Abstract

We demonstrate a high bit-rate quantum random number generator by interferometric detection of phase diffusion in a gain-switched DFB laser diode. Gain switching at few-GHz frequencies produces a train of bright pulses with nearly equal amplitudes and random phases. An unbalanced Mach-Zehnder interferometer is used to interfere subsequent pulses and thereby generate strong random-amplitude pulses, which are detected and digitized to produce a high-rate random bit string. Using established models of semiconductor laser field dynamics, we predict a regime of high visibility interference and nearly complete vacuum-fluctuation-induced phase diffusion between pulses. These are confirmed by measurement of pulse power statistics at the output of the interferometer. Using a 5.825 GHz excitation rate and 14-bit digitization, we observe 43 Gbps quantum randomness generation.

© 2014 Optical Society of America

1. Introduction

Random number generators (RNG) have been extensively studied by different groups because of the wide spectrum of their applications: secure communications [1], stochastic simulation [2] and gambling [3], among others.

Deterministic algorithms known as pseudo-random number generators can rapidly generate bit sequences with long repetition lengths, and are often used as a substitute for random numbers. Physical RNGs take as input data from a physical process believed to be random, for example electronic and thermal noise [4], chaotic semiconductor lasers [5] and amplified spontaneous emission signals [6]. A subset of physical RNGs, known as quantum RNGs (QRNGs) use a physical process with randomness derived from quantum processes. Processes used include radioactive decay [7], two-path splitting of single photons [8], photon number path entanglement [9], amplified spontaneous emission [10], measurement of the phase noise of a laser [1113], photon arrival time [14], and vacuum-seeded bistable processes [15]. These physical RNGs and QRNGs show a trade-off between speed of generation and surety of the random bits generated: chaotic and ASE sources [5, 6, 16, 17] reach hundreds of Gbps using signals that include contributions from both random and in-principle predictable sources, e.g. detector noise. In contrast, QRNGs can guarantee the quantum origin and thus the randomness of the signal [12, 14, 18], although to date at lower bit rates.

Recently, Jofre, et al. [12] and Xu, et al. [13] have reported quantum random number generation using phase diffusion in semiconductor lasers. In [12] a QRNG rate up to 1.1 Gbps was demonstrated and in [13], 6 Gbps QRNG rate. The main difference is that Jofre et al. strongly modulate the laser diode, taking it below and above threshold, while Xu et al. directly detect phase fluctuations in an above-threshold continuous wave regime. For the same mean power, pulsing accelerates the phase diffusion rate, which is proportional to the spontaneous emission rate over the intra-cavity photon number. Here we report improvements in both speed and surety in the Jofre et al. method. We achieve 43 Gbps by using a higher bandwidth laser diode and a faster modulating system. Also, we demonstrate a new method to lower-bound the quantum contribution to the randomness of the observed pulse distribution.

2. System configuration

A 1550 nm distributed feedback (DFB) laser diode (LD) with 10 Gbps modulation bandwidth is driven through its DC input with a constant current of 15 mA, and through its AC-coupled RF input with a waveform modulated at 5.825 GHz. These two sources, combined with a bias-tee and recorded on an oscilloscope, are shown in Fig. 1. Because the RF drive amplitude exceeds the bias, the diode is reverse-biased for about 40% of the cycle, creating strong attenuation within the material between periods of high gain. The resulting optical output consists of pulses of ∼ 85 ps width and ∼ 7.65 mW peak power. To avoid back reflections into the oscillator cavity, the LD has an internal 30 dB optical isolator.

 

Fig. 1 Electrical and optical pulse trains. Magenta, (upper trace): electrical current drive applied to the laser, with PRF of 172 ps. Blue, (lower trace): photo-detected optical pulses of 85 ps time width and 7.65 mW peak power and (black, dashed line) 9 mA LD current threshold. Simulation is a conservative fitting of the rate equations such that the predicted detected output power vs. time is always larger than the observed output power vs. time.

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The linearly polarized optical pulses pass through an unbalanced Mach-Zehnder interferometer (U-MZI) composed of two 50/50 polarization maintaining couplers (PMC) with arm-length difference related to the pulse repetition frequency (PRF), see Fig. 2. The 3.55 cm path delay length is designed to interfere pulses at a PRF of 5.825 GHz. The optical output is detected by a 14-bit oscilloscope (DSA8200 with module 80C02), with 12.5 GHz bandwidth and triggered by the system clock reference, taken from the electrical pulse generator (Anritsu MP1800A).

 

Fig. 2 Unbalanced Mach-Zehnder interferometer (U-MZI). Phase-randomized coherent optical pulses interfering at the output of the U-MZI produce random intensities. (Pulse driver) denotes the electrical pulse generator that directly modulates the laser, (LD) laser diode, (PMC) polarization maintaining coupler, (PMF) polarization maintaining fiber, (θ0–3) optical phases of different consecutive pulses, (θloop) phase introduced by the delay line and (PD) a fast photodetector.

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3. Principle of operation

The field detected at the output of the interferometer is

(out)(t)=ε11(1)ε11(2)(laser)(tt1)+ε12(1)ε21(2)(laser)(tt2),
where εij(k) is the field transmission coefficient of the kth PMC from input port i to output port j, (laser) is the field emitted by the laser and t1,2 are the delays caused by arms 1,2 of the U-MZI, respectively. For convenience, we write (laser)(t) = A(t)exp[−iωt]exp[j] where ω is the central frequency of the laser emission and θj is the phase of the jth pulse. Defining the powers u(out)(t) ≡ |𝒠(out)(t)|2, u1(t)|ε11(1)ε11(2)(laser)(tt1)|2 and u2(t)|ε12(1)ε21(2)(laser)(tt2)|2 we have the observed signal
u(out)(t)=u1(t)+u2(t)+2|g(1)(t)|u1(t)u2(t)cos(θjθj1+Δϕ)+unoise,
where Δϕω(t1t2) is the relative phase delay of the two arms, g(1)(t)*(laser)(tt1)(laser)(tt2)/|(laser)(tt1)|2|(laser)(tt2)|2 is the degree of first-order coherence (the visibility) and here 〈·〉 indicates an average over the response time implied by the finite bandwidth of the detector and recording electronics. unoise is noise contribution from the detection and digitization electronics. Fluctuation of Δϕ due to changes in t1t2, e.g. stretching of the fiber, were measured by above-threshold continous-wave operation of the laser, which showed RMS phase fluctuations, between points separated by 1/PRF, of ≤ 2 × 10−7 rad, which is negligible on the scale of the quantum fluctuations described below.

