðber; PC1 and PC2, two polarization controllers; PDI, polarization-dependent isolator; OC, optical coupler; WDM, wavelength-division-multiplexing coupler; Pump, pump light; 3 dB, 3-dB coupler; EDFA, erbium-doped fiber amplifier; Att., optical attenuator; FDL, ðber delay line; CW, continuous-wave light; DFB, distributed-feedback laser diode; BPF, optical bandpass ðlter; OSC, oscilloscope.
2.1 Pulse amplitude chaos generation and its characteristics
As the stochastic source of the RNG, pulse amplitude chaos is generated by a MLFRL as illustrated in Fig. 1. The laser is a ring laser configuration using the nonlinear polarization rotation (NPR) technique. The cavity is composed of a polarization-dependent isolator (PDI) and two polarization controllers (PC1 and PC2) in a single mode fiber (SMF) and an erbium-doped fiber (EDF), which is pumped via a wavelength-division multiplexed coupler (WDM) at 1480 nm and provides gain to the cavity. The mode-locked pulse stream is finally coupled out through an optical coupler (OC).
Pulse amplitude chaos is an intrinsic feature of NPR MLFRLs, which has been theoretically and experimentally demonstrated by Zhao et al. [22, 23] and our group [24], respectively. Herein, we focus on analyzing the characteristics of the pulse amplitude chaos and confirm that it can be an ideal stochastic source for random number generation. The pulse amplitude chaos generation can be well described by the extended coupled complex nonlinear Schrödinger equations [23–27]:
In the above equations, u and v are the normalized envelopes of the optical pulses along the two orthogonal polarized modes of the optical ðber. Δβ = 2π/LB is the wave-number difference between the two modes, where LB is the beat length. δ = β1x-β1y is the linear group-velocity difference between the two orthogonal polarization modes, where β1x and β1y are the linear group-velocity related to the two orthogonal polarized modes. β2 and β3 are the group velocity dispersion (GVD) parameter and the third-order dispersion coefficient, respectively. γ represents the nonlinearity parameter of the ðbers. g is the saturable gain coefficient of the ðbers and Ωg is the bandwidth of the laser gain. For single mode ðbers, g = 0. For EDF, g = G · exp[-∫(|u|2 + |v|2)dt/Psat], where G is the small signal gain coefficient and Psat is the saturation energy. In our simulation, the whole cavity length L is set as 10 m, which consists of a 2 m-long EDF with β2 = 50 ps/nm/km and two sections of 4 m-long SMF with β2 = −30 ps/nm/km. Other parameters are set as follows: γ = 4 W−1km−1, β3 = 0.1 ps2/nm/km, Ωg = 25 nm, LB = L/2, Psat = 250 and the orientation of passive polarizer to the ðber fast axis θ = 0.125π. More simulation details see Ref [24].
With the above parameter selection, this MLFRL can emit mode-locked pulses in an appropriate linear cavity phase delay bias range which corresponds to the orientations of the polarization controllers. With a fixed linear cavity phase delay bias but different pump power, stable uniform pulse train can always be obtained. However, once the pulse powers exceed a certain threshold value, the MLFRL will experience quasi-periodic route to chaos [22–24]. Figure 2 shows an example of quasi-periodic route to chaos. Here, the linear cavity phase delay bias is fixed at 1.6 π. When pump power is relatively weak (G = 338 km–1), a stable pulse train with uniform amplitude can be obtained [Fig. 2(a(I))]. With further increasing the pump power to a certain value (G = 342 km–1), the MLFRL will be operated in period-2 state and the intensity of the pulse alters between two different values [Fig. 2(b(I))]. Further slightly increasing the pump power (G = 346 km–1), a multi-periodic state appears as shown in Fig. 2(c(I)). Eventually, the MLFRL will snap into the chaos state as the pump power are sufficient strong (G = 348 km–1) [Fig. 2(d(I))]. In order to display the quasi-periodic route to chaos more directly, we extract the output pulse powers in each round when the laser works at different state and show them in the right column of Fig. 2, where the blue dots represent the values of output pulse powers in each round.
Fig. 2 Quasi-periodic route to chaos with a fixed linear cavity phase delay bias of 1.6 π under different pump strength which are given in three-dimensional form (left column) and two-dimensional form (right column). (a) Period-1 state, G = 338 km−1; (b) Period-2 state, G = 342 km−1; (c) Multi-periodic state, G = 346 km−1; (d) Chaotic state, G = 348 km−1.
