Ultra-fast quantum randomness generation by accelerated phase diffusion in a pulsed laser diode

C. Abellán; W. Amaya; M. Jofre; M. Curty; A. Acín; J. Capmany; V. Pruneri; M. W. Mitchell

doi:10.1364/OE.22.001645

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 [11–13], 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, (θ_0–3) optical phases of different consecutive pulses, (θ_loop) 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

ℰ^{(out)} (t) = ε_{11}^{(1)} ε_{11}^{(2)} ℰ^{(laser)} (t - t_{1}) + ε_{12}^{(1)} ε_{21}^{(2)} ℰ^{(laser)} (t - t_{2}),

where

ε_{i j}^{(k)}

is the field transmission coefficient of the kth PMC from input port i to output port j, ℰ^(laser) is the field emitted by the laser and t_1,2 are the delays caused by arms 1,2 of the U-MZI, respectively. For convenience, we write ℰ^(laser)(t) = A(t)exp[−iωt]exp[iθ_j] where ω is the central frequency of the laser emission and θ_j is the phase of the jth pulse. Defining the powers u^(out)(t) ≡ |𝒠^(out)(t)|²,

u_{1} (t) \equiv {| ε_{11}^{(1)} ε_{11}^{(2)} ℰ^{(laser)} (t - t_{1}) |}^{2}

and

u_{2} (t) \equiv {| ε_{12}^{(1)} ε_{21}^{(2)} ℰ^{(laser)} (t - t_{2}) |}^{2}

we have the observed signal

u^{(out)} (t) = u_{1} (t) + u_{2} (t) + 2 | g^{(1)} (t) | \sqrt{u_{1} (t) u_{2} (t)} cos (θ_{j} - θ_{j - 1} + Δ ϕ) + u_{noise},

where Δϕ ≡ ω(t₁ − t₂) is the relative phase delay of the two arms,

g^{(1)} (t) \equiv 〈 ℰ^{* (laser)} (t - t_{1}) ℰ^{(laser)} (t - t_{2}) 〉 / \sqrt{〈 {| ℰ^{(laser)} (t - t_{1}) |}^{2} 〉 〈 {| ℰ^{(laser)} (t - t_{2}) |}^{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. u_noise is noise contribution from the detection and digitization electronics. Fluctuation of Δϕ due to changes in t₁ − t₂, 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 [19–21], where an expression for the average phase diffusion 〈Δθ²(t)〉 is derived under various conditions from the Langevin equation

\dot{θ} = \frac{α}{2} G_{N} (n - n_{th}) - \frac{β_{SE}}{2} \frac{G_{N} (n - n_{0}) p}{1 + \sqrt{1 + p}} + F_{θ} (t),

where F_θ 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]

\frac{d}{d t} 〈 Δ θ {(t)}^{2} 〉 = \frac{R_{sp}}{2 s} (1 + α^{2}),

gives a lower bound on the phase diffusion in an interval of time t. Here R_sp 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 R_sp and s are described by differential equations, given in [21], Eqs. (13)–(17), which couple s to n.

\dot{s} = G_{N} (\frac{n - n_{0}}{\sqrt{1 + s / s_{sat}}} - (n_{th} - n_{0})) s + R_{sp},

\dot{n} = I / q - γ_{e} n - G_{N} \frac{n - n_{0}}{\sqrt{1 + s / s_{sat}}} s,

where G_N is the gain per carrier, n₀ is the number of carriers at transparency, n_th is the number of carriers at threshold, s_sat is the saturation photon number, γ_e is the carrier recombination lifetime = 1/τ_e ≈ 1 × 10⁹, I the injection current and q is the electronic charge. To give values to these parameters, we note that G_N = γ/(n_th − n₀), where γ = c(α_m + α_s)/n̄ is the cavity decay rate where c is the speed of light in vacuum, n̄ = 4.33 is the effective refractive index, α_s = 4.5 cm⁻¹ describes losses due to scattering and absorption and α_m ≈ 1.4/L is the cavity escape loss, where L is the cavity length. Because s_sat, L and n₀ are unknown to us, we choose them by fitting to the observed pulse-shape (see below). R_sp can be written R_sp = nγ_eR₀, where R₀ = K_totΓ_confβ_SE is a constant containing Γ_conf the “confinement factor”, β_SE the fraction of spontaneous emission coupled to the lasing mode, and K_tot 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 R_sp 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 s_sat, L, n_th, G_N, 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 τ_f = 0.35/BW_osc, where BW_osc = 12.5 GHz is the bandwidth of the oscilloscope.

A recursive method is used to extract these parameters:

We set s_sat to a reasonable value, and L to one of 100, 200, 500 or 1000 μm.
We solve the steady-state solution of the differential equations, Eq. (7) and Eq. (8), near threshold, i.e., with n ≈ n_th. The DFB LD emits s_th′ = 0.3 mW when it is biased with a constant current of I_th′ = 10 mA. (The subscript th_′ refers to steady-state solution slightly above threshold). Eqs. (5) and (6) then take the form $γ (\frac{1}{\sqrt{1 + s_{{th}^{'}} / s_{sat}}} - 1) s_{{th}^{'}} + R_{0} γ_{e} n_{th} = 0,$ $\frac{I}{q} - γ_{e} n_{th} - γ s_{{th}^{'}} \frac{1}{\sqrt{1 + s_{{th}^{'}} / s_{sat}}} = 0,$ from which we extract n_th and R₀.
We choose the maximum G_N, 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 G_N directly specifies n₀, via G_N = γ/(n_th − n₀).
We repeat steps (1)–(3) to find the values of s_sat and L that minimize the rms deviation of the simulation from the observed power curves.