4. Phase diffusion

The experiment is performed under gain-switching conditions, and the field within the cavity alternately experiences two working regimes: far above threshold and far below threshold. During the former, stimulated emission dominates and coherent optical pulses are emitted. During the latter, the intracavity field is strongly attenuated and significant phase diffusion occurs. The stochastic nature of this process has been studied in detail by C. and G. Agrawal in [1921], where an expression for the average phase diffusion 〈Δθ2(t)〉 is derived under various conditions from the Langevin equation

θ˙=α2GN(nnth)βSE2GN(nn0)p1+1+p+Fθ(t),
where Fθ is a Langevin force responsible for the phase diffusion process, α = 5.4 is the linewidth enhancement factor first derived by C. Henry [19] and all other parameters are described below. The linear approximation proposed in [20]
ddtΔθ(t)2=Rsp2s(1+α2),
gives a lower bound on the phase diffusion in an interval of time t. Here Rsp is the spontaneous emission rate, which depends (see next paragraph) on the number of carriers n, and s is the number of photons in the cavity [19]. Hence, the dynamical behavior of Rsp and s are described by differential equations, given in [21], Eqs. (13)(17), which couple s to n.
s˙=GN(nn01+s/ssat(nthn0))s+Rsp,
n˙=I/qγenGNnn01+s/ssats,
where GN is the gain per carrier, n0 is the number of carriers at transparency, nth is the number of carriers at threshold, ssat is the saturation photon number, γe is the carrier recombination lifetime = 1/τe ≈ 1 × 109, I the injection current and q is the electronic charge. To give values to these parameters, we note that GN = γ/(nthn0), where γ = c(αm + αs)/ is the cavity decay rate where c is the speed of light in vacuum, = 4.33 is the effective refractive index, αs = 4.5 cm−1 describes losses due to scattering and absorption and αm ≈ 1.4/L is the cavity escape loss, where L is the cavity length. Because ssat, L and n0 are unknown to us, we choose them by fitting to the observed pulse-shape (see below). Rsp can be written Rsp = eR0, where R0 = KtotΓconfβSE is a constant containing Γconf the “confinement factor”, βSE the fraction of spontaneous emission coupled to the lasing mode, and Ktot the “total enhancement factor” in a DFB LD, i.e. the enhancement in the fraction of spontaneous emission actually coupled to the lasing mode because of the geometry. We also do not have access to these parameters, but their combined effect is measurable using the steady-state power (see below).

The solution of the rate equations, i.e. s(t) and n(t), permits the calculation of Rsp and therefore the dynamical evolution of the average phase diffusion 〈Δθ2〉 in Eq. (4). In order to fit a solution of these equations to the measurements on our DFB LD, we first need to estimate the parameters ssat, L, nth, GN, n0 and R0 (these are not all independent). Also, the solution is low pass filtered with a single-pole recursive filter with a time constant of τf = 0.35/BWosc, where BWosc = 12.5 GHz is the bandwidth of the oscilloscope.

A recursive method is used to extract these parameters:

  1. We set ssat to a reasonable value, and L to one of 100, 200, 500 or 1000 μm.
  2. We solve the steady-state solution of the differential equations, Eq. (7) and Eq. (8), near threshold, i.e., with nnth. The DFB LD emits sth′ = 0.3 mW when it is biased with a constant current of Ith′ = 10 mA. (The subscript th refers to steady-state solution slightly above threshold). Eqs. (5) and (6) then take the form
    γ(11+sth/ssat1)sth+R0γenth=0,
    Iqγenthγsth11+sth/ssat=0,
    from which we extract nth and R0.
  3. We choose the maximum GN, which mainly controls the speed of the dynamics, such that the predicted detected output power vs. time is always larger than the observed output power vs. time (from Fig. 1). Note that setting GN directly specifies n0, via GN = γ/(nthn0).
  4. We repeat steps (1)(3) to find the values of ssat and L that minimize the rms deviation of the simulation from the observed power curves.

After these steps, ssat = 7.7 × 105, L = 500 μm, nth = 5.62 × 107, R0 = 8.8 × 10−4, GN = 2.3 × 104 and n0 = 3.46 × 107. The result is shown in Fig. 1. Note that phase diffusion is larger for low optical intensities. Hence, the fitting is conservative because the simulated optical power is always larger than the measurement.

Finally, Eq. (4) is numerically integrated to find 〈Δθ2〉 > (9.45 rad)2. For all practical purposes this value describes a full randomization. We note that cosθ is unchanged under θθ + 2πn (n integer) and under θ → −θ. If θ is described by a Gaussian PDF G(θ) with width 〈Δθ2〉, then an equivalent distribution, for the purposes of the cosine, is Gπ(θ)s=±1n=G(sθ+2πn) for 0 ≤ θ < π and Gπ(θ) ≡ 0 otherwise. Already with 〈Δθ2〉 = (2π)2, Gπ approximates a uniform distribution on [0, π) with a fractional error below 10−8.

5. Raw data analysis

A RNG should produce an uncorrelated and uniformly distributed string of numbers. The un-correlation avoids the capability to predict future numbers while the uniformity ensures the same apparition probability to all of them. Sampling the interfered field, an uncorrelated but not equally distributed sequence of numbers is achieved. The nearly complete phase diffusion is expected to provide uncorrelated pulses, while the uniformity will be achieved using a randomness extractor over the raw data.

For statistical testing, 120 × 106 pulses were recorded on a fast 14-bit sampling oscilloscope. From each pulse a single sample was recorded, taken ∼ 13 ps after the pulse peak, to produce 120 × 106 14-bit numbers. This delay was chosen experimentally to give a broad distribution in Fig. 3(a). We observe experimentally that the distribution takes on the expected shape after power saturation occurs, i.e. after the pulse peak, presumably due to damping of transients associated with the rapid turn-on. These data were taken over a five day period as a test of stability and repeatability.

 

Fig. 3 In Fig. 3(a), input power distributions (left y axis) and the resultant output power distribution (right y axis). The visibility achieved for the interferometer is 0.9. The power distribution has clearly widened due to the random phase generated by amplified spontaneous emission (ASE). In Fig. 3(b), normalized correlation of 50 subsequent sampled pulses. The autocorrelation has been evaluated with 120 × 106 14-bit samples, but just the first 50 terms are shown. As expected, it follows a delta-function like behaviour indicating the random nature of the process.

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The power from each branch of the U-MZI, as measured at the output, are 0.97 mW and 0.9 mW on average, with a standard deviation of ∼ 45 μW. This deviation around the mean arises from background noise and spontaneous emission events, producing narrow distributions for the interfering pulses. In contrast, due to interference of equal power optical pulses with random phases, the output power distribution is expected to broaden. The fact that the optical pulses have a predictable power and true random phase implies that when interfering these fields, random input phases of quantum origin are converted into macroscopically measurable random output intensities. This behaviour is shown in Fig. 3(a).

We evaluate the normalized autocorrelation function to confirm the intuition that the process is intrinsically uncorrelated. As seen in Fig. 3(b), the non-shifted correlation values are at the 40 dB level.

6. Randomness extraction and post-processing analysis

Randomness extractors transform non-uniformly-distributed sequences into uniformly-distributed sequences [22] at the cost of losing a fraction of the bits. Given a random variable X distributed according to the distribution P[X = xi], the min-entropy

H(X)log2(maxxiP[X=xi]),
quantifies the amount of extractable randomness, in the sense that from a sequence {X1,...,XN} with N ≫ 1, a uniform random bit sequence of length NH(X) can be extracted. For our digitized input with b bits of resolution, we define the reduction factor RF ≡ b/H(X).