In our simulation, the MLFRL can actually generate pulse amplitude chaos in a pump power range of 348 km−1 < G < 350 km−1 under the above-mentioned condition. We arbitrarily select an operating point in the chaos region with G = 349 km−1 in order to analyze the characteristics of pulse amplitude chaos. Figure 3 are the autocorrelation function curve, first return map and histogram of the extracted chaotic pulse powers, respectively. No apparent harmonic peak in the autocorrelation characteristics is found [Fig. 3(a)]. This indicates the correlation of the generated pulse amplitude chaos is statistically insignificant and has no periodic components harmful to true random number generation. The first return map of the chaotic pulse powers is shown in Fig. 3(b) which has no discernable structure. It reveals that there is no correlation between one pulse at one round to the neighboring round and the trajectory in phase space move in a disordered form. Finally, we consider the distribution of the pulse amplitude chaos which has great effect on the level of complexity in equalizing the ratio of “0” and “1” in the following quantization procedure (Section 2.2). As Fig. 3(c) shown, the distribution of the obtained pulse amplitude chaos in our method is highly symmetric. For such a distribution, one can easily perform an even division of the chaotic pulses into “0” and “1” without bias, which is very useful to the realization of a TRNG.
Fig. 3 Characteristics of pulse amplitude chaos. (a) Auto-correlation curve of the chaotic pulse power; (b) First return map of the chaotic pulse power; (c) Stochastic histogram of the chaotic pulse power.
Although an aperiodic oscilloscope trace and a spike-like autocorrelation curve are necessary for chaos, they are not sufficient. Stochastic noise can also have these features. A careful analysis is necessary to determine whether the final state of MLFRL is chaos. Our analysis starts with the reconstruction of a pseudo phase space from the original one-dimension chaotic data in the final state of MLFRL via the delay-coordinate technique. We then use Grassberger-Procaccia algorithem (GPA) [28] to obtain the correlation dimensions (CD2) and embedding dimensions (m) for the original data. A graph of CD2 as a function of m based on 20 000 data points is shown in Fig. 4 . If the final state of MLFRL is chaos, CD2 will be expected to converge to a value, i.e., become independent of m at least for large m, and that value is usually considered as the estimate for the correlation dimension. In contrast, a white noise has m that monotonically increases with CD2. As Fig. 4 showing, CD2 clearly converges, indicating that the final state of MLFRL is not white noise. However, the GPA analysis is sensitive to linear as well as nonlinear correlations. Thus a further check must be performed to ensure that colored noise is not responsible for the aperiodicity. It has been pointed out that a comparison with so-called “surrogate” data [29], which have the same linear correlations as the original data, can provide such a check. The surrogate data are produced by Fourier transforming the original data, randomizing the phases, and then reversing the Fourier transform. A comparison of the GPA analysis of the surrogate data with the original data [Fig. 4] clearly shows convergence in the original data which is absent in the surrogate data. Thus, we confirm that the final state of MLFRL is indeed chaotic and its correlation dimension is a fractional number around 4.
Fig. 4 Result of Grassberger-Procaccia algorithem (GPA) analysis on chaotic data set shown in Fig. 3. Black squares: GPA analysis of original data. Red circles: GPA analysis of surrogate data.
2.2 All-optical flip-flop and random bits generation
The quantization to pulse amplitude chaos is similar to that of Refs [18], which is realized through an all-optical flip-flop (AOFF) proposed by Huybrechts et al. [30, 31]. The AOFF is realized by means of the bistability existing in a single λ/4-shifted DFB laser (λ/4 DFB), which can be described as below: A λ/4 DFB with antireflection-coated facets is biased above threshold. When an external light outside the stop-band of the grating is injected into the laser, the laser can operate at two different stable states: switching on and switching off.