After these steps, s_sat = 7.7 × 10⁵, L = 500 μm, n_th = 5.62 × 10⁷, R₀ = 8.8 × 10⁻⁴, G_N = 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 θ → θ + 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 $G_{π} (θ) \equiv \sum_{s = \pm 1} \sum_{n = - \infty}^{\infty} G (s θ + 2 π n)$ for 0 ≤ θ < π and G_π(θ) ≡ 0 otherwise. Already with 〈Δθ²〉 = (2π)², G_π 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 = x_i], the min-entropy

H_{\infty} (X) \equiv - {log}_{2} (max_{\forall x_{i}} P [X = x_{i}]),

quantifies the amount of extractable randomness, in the sense that from a sequence {X₁,...,X_N} 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 $\sqrt{u_{1} u_{2}} cos ϕ$ , 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, $X_{3} = \sqrt{u_{1} u_{2}} cos θ$ and X₄ = u_noise, where X₃ is the product of three random variables $\sqrt{u_{1}}$ , $\sqrt{u_{2}}$ and cosθ. Assuming that u₁, u₂, cosθ and u_noise are independent, does not imply that the four random variables X₁, X₂, X₃, X₄ are independent. In fact, $\sqrt{u_{1} u_{2}} cos θ$ 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) = \frac{1}{π \sqrt{(x - a) (b - x)}}; μ_{x} = \frac{a + b}{2}; var [x] = \frac{1}{8} {(b - a)}^{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 (u_{1}) + var (u_{2}) + var (u_{noise}) + 4 {| g (t_{loop}) |}^{2} var (\sqrt{u_{1} u_{2}} cos θ) .

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 (t_{loop}) |}^{2} \approx \frac{var (u^{(out)}) - var (u_{1}) - var (u_{2}) - var (u_{noise})}{2 E {[\sqrt{u_{1}}]}^{2} E {[\sqrt{u_{2}}]}^{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, $E {[\sqrt{u_{1}}]}^{2} = 0.97 mW$ , and $E {[\sqrt{u_{2}}]}^{2} = 0.90 mW$ . From measurements with the laser off, we find the contribution of detection and digitization noise var(u_noise) = 1.45 × 10⁻⁴ mW². Applying Eq. (12) we find |g(t_loop)| = 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 u_min and u_max, where

u_{min} \equiv E [u_{1} + u_{2} - 2 | g (t_{loop}) | \sqrt{u_{1} u_{2}}],

u_{max} \equiv E [u_{1} + u_{2} + 2 | g (t_{loop}) | \sqrt{u_{1} u_{2}}] .

Because the arcsine distribution is peaked at x = u_min, 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 A_ADC is the dynamic range of the analog-to-digital converter (ADC), b its resolution in bits so that Δu = A_ADC/2^b is the bin size, the probability mass of the first bin is

\begin{array}{l} P [X = x_{1}] & = & \frac{1}{π} \int_{u_{min}}^{u_{min} + Δ u} \frac{1}{\sqrt{(u - u_{min}) (u_{max} - u)}} d u \\ = & \frac{2}{π} arcsin \sqrt{\frac{A_{ADC}}{2^{b} (u_{max} - u_{min})} .} \end{array}

Using the fact that arcsinx ≈ x 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_{\infty} (X) \approx \frac{b}{2} - \frac{1}{2} {log}_{2} (\frac{4 A_{ADC}}{π^{2} (u_{max} - u_{min})}) .

In the experiment, where A_ADC = 5 mW, b = 14 bits,

u_{max} - u_{min} = 4 | g (t_{loop}) | \sqrt{E [u_{1}] E [u_{2}]} = 3.34 mW

, 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 δ_eu = P_i − 1/128, where P_i 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 (α_SL) 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 P_{value_T} = Γ(9/2, χ²/2) for each test. All P_{value_T} > 10⁻⁴. The smallest one is for the non-periodic template matching test, giving a value of P_{value_T} = 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 $[1 - α_{SL} \pm 3 \sqrt{(1 - α_{SL}) α_{SL} / m}]$ , 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 P_{value_T}, 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).

χ^{2} = \frac{s}{10} \sum_{i = 1}^{10} {(F_{i} - \frac{s}{10})}^{2},

where s is the number of P-values per test and F_i 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.

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Ultra-fast quantum randomness generation by accelerated phase diffusion in a pulsed laser diode

Abstract

1. Introduction

2. System configuration

3. Principle of operation

4. Phase diffusion

5. Raw data analysis

6. Randomness extraction and post-processing analysis

6.1. Procedure

6.2. Min-entropy estimation

6.3. Statistical testing

7. Conclusions

Acknowledgments

References and links

Cited By

Figures (5)

Equations (17)

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