The detected signal given by Eq. (2) contains a large random component through the last term u1u2cosϕ, where ϕ is the random phase due to phase diffusion. If this were the only fluctuating term, the observed signal would obey an arcsine distribution, the distribution of the cosine of a uniformly-distributed phase. Other fluctuating contributions to the signal, including photo-detection noise, laser current fluctuations, and digitization errors, broaden and smooth this ideal distribution to give the observed distribution shown in Fig. 3. These other noise sources are not guaranteed to be random. Fortunately, the randomness extraction step will remove any non-random effects of these other noise sources, provided that the reduction factor is chosen using the min-entropy of the quantum contribution. In what follows, we show that this quantum contribution can be determined from second order statistics of the measured observable and noise sources, plus our knowledge about the distribution of the phase.

6.1. Procedure

The interfering process described in Eq. (2) contains four random variables: X1 = u1, X2 = u2, X3=u1u2cosθ and X4 = unoise, where X3 is the product of three random variables u1, u2 and cosθ. Assuming that u1, u2, cosθ and unoise are independent, does not imply that the four random variables X1, X2, X3, X4 are independent. In fact, u1u2cosθ depends on u1 and u2. Nevertheless, the fact that cosθ is equally likely to be positive or negative implies that the four random variables are uncorrelated.

The arcsine probability distribution function, its mean and its variance are given by:

PDF(x)=1π(xa)(bx);μx=a+b2;var[x]=18(ba)2.

In order to estimate the visibility from the observed distributions of u1, u2 and u(out), the variance at both sides of Eq. (2) is calculated. Assuming uncorrelation:

var(u(out))=var(u1)+var(u2)+var(unoise)+4|g(tloop)|2var(u1u2cosθ).
From laser physics, we know that the intra-cavity field describes a diffusion process and therefore θ is gaussianly distributed. Also, using measures and well-established models of semiconductor DFB lasers, an standard deviation larger than π was estimated in Section 4. As described above, this justifies considering θ to be uniformly distributed on [0, π), and thus that cosθ is described by an arcsine distribution. From Eq. (10) and Eq. (11), it then follows that
|g(tloop)|2var(u(out))var(u1)var(u2)var(unoise)2E[u1]2E[u2]2,
where E[·] indicates the mean value.

From the data shown in Fig. 3, we obtain var(u(out)) = 1.4 mW2, var(u1) = 2.0 × 10−3 mW2, var(u2) = 2.1 × 10−3 mW2, E[u1]2=0.97mW, and E[u2]2=0.90mW. From measurements with the laser off, we find the contribution of detection and digitization noise var(unoise) = 1.45 × 10−4 mW2. Applying Eq. (12) we find |g(tloop)| = 0.90.

6.2. Min-entropy estimation

The distribution produced by the random phase, i.e. without considering undesired random effects, is given by an arcsine distribution between umin and umax, where

uminE[u1+u22|g(tloop)|u1u2],
umaxE[u1+u2+2|g(tloop)|u1u2].
Because the arcsine distribution is peaked at x = umin, and because the min-entropy only depends on the weight of the most probable event, the min-entropy of the digitized arcsine distribution only depends on the probability of the first bin. If AADC is the dynamic range of the analog-to-digital converter (ADC), b its resolution in bits so that Δu = AADC/2b is the bin size, the probability mass of the first bin is
P[X=x1]=1πuminumin+Δu1(uumin)(umaxu)du=2πarcsinAADC2b(umaxumin).
Using the fact that arcsinxx for x small, it can easily be seen that the probability of the first bin decreases exponentially with b and therefore the min-entropy, given by Eq. (16), increases linearly with the resolution of the digitization.
H(X)b212log2(4AADCπ2(umaxumin)).
In the experiment, where AADC = 5 mW, b = 14 bits, umaxumin=4|g(tloop)|E[u1]E[u2]=3.34mW, the min entropy is 7.33 bits. With the PRF of 5.825 Gpulses/s, this implies a quantum randomness generation rate of 43 Gbps.

In order to extract the randomness arising from amplification of the vacuum field, a hash function with a reduction factor RF = 14/7.33 ≈ 1.9 is applied to the raw data, reducing 120 × 106 14-bit numbers to 125 × 106 7-bit numbers. The hashing is performed by the Whirlpool hash function [23]. As seen in Fig. 4, correlations within the resulting data are at the −40 dB level, which coincides with the statistical uncertainty given the quantity of output numbers. To check uniformity, we compute the frequencies of the 7-bit numbers. Fig. 4(b) shows the deviation from the ideal 7-bit uniform distribution δeu = Pi − 1/128, where Pi is the probability of the i-th bin. No statistically significant deviation from uniformity is observed.

 

Fig. 4 Statistical characterization of 125 × 106 7-bit numbers produced by hashing the experimental data. (a) autocorrelation of the hashed data. Autocorrelation at the −40 dB level is seen, corresponding to the expected statistical variation. (b) deviation from the ideal 7-bit uniform distribution. Dashed lines show plus/minus 1 sigma expected variation.

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6.3. Statistical testing

The 15-test battery proposed by NIST is applied on the hashed data to assess its randomness. The significance level (αSL) is set at 0.01. It means that 1 in a 100 sequences is expected to be rejected even if it is produced by a fair random generator. Using suggestions in [24], the proportion of accepted/rejected sequences and the uniformity of the P-values are computed. As shown in Fig. 5, all tests are passed with a sequence of 1.5 Gbits.

 

Fig. 5 Summary of the results of the NIST test suite to assess randomness. Dashed lines in Fig. 5(a) represent the confidence interval where the proportion of sequences accepted/rejected per test must fall in. In Fig. 5(b) is plot PvalueT = Γ(9/2, χ2/2) for each test. All PvalueT > 10−4. The smallest one is for the non-periodic template matching test, giving a value of PvalueT = 0.0926.

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In order to evaluate the results of the tests, two statistics are calculated. First, the ratio of accepted to rejected sequences is calculated and must fall within the confidence interval defined by [1αSL±3(1αSL)αSL/m], where m = 1500 is the number of 1 Mbit sequences tested, see Fig. 5(a). Second, the ε-uniformity of the P-values is examined. The idea is to compute PvalueT, a ‘P-value of P-values’. The procedure is as follows: for each test, (i) calculate a 10-bit histogram of P-values, (ii) compute the χ2 defined in Eq. (17) and (iii) calculate the incomplete gamma statistical function Γ(9/2, χ2/2), which must be larger than 10−4, see Fig. 5(b).

χ2=s10i=110(Fis10)2,
where s is the number of P-values per test and Fi is the number of P-values in the i-th bin.

7. Conclusions

We have demonstrated a quantum random number generator (QRNG) based on interferometric detection of phase diffusion in a high bandwidth gain-switched DFB laser diode. The method uses interferometry rather than single-photon or shot-noise-limited detection to convert microscopic quantum observables to macroscopically detectable signals, and is demonstrated with commercially available telecommunications components. These features make it robust, low cost, compact and low power consumption. Using a 5.825 GHz excitation rate and 14-bit digitization, we observe 43 Gbps quantum randomness generation. We compute a conservative bound on the phase diffusion between pulses and find diffusion by at least 9.45 radians rms, equivalent to complete phase randomness for all practical purposes. The generated bit strings show no detectable correlations or deviations from uniformity, and pass all tests in the NIST 15-test battery. We provide a new method to determine the quantum contribution to the detected random signals, and show that the quantum contribution to min-entropy grows proportional to the digitization resolution in bits, suggesting a straightforward route to even higher QRNG bit rates. The high rate as well as the quantum nature of the randomness generation make this scheme attractive for cryptography, gambling and quantum key distribution.