Figure 5 is one of Huybrechts’ static experimental results under the injection of only continuous-wave (CW) light [31]. From Fig. 5, we can clearly see that only when injection light power is above Pth2, the output power of lasing light can jump down to a tiny level of nearly 0 mW. While the injection light power is below Pth1, the output power of lasing light will maintain a higher level around 1 mW. To obtain flip-flop operation, the DFB must operate in the bistable regime through injecting a CW light as the holding beam. The switching of the two states of the bistability can be controlled by the injected optical pulses from the left and right facet of the DFB. The left-hand pulses switch the laser off by disturbing the uniformity of the carrier distribution. The right-hand pulses switch the laser on by restore the uniformity in the cavity again. To trigger the AOFF, the pulse power should be higher above a certain threshold so as to induce the conversion of the output state of the DFB. In our simulation about the AOFF, the power of CW light was set to be Pth1 = 1.6 mW so that the threshold (i.e. ΔP = Pth2 – Pth1) equals to 0.1 mW, the average power of the chaotic pulse trains. Thus, the quantizing to pulse amplitude chaos can be achieved, as illustrated in Fig. 6 . Figure 6(a) is a time series of pulse amplitude chaos generated by the MLFRL, which consists of a chaotic pulse train with a fixed frequency of 20 MHz but randomly distributed pulse powers. Figure 6(b) shows the chaotic pulse train after the 3-dB coupler, which is injected into the DFB from its left facet and with a mean power about 0.1 mW. Figure 6(c) is the identical pulse train delayed by 25 ns on the right side of the DFB. Figure 6(d) is an output waveform of the AOFF (the DFB) through the BPF transmitting only the lasing light of the DFB. The output is coded in the way as Fig. 6(d) shown. From it, one can see that each bit of “0” or “1” occupies a duration time of 50 ns and the extinction ratio between “0” and “1” is as high as 30 dB.
Fig. 5 Bistability curve: laser output power as a function of the power of the injected light [31]. There are two typical threshold values: Pth1 is 1.6 mW and. Pth2 is 1.7 mW.
Fig. 6 Temporal waveforms: (a) Pulse amplitude chaos from the MLFRL. (b) Pulse amplitude chaos injected into the left-hand of the AOFF. (c) Delayed pulse amplitude chaos injected into the right-hand of the AOFF. (d) Random bit sequence.
3. Randomness verification
After the above processes, we got a train of random bits with a rate of 20 Mbps which is determined by the repetition rate of the MLFRL. A rough verification of its statistical randomness can be characterized by a bitmap image [Fig. 7 ] constructed from 300 × 300 bits of the generated random sequences. As shown in Fig. 7, we can see clearly the bitmap image exhibits no apparent pattern or bias.
Fig. 7 Random bit patterns with 300 × 300 bits are shown in a two-dimensional plane. Bits “1” and “0” are converted to white and black dots, respectively, and placed from left to right (and from top to bottom).
Further, to better qualify the statistical randomness of the random bits, we used the industry-standard statistical test suite of the National Institute of Standards and Technology (NIST) [20] and the Diehard test suite [21]. The NIST test suite consists of 15 statistical tests as shown in Table 1 , and each test is performed using 1000 samples of 1 Mb data and significance level α = 0.01. The passing criteria are that the proportion of the sequences satisfying condition for the p-value, p > α, should be in the range of 0.99 ± 0.0094392, and the P-value of the uniformity of the p-values should be larger than 0.0001. Diehard test suite consist of 18 statistical tests as shown in Table 2 , which are performed using 74 Mb data and significant level α = 0.01. For “success”, the P-value (uniformity of the p-values) of each test should be within [0.01, 0.99]. As shown in Table 1 and 2, the bit sequences generated in our method can pass all tests of both the NIST and Diehard tests. These results confirm that the generated random bits can be statistically regarded as truly independent random bits.
Table 1. Typical results of NIST statistical tests. Using 1000 samples of 1 Mb data and significance level α = 0.01, for “Success”, the P-value (uniformity of p-values) should be larger than 0.0001 and the proportion should be greater than 0.9805608.
Table 2. Typical results of Diehard statistical tests. Using 74 Mb data and significance level α = 0.01, for “Success”, the P-value (uniformity of p-values) should be larger than 0.0001. “KS” indicates that single P-value is obtained by the Kolmogorov-Smirnov (KS) test.
4. Discussions
4.1 Generation rate of RNG and its improvements
One critical factor for many applications of RNG is a high bit generation rate. The rate in our RNG is primarily determined by the repetition rate of the pulse amplitude chaos corresponding to the cavity length of the MLFRL. Huybrechts et al. [32] demonstrated that the rising time of the AOFF can be down to 40 ps in experiment and even 10 ps in theory. It indicates the bandwidth of AOFF can be larger than 30 GHz. Therefore, if the repetition rate of the pulse amplitude chaos generated by the MLFRL is increased, our RNG has great potential to work at a tremendous bit generation rate.
There are two ways to improve the repetition rate of the pulse amplitude chaos generated by the MLFRL. The first one is the introduction of time-division multiplexing (TDM) technology into the proposed RNG. Using a large number of independent MLFRLs which works at different conditions with each other and multiplexing their output signal (pulse amplitude chaos) before the quantization procedure, we can yield high repetition rate of chaotic pulses and then achieve random bits generation at a rate of several Gbps or higher. The second one is to shorten the cavity length of the MLFRL. For example, we can replace the EDF with the small size SOA and substitute ordinary SMFs with high birefringence optical fibers with ultra-short beat length.