Acknowledgments

This work was supported by the ERC under project MAMBO (Proof of Concept of PERCENT) and project AQUMET, MINECO under projects FIS2011-23520, TEC2010-14832, and Explora INTRINQRA, Galician Regional Government under projects CN2012/279 and CN2012/260 “Consolidation of research units: AtlantTIC” and FEDER under project Ref: UPVOV10-3E-492.

References and links

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2. X. Cai and X. Wang, “Stochastic modeling and simulation of gene networks - a review of the state-of-the-art research on stochastic simulations,” IEEE Signal Process. Mag. 24, 27–36 (2007). [CrossRef]  

3. C. Hall and B. Schneier, “Remote electronic gambling,” in “13th Annual Computer Security Applications Conference” (1997), pp. 232–238. [CrossRef]  

4. C. Petrie and J. Connelly, “A noise-based ic random number generator for applications in cryptography,” IEEE Trans. Circuits Syst. I, Fundam. Theory Appl. 47, 615–621 (2000). [CrossRef]  

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7. Y. Yoshizawa, H. Kimura, H. Inoue, K. Fujita, M. Toyama, and O. Miyatake, “Physical random numbers generated by radioactivity,” J. Jpn. Soc. Comput. Stat. 12, 67–81 (1999).

8. T. Jennewein, U. Achleitner, G. Weihs, H. Weinfurter, and A. Zeilinger, “A fast and compact quantum random number generator,” Rev. Sci. Intrum. 71, 1675–1680 (2000). [CrossRef]  

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13. F. Xu, B. Qi, X. Ma, H. Xu, H. Zheng, and H.-K. Lo, “Ultrafast quantum random number generation based on quantum phase fluctuations,” Opt. Express 20, 12366–12377 (2012). [CrossRef]   [PubMed]  

14. 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, 171105 (2011). [CrossRef]  

15. A. Marandi, N. C. Leindecker, K. L. Vodopyanov, and R. L. Byer, “All-optical quantum random bit generation from intrinsically binary phase of parametric oscillators,” Opt. Express 20, 19322–19330 (2012). [CrossRef]   [PubMed]  

16. X.-Z. Li and S.-C. Chan, “Random bit generation using an optically injected semiconductor laser in chaos with oversampling,” Opt. Lett. 37, 2163–2165 (2012). [CrossRef]   [PubMed]  

17. 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, 20452–20462 (2013). [CrossRef]   [PubMed]  

18. S. Pironio, A. Acín, S. Massar, A. B. de la Giroday, D. N. Matsukevich, P. Maunz, S. Olmschenk, D. Hayes, L. Luo, T. A. Manning, and C. Monroe, “Random numbers certified by Bell’s theorem,” Nature 464, 1021–1024 (2010). [CrossRef]   [PubMed]  

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23. V. Rijmen and P. S. L. M. Barreto, “The WHIRLPOOL hash function,” http://www.larc.usp.br/~pbarreto/WhirlpoolPage.html (2008).

24. A. Rukhin, J. Soto, J. Nechvatal, E. Barker, S. Leigh, M. Levenson, M. Vangel, D. Banks, A. Heckert, J. Dray, and S. Vo, “A statistical test suite for random and pseudorandom number generators for cryptographic applications,” NIST Special Publication 800-22 revision 1a (2010).

References

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  1. A. Tajima, A. Tanaka, W. Maeda, S. Takahashi, and A. Tomita, “Practical quantum cryptosystem for metro area applications,” IEEE J. Sel. Top. Quantum Electron. 13, 1031–1038 (2007).
    [Crossref]
  2. X. Cai and X. Wang, “Stochastic modeling and simulation of gene networks - a review of the state-of-the-art research on stochastic simulations,” IEEE Signal Process. Mag. 24, 27–36 (2007).
    [Crossref]
  3. C. Hall and B. Schneier, “Remote electronic gambling,” in “13th Annual Computer Security Applications Conference” (1997), pp. 232–238.
    [Crossref]
  4. C. Petrie and J. Connelly, “A noise-based ic random number generator for applications in cryptography,” IEEE Trans. Circuits Syst. I, Fundam. Theory Appl. 47, 615–621 (2000).
    [Crossref]
  5. I. Kanter, Y. Aviad, I. Reidler, E. Cohen, and M. Rosenbluh, “An optical ultrafast random bit generator,” Nat. Photonics 4, 58–61 (2010).
    [Crossref]
  6. 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, 1329–1334 (2012).
    [Crossref]
  7. Y. Yoshizawa, H. Kimura, H. Inoue, K. Fujita, M. Toyama, and O. Miyatake, “Physical random numbers generated by radioactivity,” J. Jpn. Soc. Comput. Stat. 12, 67–81 (1999).
  8. T. Jennewein, U. Achleitner, G. Weihs, H. Weinfurter, and A. Zeilinger, “A fast and compact quantum random number generator,” Rev. Sci. Intrum. 71, 1675–1680 (2000).
    [Crossref]
  9. O. Kwon, Y.-W. Cho, and Y.-H. Kim, “Quantum random number generator using photon-number path entanglement,” Appl. Opt. 48, 1774–1778 (2009).
    [Crossref] [PubMed]
  10. 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, 23584–23597 (2010).
    [Crossref] [PubMed]
  11. 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 81, 051137 (2010).
    [Crossref]
  12. M. Jofre, M. Curty, F. Steinlechner, G. Anzolin, J. P. Torres, M. W. Mitchell, and V. Pruneri, “True random numbers from amplified quantum vacuum,” Opt. Express 19, 20665–20672 (2011).
    [Crossref] [PubMed]
  13. F. Xu, B. Qi, X. Ma, H. Xu, H. Zheng, and H.-K. Lo, “Ultrafast quantum random number generation based on quantum phase fluctuations,” Opt. Express 20, 12366–12377 (2012).
    [Crossref] [PubMed]
  14. 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, 171105 (2011).
    [Crossref]
  15. A. Marandi, N. C. Leindecker, K. L. Vodopyanov, and R. L. Byer, “All-optical quantum random bit generation from intrinsically binary phase of parametric oscillators,” Opt. Express 20, 19322–19330 (2012).
    [Crossref] [PubMed]
  16. X.-Z. Li and S.-C. Chan, “Random bit generation using an optically injected semiconductor laser in chaos with oversampling,” Opt. Lett. 37, 2163–2165 (2012).
    [Crossref] [PubMed]
  17. 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, 20452–20462 (2013).
    [Crossref] [PubMed]
  18. S. Pironio, A. Acín, S. Massar, A. B. de la Giroday, D. N. Matsukevich, P. Maunz, S. Olmschenk, D. Hayes, L. Luo, T. A. Manning, and C. Monroe, “Random numbers certified by Bell’s theorem,” Nature 464, 1021–1024 (2010).
    [Crossref] [PubMed]
  19. C. Henry, “Theory of the linewidth of semiconductor lasers,” IEEE J. Quantum Electron. 18, 259–264 (1982).
    [Crossref]
  20. C. Henry, “Phase noise in semiconductor lasers,” J. Lightwave Technol. 4, 298–311 (1986).
    [Crossref]
  21. G. Agrawal, “Effect of gain and index nonlinearities on single-mode dynamics in semiconductor lasers,” IEEE J. Quantum Electron. 26, 1901–1909 (1990).
    [Crossref]
  22. N. Nisan and A. Ta-Shma, “Extracting randomness: A survey and new constructions,” J. Comput. Sci. Tech. 58, 148–173 (1999).
  23. V. Rijmen and P. S. L. M. Barreto, “The WHIRLPOOL hash function,” http://www.larc.usp.br/~pbarreto/WhirlpoolPage.html (2008).
  24. A. Rukhin, J. Soto, J. Nechvatal, E. Barker, S. Leigh, M. Levenson, M. Vangel, D. Banks, A. Heckert, J. Dray, and S. Vo, “A statistical test suite for random and pseudorandom number generators for cryptographic applications,” NIST Special Publication 800-22 revision 1a (2010).

2013 (1)

2012 (4)

2011 (2)

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, 171105 (2011).
[Crossref]

M. Jofre, M. Curty, F. Steinlechner, G. Anzolin, J. P. Torres, M. W. Mitchell, and V. Pruneri, “True random numbers from amplified quantum vacuum,” Opt. Express 19, 20665–20672 (2011).
[Crossref] [PubMed]

2010 (4)

S. Pironio, A. Acín, S. Massar, A. B. de la Giroday, D. N. Matsukevich, P. Maunz, S. Olmschenk, D. Hayes, L. Luo, T. A. Manning, and C. Monroe, “Random numbers certified by Bell’s theorem,” Nature 464, 1021–1024 (2010).
[Crossref] [PubMed]

I. Kanter, Y. Aviad, I. Reidler, E. Cohen, and M. Rosenbluh, “An optical ultrafast random bit generator,” Nat. Photonics 4, 58–61 (2010).
[Crossref]

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, 23584–23597 (2010).
[Crossref] [PubMed]

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 81, 051137 (2010).
[Crossref]

2009 (1)

2007 (2)

A. Tajima, A. Tanaka, W. Maeda, S. Takahashi, and A. Tomita, “Practical quantum cryptosystem for metro area applications,” IEEE J. Sel. Top. Quantum Electron. 13, 1031–1038 (2007).
[Crossref]

X. Cai and X. Wang, “Stochastic modeling and simulation of gene networks - a review of the state-of-the-art research on stochastic simulations,” IEEE Signal Process. Mag. 24, 27–36 (2007).
[Crossref]

2000 (2)

C. Petrie and J. Connelly, “A noise-based ic random number generator for applications in cryptography,” IEEE Trans. Circuits Syst. I, Fundam. Theory Appl. 47, 615–621 (2000).
[Crossref]

T. Jennewein, U. Achleitner, G. Weihs, H. Weinfurter, and A. Zeilinger, “A fast and compact quantum random number generator,” Rev. Sci. Intrum. 71, 1675–1680 (2000).
[Crossref]

1999 (2)

Y. Yoshizawa, H. Kimura, H. Inoue, K. Fujita, M. Toyama, and O. Miyatake, “Physical random numbers generated by radioactivity,” J. Jpn. Soc. Comput. Stat. 12, 67–81 (1999).

N. Nisan and A. Ta-Shma, “Extracting randomness: A survey and new constructions,” J. Comput. Sci. Tech. 58, 148–173 (1999).

1990 (1)

G. Agrawal, “Effect of gain and index nonlinearities on single-mode dynamics in semiconductor lasers,” IEEE J. Quantum Electron. 26, 1901–1909 (1990).
[Crossref]

1986 (1)

C. Henry, “Phase noise in semiconductor lasers,” J. Lightwave Technol. 4, 298–311 (1986).
[Crossref]

1982 (1)

C. Henry, “Theory of the linewidth of semiconductor lasers,” IEEE J. Quantum Electron. 18, 259–264 (1982).
[Crossref]

Achleitner, U.

T. Jennewein, U. Achleitner, G. Weihs, H. Weinfurter, and A. Zeilinger, “A fast and compact quantum random number generator,” Rev. Sci. Intrum. 71, 1675–1680 (2000).
[Crossref]

Acín, A.

S. Pironio, A. Acín, S. Massar, A. B. de la Giroday, D. N. Matsukevich, P. Maunz, S. Olmschenk, D. Hayes, L. Luo, T. A. Manning, and C. Monroe, “Random numbers certified by Bell’s theorem,” Nature 464, 1021–1024 (2010).
[Crossref] [PubMed]

Agrawal, G.

G. Agrawal, “Effect of gain and index nonlinearities on single-mode dynamics in semiconductor lasers,” IEEE J. Quantum Electron. 26, 1901–1909 (1990).
[Crossref]

Anzolin, G.

Argyris, A.

Aviad, Y.

I. Kanter, Y. Aviad, I. Reidler, E. Cohen, and M. Rosenbluh, “An optical ultrafast random bit generator,” Nat. Photonics 4, 58–61 (2010).
[Crossref]

Banks, D.

A. Rukhin, J. Soto, J. Nechvatal, E. Barker, S. Leigh, M. Levenson, M. Vangel, D. Banks, A. Heckert, J. Dray, and S. Vo, “A statistical test suite for random and pseudorandom number generators for cryptographic applications,” NIST Special Publication 800-22 revision 1a (2010).

Barker, E.

A. Rukhin, J. Soto, J. Nechvatal, E. Barker, S. Leigh, M. Levenson, M. Vangel, D. Banks, A. Heckert, J. Dray, and S. Vo, “A statistical test suite for random and pseudorandom number generators for cryptographic applications,” NIST Special Publication 800-22 revision 1a (2010).

Benson, O.

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, 171105 (2011).
[Crossref]

Berlin, M.

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, 171105 (2011).
[Crossref]

Byer, R. L.

Cai, X.

X. Cai and X. Wang, “Stochastic modeling and simulation of gene networks - a review of the state-of-the-art research on stochastic simulations,” IEEE Signal Process. Mag. 24, 27–36 (2007).
[Crossref]

Chan, S.-C.

Cho, Y.-W.

Cohen, E.

I. Kanter, Y. Aviad, I. Reidler, E. Cohen, and M. Rosenbluh, “An optical ultrafast random bit generator,” Nat. Photonics 4, 58–61 (2010).
[Crossref]

Connelly, J.

C. Petrie and J. Connelly, “A noise-based ic random number generator for applications in cryptography,” IEEE Trans. Circuits Syst. I, Fundam. Theory Appl. 47, 615–621 (2000).
[Crossref]

Curty, M.

de la Giroday, A. B.

S. Pironio, A. Acín, S. Massar, A. B. de la Giroday, D. N. Matsukevich, P. Maunz, S. Olmschenk, D. Hayes, L. Luo, T. A. Manning, and C. Monroe, “Random numbers certified by Bell’s theorem,” Nature 464, 1021–1024 (2010).
[Crossref] [PubMed]

Deligiannidis, S.

Dray, J.