4.2 Tolerance of RNG and its improvements
Another critical factor for the applications of RNG is the robustness or tolerance of the system. Here, we discuss the tolerance of our RNG by analyzing the effect of threshold bias on the randomness of the generated random numbers. Notice that here the threshold bias represents the difference between the threshold of the all-optical flip-flop (AOFF) and the average power of the chaotic pulse trains. In our system, the threshold (i.e. ΔP = Pth2 – Pth1) of AOFF is fixed at 0.1 mW and the average power of the chaotic pulse trains can be adjusted by the erbium-doped fiber amplifiers (EDFA) and attenuators (Att.) as shown in Fig. 1.
Figure 8 shows the frequency of “0” in a random bit sequence and the number of passed NIST tests as a function of the threshold bias. A value of “15” for the number of passed tests means that all the tests are passed. The frequency of “0” can play the role of indicator for the quality of the randomness. Figure 8 indicates that the frequency of “0” decreases almost linearly with the increase of threshold bias and only the random sequences having a frequency of “0” in the range from 49.87% to 50.13% can passed all the NIST tests. To get a high-quality random numbers in this condition, the threshold bias should be within a range between −0.001 and 0.001 mW.
Fig. 8 The frequency of “0” in a random bit sequence (black squares) and the number of passed NIST tests (blue circles) as a function of the threshold bias. Here the threshold bias represents the difference between the threshold of the all-optical flip-flop (AOFF) which is fixed at 0.1 mW and the average power of the chaotic pulse trains.
However, it should be noticed that the tolerance of threshold bias is so small, because the threshold (i.e. ΔP = Pth2 – Pth1) of AOFF in our system is fixed at 0.1 mW. This causes the average power of the chaotic pulses has to be attenuated to a level around 0.1 mW. If the threshold of AOFF (i.e. the hysteresis curve width expressed asΔP = Pth2 – Pth1) is big enough, the average power of chaotic pulses will be able to vary on a much larger scale and without any loss of randomness and thus the tolerance of our system can be greatly improved. As a matter of fact, the threshold of AOFF indeed can be easily enhanced by adjusting the bias current of AOFF. This point has been successfully demonstrated by Huybrechts numerically and experimentally [30,31]. So, there is still room for improvement in the robustness of our RNG.
In addition, we want to point out that the post-processing procedure such as XOR operation in the previous RNGs based on continuous analog chaotic light from the optical feedback semiconductor laser is not only to improve the 1/0 ratio in random bit sequences and then allow a much larger threshold bias, but, more importantly, to eliminate the inherent weak periodicity introduced by the external feedback cavity [11–18]. For this kind of chaotic signal, the random numbers generated based on it and without post-processing procedure cannot pass industry-standard statistical tests (such as NIST and Diehard tests) even if the random numbers has an even 1/0 ratio in practice. Different with them, our pulse amplitude chaos has no periodicity [Fig. 3(a)] and processes a highly symmetric distribution [Fig. 3(c)]. Therefore, our method does not require post-processing procedures in theory. In this sense, we believe that our method also provides a clue in theory to the implementation of RNG with no need of post-processing procedures.
5. Conclusions
One direct method for all-optical random bit generation using discrete pulse amplitude chaos in a mode-locked fiber ring laser is theoretically demonstrated. Our simulation results show that the pulse amplitude chaos consisting of a stream of pulses processes several excellent characteristics favorable for high-quality true random bits generation, such as aperiodicity and symmetric stationary distribution. Different from previously reported random number generators, our RNG do not require the sampling procedure but directly quantizing the chaotic pulse train to random bit sequences. So it can bypass the possible aliasing problem caused by sampling procedure and reduce hardware complexity. Free from post-processing, the generated random bit sequences pass successfully the standard statistical tests for randomness. In addition, the RNG in this paper is a conceptual protocol and its bit rate can be further increased by using MLFRLs with higher repetition rate or multiplexing technology.
Acknowledgments
We acknowledge Koen Huybrechts from Ghent University for providing the data about the all-optical flip-flop. We thank Han Zhang from Université libre de Bruxelles for helpful discussions and comments. This work is supported partially by the Key Program of National Natural Science Foundation of China (Grant 60927007 and Grant 61001114) and in part by the open subject of the State Key Laboratory of Quantum Optics and Quantum Optics devices of China (Grant 200903).
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