A. Rukhin, J. Soto, J. Nechvatal, E. Barker, S. Leigh, M. Levenson, M. Vangel, D. Banks, A. Heckert, J. Dray, and S. Vo, “A statistical test suite for random and pseudorandom number generators for cryptographic applications,” NIST Special Publication 800-22 revision 1a (2010).

Fujita, K.

Y. Yoshizawa, H. Kimura, H. Inoue, K. Fujita, M. Toyama, and O. Miyatake, “Physical random numbers generated by radioactivity,” J. Jpn. Soc. Comput. Stat. 12, 67–81 (1999).

Guo, H.

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 81, 051137 (2010).
[Crossref]

Hall, C.

C. Hall and B. Schneier, “Remote electronic gambling,” in “13th Annual Computer Security Applications Conference” (1997), pp. 232–238.
[Crossref]

Hayes, D.

S. Pironio, A. Acín, S. Massar, A. B. de la Giroday, D. N. Matsukevich, P. Maunz, S. Olmschenk, D. Hayes, L. Luo, T. A. Manning, and C. Monroe, “Random numbers certified by Bell’s theorem,” Nature 464, 1021–1024 (2010).
[Crossref] [PubMed]

Heckert, A.

A. Rukhin, J. Soto, J. Nechvatal, E. Barker, S. Leigh, M. Levenson, M. Vangel, D. Banks, A. Heckert, J. Dray, and S. Vo, “A statistical test suite for random and pseudorandom number generators for cryptographic applications,” NIST Special Publication 800-22 revision 1a (2010).

Henry, C.

C. Henry, “Phase noise in semiconductor lasers,” J. Lightwave Technol. 4, 298–311 (1986).
[Crossref]

C. Henry, “Theory of the linewidth of semiconductor lasers,” IEEE J. Quantum Electron. 18, 259–264 (1982).
[Crossref]

Inoue, H.

Y. Yoshizawa, H. Kimura, H. Inoue, K. Fujita, M. Toyama, and O. Miyatake, “Physical random numbers generated by radioactivity,” J. Jpn. Soc. Comput. Stat. 12, 67–81 (1999).

Jennewein, T.

T. Jennewein, U. Achleitner, G. Weihs, H. Weinfurter, and A. Zeilinger, “A fast and compact quantum random number generator,” Rev. Sci. Intrum. 71, 1675–1680 (2000).
[Crossref]

Jofre, M.

Kanter, I.

I. Kanter, Y. Aviad, I. Reidler, E. Cohen, and M. Rosenbluh, “An optical ultrafast random bit generator,” Nat. Photonics 4, 58–61 (2010).
[Crossref]

Kim, Y.-H.

Kimura, H.

Y. Yoshizawa, H. Kimura, H. Inoue, K. Fujita, M. Toyama, and O. Miyatake, “Physical random numbers generated by radioactivity,” J. Jpn. Soc. Comput. Stat. 12, 67–81 (1999).

Kwon, O.

Leifgen, M.

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, 171105 (2011).
[Crossref]

Leigh, S.

A. Rukhin, J. Soto, J. Nechvatal, E. Barker, S. Leigh, M. Levenson, M. Vangel, D. Banks, A. Heckert, J. Dray, and S. Vo, “A statistical test suite for random and pseudorandom number generators for cryptographic applications,” NIST Special Publication 800-22 revision 1a (2010).

Leindecker, N. C.

Levenson, M.

A. Rukhin, J. Soto, J. Nechvatal, E. Barker, S. Leigh, M. Levenson, M. Vangel, D. Banks, A. Heckert, J. Dray, and S. Vo, “A statistical test suite for random and pseudorandom number generators for cryptographic applications,” NIST Special Publication 800-22 revision 1a (2010).

Li, L.

Li, P.

Li, X.

Li, X.-Z.

Liu, Y.

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 81, 051137 (2010).
[Crossref]

Lo, H.-K.

Luo, L.

S. Pironio, A. Acín, S. Massar, A. B. de la Giroday, D. N. Matsukevich, P. Maunz, S. Olmschenk, D. Hayes, L. Luo, T. A. Manning, and C. Monroe, “Random numbers certified by Bell’s theorem,” Nature 464, 1021–1024 (2010).
[Crossref] [PubMed]

Ma, X.

Maeda, W.

A. Tajima, A. Tanaka, W. Maeda, S. Takahashi, and A. Tomita, “Practical quantum cryptosystem for metro area applications,” IEEE J. Sel. Top. Quantum Electron. 13, 1031–1038 (2007).
[Crossref]

Manning, T. A.

S. Pironio, A. Acín, S. Massar, A. B. de la Giroday, D. N. Matsukevich, P. Maunz, S. Olmschenk, D. Hayes, L. Luo, T. A. Manning, and C. Monroe, “Random numbers certified by Bell’s theorem,” Nature 464, 1021–1024 (2010).
[Crossref] [PubMed]

Marandi, A.

Massar, S.

S. Pironio, A. Acín, S. Massar, A. B. de la Giroday, D. N. Matsukevich, P. Maunz, S. Olmschenk, D. Hayes, L. Luo, T. A. Manning, and C. Monroe, “Random numbers certified by Bell’s theorem,” Nature 464, 1021–1024 (2010).
[Crossref] [PubMed]

Matsukevich, D. N.

S. Pironio, A. Acín, S. Massar, A. B. de la Giroday, D. N. Matsukevich, P. Maunz, S. Olmschenk, D. Hayes, L. Luo, T. A. Manning, and C. Monroe, “Random numbers certified by Bell’s theorem,” Nature 464, 1021–1024 (2010).
[Crossref] [PubMed]

Maunz, P.

S. Pironio, A. Acín, S. Massar, A. B. de la Giroday, D. N. Matsukevich, P. Maunz, S. Olmschenk, D. Hayes, L. Luo, T. A. Manning, and C. Monroe, “Random numbers certified by Bell’s theorem,” Nature 464, 1021–1024 (2010).
[Crossref] [PubMed]

Mitchell, M. W.

Miyatake, O.

Y. Yoshizawa, H. Kimura, H. Inoue, K. Fujita, M. Toyama, and O. Miyatake, “Physical random numbers generated by radioactivity,” J. Jpn. Soc. Comput. Stat. 12, 67–81 (1999).

Monroe, C.

S. Pironio, A. Acín, S. Massar, A. B. de la Giroday, D. N. Matsukevich, P. Maunz, S. Olmschenk, D. Hayes, L. Luo, T. A. Manning, and C. Monroe, “Random numbers certified by Bell’s theorem,” Nature 464, 1021–1024 (2010).
[Crossref] [PubMed]

Murphy, T. E.

Nechvatal, J.

A. Rukhin, J. Soto, J. Nechvatal, E. Barker, S. Leigh, M. Levenson, M. Vangel, D. Banks, A. Heckert, J. Dray, and S. Vo, “A statistical test suite for random and pseudorandom number generators for cryptographic applications,” NIST Special Publication 800-22 revision 1a (2010).

Nisan, N.

N. Nisan and A. Ta-Shma, “Extracting randomness: A survey and new constructions,” J. Comput. Sci. Tech. 58, 148–173 (1999).

Olmschenk, S.

S. Pironio, A. Acín, S. Massar, A. B. de la Giroday, D. N. Matsukevich, P. Maunz, S. Olmschenk, D. Hayes, L. Luo, T. A. Manning, and C. Monroe, “Random numbers certified by Bell’s theorem,” Nature 464, 1021–1024 (2010).
[Crossref] [PubMed]

Petrie, C.

C. Petrie and J. Connelly, “A noise-based ic random number generator for applications in cryptography,” IEEE Trans. Circuits Syst. I, Fundam. Theory Appl. 47, 615–621 (2000).
[Crossref]

Pikasis, E.

Pironio, S.

S. Pironio, A. Acín, S. Massar, A. B. de la Giroday, D. N. Matsukevich, P. Maunz, S. Olmschenk, D. Hayes, L. Luo, T. A. Manning, and C. Monroe, “Random numbers certified by Bell’s theorem,” Nature 464, 1021–1024 (2010).
[Crossref] [PubMed]

Pruneri, V.

Qi, B.

Rahn, H.-J.

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, 171105 (2011).
[Crossref]

Reidler, I.

I. Kanter, Y. Aviad, I. Reidler, E. Cohen, and M. Rosenbluh, “An optical ultrafast random bit generator,” Nat. Photonics 4, 58–61 (2010).
[Crossref]

Röhlicke, T.

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, 171105 (2011).
[Crossref]

Rosenbluh, M.

I. Kanter, Y. Aviad, I. Reidler, E. Cohen, and M. Rosenbluh, “An optical ultrafast random bit generator,” Nat. Photonics 4, 58–61 (2010).
[Crossref]

Roy, R.

Rukhin, A.

A. Rukhin, J. Soto, J. Nechvatal, E. Barker, S. Leigh, M. Levenson, M. Vangel, D. Banks, A. Heckert, J. Dray, and S. Vo, “A statistical test suite for random and pseudorandom number generators for cryptographic applications,” NIST Special Publication 800-22 revision 1a (2010).

Salevan, J. C.

Schneier, B.

C. Hall and B. Schneier, “Remote electronic gambling,” in “13th Annual Computer Security Applications Conference” (1997), pp. 232–238.
[Crossref]

Soto, J.

A. Rukhin, J. Soto, J. Nechvatal, E. Barker, S. Leigh, M. Levenson, M. Vangel, D. Banks, A. Heckert, J. Dray, and S. Vo, “A statistical test suite for random and pseudorandom number generators for cryptographic applications,” NIST Special Publication 800-22 revision 1a (2010).

Steinlechner, F.

Syvridis, D.

Tajima, A.

A. Tajima, A. Tanaka, W. Maeda, S. Takahashi, and A. Tomita, “Practical quantum cryptosystem for metro area applications,” IEEE J. Sel. Top. Quantum Electron. 13, 1031–1038 (2007).
[Crossref]

Takahashi, S.

A. Tajima, A. Tanaka, W. Maeda, S. Takahashi, and A. Tomita, “Practical quantum cryptosystem for metro area applications,” IEEE J. Sel. Top. Quantum Electron. 13, 1031–1038 (2007).
[Crossref]

Tanaka, A.

A. Tajima, A. Tanaka, W. Maeda, S. Takahashi, and A. Tomita, “Practical quantum cryptosystem for metro area applications,” IEEE J. Sel. Top. Quantum Electron. 13, 1031–1038 (2007).
[Crossref]

Tang, W.

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 81, 051137 (2010).
[Crossref]

Ta-Shma, A.

N. Nisan and A. Ta-Shma, “Extracting randomness: A survey and new constructions,” J. Comput. Sci. Tech. 58, 148–173 (1999).

Tomita, A.

A. Tajima, A. Tanaka, W. Maeda, S. Takahashi, and A. Tomita, “Practical quantum cryptosystem for metro area applications,” IEEE J. Sel. Top. Quantum Electron. 13, 1031–1038 (2007).
[Crossref]

Torres, J. P.

Toyama, M.

Y. Yoshizawa, H. Kimura, H. Inoue, K. Fujita, M. Toyama, and O. Miyatake, “Physical random numbers generated by radioactivity,” J. Jpn. Soc. Comput. Stat. 12, 67–81 (1999).

Vangel, M.

A. Rukhin, J. Soto, J. Nechvatal, E. Barker, S. Leigh, M. Levenson, M. Vangel, D. Banks, A. Heckert, J. Dray, and S. Vo, “A statistical test suite for random and pseudorandom number generators for cryptographic applications,” NIST Special Publication 800-22 revision 1a (2010).

Vo, S.

A. Rukhin, J. Soto, J. Nechvatal, E. Barker, S. Leigh, M. Levenson, M. Vangel, D. Banks, A. Heckert, J. Dray, and S. Vo, “A statistical test suite for random and pseudorandom number generators for cryptographic applications,” NIST Special Publication 800-22 revision 1a (2010).

Vodopyanov, K. L.

Wahl, M.

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, 171105 (2011).
[Crossref]

Wang, A.

Wang, X.

X. Cai and X. Wang, “Stochastic modeling and simulation of gene networks - a review of the state-of-the-art research on stochastic simulations,” IEEE Signal Process. Mag. 24, 27–36 (2007).
[Crossref]

Wang, Y.

Wei, W.

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 81, 051137 (2010).
[Crossref]

Weihs, G.

T. Jennewein, U. Achleitner, G. Weihs, H. Weinfurter, and A. Zeilinger, “A fast and compact quantum random number generator,” Rev. Sci. Intrum. 71, 1675–1680 (2000).
[Crossref]

Weinfurter, H.

T. Jennewein, U. Achleitner, G. Weihs, H. Weinfurter, and A. Zeilinger, “A fast and compact quantum random number generator,” Rev. Sci. Intrum. 71, 1675–1680 (2000).
[Crossref]

Williams, C. R. S.

Xu, F.

Xu, H.

Yoshizawa, Y.

Y. Yoshizawa, H. Kimura, H. Inoue, K. Fujita, M. Toyama, and O. Miyatake, “Physical random numbers generated by radioactivity,” J. Jpn. Soc. Comput. Stat. 12, 67–81 (1999).

Zeilinger, A.

T. Jennewein, U. Achleitner, G. Weihs, H. Weinfurter, and A. Zeilinger, “A fast and compact quantum random number generator,” Rev. Sci. Intrum. 71, 1675–1680 (2000).
[Crossref]

Zhang, J.

Zheng, H.

Appl. Opt. (1)

Appl. Phys. Lett. (1)

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, 171105 (2011).
[Crossref]

IEEE J. Quantum Electron. (2)

C. Henry, “Theory of the linewidth of semiconductor lasers,” IEEE J. Quantum Electron. 18, 259–264 (1982).
[Crossref]

G. Agrawal, “Effect of gain and index nonlinearities on single-mode dynamics in semiconductor lasers,” IEEE J. Quantum Electron. 26, 1901–1909 (1990).
[Crossref]

IEEE J. Sel. Top. Quantum Electron. (1)

A. Tajima, A. Tanaka, W. Maeda, S. Takahashi, and A. Tomita, “Practical quantum cryptosystem for metro area applications,” IEEE J. Sel. Top. Quantum Electron. 13, 1031–1038 (2007).
[Crossref]

IEEE Signal Process. Mag. (1)

X. Cai and X. Wang, “Stochastic modeling and simulation of gene networks - a review of the state-of-the-art research on stochastic simulations,” IEEE Signal Process. Mag. 24, 27–36 (2007).
[Crossref]

IEEE Trans. Circuits Syst. I, Fundam. Theory Appl. (1)

C. Petrie and J. Connelly, “A noise-based ic random number generator for applications in cryptography,” IEEE Trans. Circuits Syst. I, Fundam. Theory Appl. 47, 615–621 (2000).
[Crossref]

J. Comput. Sci. Tech. (1)

N. Nisan and A. Ta-Shma, “Extracting randomness: A survey and new constructions,” J. Comput. Sci. Tech. 58, 148–173 (1999).

J. Jpn. Soc. Comput. Stat. (1)

Y. Yoshizawa, H. Kimura, H. Inoue, K. Fujita, M. Toyama, and O. Miyatake, “Physical random numbers generated by radioactivity,” J. Jpn. Soc. Comput. Stat. 12, 67–81 (1999).

J. Lightwave Technol. (2)

Nat. Photonics (1)

I. Kanter, Y. Aviad, I. Reidler, E. Cohen, and M. Rosenbluh, “An optical ultrafast random bit generator,” Nat. Photonics 4, 58–61 (2010).
[Crossref]

Nature (1)

S. Pironio, A. Acín, S. Massar, A. B. de la Giroday, D. N. Matsukevich, P. Maunz, S. Olmschenk, D. Hayes, L. Luo, T. A. Manning, and C. Monroe, “Random numbers certified by Bell’s theorem,” Nature 464, 1021–1024 (2010).
[Crossref] [PubMed]

Opt. Express (5)

Opt. Lett. (1)

Phys. Rev. E (1)

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 81, 051137 (2010).
[Crossref]

Rev. Sci. Intrum. (1)

T. Jennewein, U. Achleitner, G. Weihs, H. Weinfurter, and A. Zeilinger, “A fast and compact quantum random number generator,” Rev. Sci. Intrum. 71, 1675–1680 (2000).
[Crossref]

Other (3)

C. Hall and B. Schneier, “Remote electronic gambling,” in “13th Annual Computer Security Applications Conference” (1997), pp. 232–238.
[Crossref]

V. Rijmen and P. S. L. M. Barreto, “The WHIRLPOOL hash function,” http://www.larc.usp.br/~pbarreto/WhirlpoolPage.html (2008).

A. Rukhin, J. Soto, J. Nechvatal, E. Barker, S. Leigh, M. Levenson, M. Vangel, D. Banks, A. Heckert, J. Dray, and S. Vo, “A statistical test suite for random and pseudorandom number generators for cryptographic applications,” NIST Special Publication 800-22 revision 1a (2010).

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Figures (5)

Fig. 1
Fig. 1 Electrical and optical pulse trains. Magenta, (upper trace): electrical current drive applied to the laser, with PRF of 172 ps. Blue, (lower trace): photo-detected optical pulses of 85 ps time width and 7.65 mW peak power and (black, dashed line) 9 mA LD current threshold. Simulation is a conservative fitting of the rate equations such that the predicted detected output power vs. time is always larger than the observed output power vs. time.
Fig. 2
Fig. 2 Unbalanced Mach-Zehnder interferometer (U-MZI). Phase-randomized coherent optical pulses interfering at the output of the U-MZI produce random intensities. (Pulse driver) denotes the electrical pulse generator that directly modulates the laser, (LD) laser diode, (PMC) polarization maintaining coupler, (PMF) polarization maintaining fiber, (θ0–3) optical phases of different consecutive pulses, (θloop) phase introduced by the delay line and (PD) a fast photodetector.
Fig. 3
Fig. 3 In Fig. 3(a), input power distributions (left y axis) and the resultant output power distribution (right y axis). The visibility achieved for the interferometer is 0.9. The power distribution has clearly widened due to the random phase generated by amplified spontaneous emission (ASE). In Fig. 3(b), normalized correlation of 50 subsequent sampled pulses. The autocorrelation has been evaluated with 120 × 106 14-bit samples, but just the first 50 terms are shown. As expected, it follows a delta-function like behaviour indicating the random nature of the process.
Fig. 4
Fig. 4 Statistical characterization of 125 × 106 7-bit numbers produced by hashing the experimental data. (a) autocorrelation of the hashed data. Autocorrelation at the −40 dB level is seen, corresponding to the expected statistical variation. (b) deviation from the ideal 7-bit uniform distribution. Dashed lines show plus/minus 1 sigma expected variation.
Fig. 5
Fig. 5 Summary of the results of the NIST test suite to assess randomness. Dashed lines in Fig. 5(a) represent the confidence interval where the proportion of sequences accepted/rejected per test must fall in. In Fig. 5(b) is plot PvalueT = Γ(9/2, χ2/2) for each test. All PvalueT > 10−4. The smallest one is for the non-periodic template matching test, giving a value of PvalueT = 0.0926.

Equations (17)

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( out ) ( t ) = ε 11 ( 1 ) ε 11 ( 2 ) ( laser ) ( t t 1 ) + ε 12 ( 1 ) ε 21 ( 2 ) ( laser ) ( t t 2 ) ,
u ( out ) ( t ) = u 1 ( t ) + u 2 ( t ) + 2 | g ( 1 ) ( t ) | u 1 ( t ) u 2 ( t ) cos ( θ j θ j 1 + Δ ϕ ) + u noise ,
θ ˙ = α 2 G N ( n n th ) β SE 2 G N ( n n 0 ) p 1 + 1 + p + F θ ( t ) ,
d d t Δ θ ( t ) 2 = R sp 2 s ( 1 + α 2 ) ,
s ˙ = G N ( n n 0 1 + s / s sat ( n th n 0 ) ) s + R sp ,
n ˙ = I / q γ e n G N n n 0 1 + s / s sat s ,
γ ( 1 1 + s th / s sat 1 ) s th + R 0 γ e n th = 0 ,
I q γ e n th γ s th 1 1 + s th / s sat = 0 ,
H ( X ) log 2 ( max x i P [ X = x i ] ) ,
PDF ( x ) = 1 π ( x a ) ( b x ) ; μ x = a + b 2 ; var [ x ] = 1 8 ( b a ) 2 .
var ( u ( out ) ) = var ( u 1 ) + var ( u 2 ) + var ( u noise ) + 4 | g ( t loop ) | 2 var ( u 1 u 2 cos θ ) .
| g ( t loop ) | 2 var ( u ( out ) ) var ( u 1 ) var ( u 2 ) var ( u noise ) 2 E [ u 1 ] 2 E [ u 2 ] 2 ,
u min E [ u 1 + u 2 2 | g ( t loop ) | u 1 u 2 ] ,
u max E [ u 1 + u 2 + 2 | g ( t loop ) | u 1 u 2 ] .
P [ X = x 1 ] = 1 π u min u min + Δ u 1 ( u u min ) ( u max u ) d u = 2 π arcsin A ADC 2 b ( u max u min ) .
H ( X ) b 2 1 2 log 2 ( 4 A ADC π 2 ( u max u min ) ) .
χ 2 = s 10 i = 1 10 ( F i s 10 ) 2 ,

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