Entropy rate of chaos in an optically injected semiconductor laser for physical random number generation

Yu Kawaguchi; Tomohiko Okuma; Kazutaka Kanno; Atsushi Uchida

doi:10.1364/OE.411694

1. Introduction

Random number generation is an essential technology for large-scale numerical simulations and information security. Physical random numbers are very important for achieving irreproducibility and unpredictability in information security applications. Several techniques of physical random number generation have been proposed in the last decade using chaotic semiconductor lasers [1–20], quantum noise [21–25], and spontaneous emission noise [26–30]. In particular, optically injected chaotic semiconductor lasers have been used as fast entropy sources for physical random number generation [10–17]. The use of chaotic semiconductor lasers has an advantage in terms of the generation speed, which exceeds one terabit per second (Tb/s) [16–19]. In addition, compact physical random number generators have been experimentally demonstrated using photonic integrated circuits [31–36].

An important issue regarding physical random number generation is evaluation of the entropy rate, which is the rate of producing uncertainty (i.e., unpredictability). Fast random number generation has been demonstrated without careful consideration of the entropy rate of physical entropy sources. Several attempts to evaluate the entropy rate for physical random number generation have been proposed, such as the use of the Lyapunov exponents of a reconstructed attractor [37], the Kolmogorov–Sinai (KS) entropy estimated from linearized model equations [38,39], permutation entropy from chaotic time series [40–43], block entropy [44], statistical tests of entropy sources known as NIST Special Publication 800–90B [45,46], and (ɛ, τ) entropy [47,48].

The (ɛ, τ) entropy is a method for estimating entropy from chaotic temporal waveforms that are quantized by the vertical resolution ɛ and the sampling time τ [49,50]. The (ɛ, τ) entropy strongly depends on the values of ɛ and τ, which is the case for random bits generated by discretizing chaotic temporal waveforms from entropy sources. Therefore, the (ɛ, τ) entropy is a suitable measure for estimating the entropy rate for random number generation using chaotic entropy sources. By contrast, the KS entropy obtained from the sum of positive Lyapunov exponents indicates the maximum entropy of high-dimensional dynamical systems. The KS entropy is calculated from linearized equations by numerical simulations [20]; it is difficult to estimate accurately from experimental time series. Thus, the (ɛ, τ) entropy is a good candidate for evaluating the entropy of experimental data.

However, calculating the (ɛ, τ) entropy is time-consuming for high-dimensional systems because the entropy is evaluated from all the variables of the chaotic attractor in the phase space. More simplified methods have been proposed, such as the approximate entropy and sample entropy [51,52]. The temporal waveforms of only one variable are required to calculate the sample entropy, which can significantly reduce the calculation time.

Despite the existence of a wide variety of studies on entropy evaluation, no comprehensive study on the correspondence between the entropy rate and the rate of random number generation has yet been reported, except for the simplest measure of the min-entropy [36]. Specifically, how much entropy can be extracted from physical entropy sources and how to generate random bit sequences without losing the entropy of physical entropy sources have not been clarified.

In this study, we evaluate the (ɛ, τ) entropy of a chaotic time series in an optically injected semiconductor laser to certify physical random number generation. We optimize the parameter values of the vertical resolution ɛ and the sampling time τ by comparing the (ɛ, τ) entropy with the KS entropy in numerical simulations. We also evaluate the (ɛ, τ) entropy of the chaotic time series obtained experimentally from the optically injected laser system. Finally, we investigate a method to generate random bit sequences that have an entropy of one bit per sample from a chaotic time series.

2. Numerical results

2.1 Numerical model and chaotic temporal waveforms

First, we present a numerical model for an optically injected chaotic semiconductor laser, a schematic of which is illustrated in Fig. 1. The stable output of one laser (Laser 1) is unidirectionally injected into the other laser (Laser 2) through an optical isolator. The dynamics of the output of Laser 2 can be controlled by changing the optical injection strength κ_inj and the optical frequency detuning Δf between Lasers 1 and 2 (i.e., Δf = f₁ – f₂, where f₁ and f₂ are the optical carrier frequencies of Lasers 1 and 2, respectively). Chaotic temporal waveforms of the laser output can be obtained when Δf is set close to the relaxation oscillation frequency f_r of Laser 2 for an intermediate value of κ_inj. We select this optically injected laser system because of its relatively low dimensionality (i.e., six variables in the coupled semiconductor laser system), rather than using a semiconductor laser with time-delayed optical feedback (i.e., infinite dimensionality in general).

Fig. 1. Schematic of optically injected chaotic semiconductor laser.

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The dynamics of the optically injected semiconductor laser (Laser 2) can be expressed by the rate equations as follows [53–55]:

(1)$$\frac{{dE(t )}}{{dt}} = \frac{{1 + i\alpha }}{2}\left[ {\frac{{{G_N}({N(t )- {N_0}} )}}{{1 + {\varepsilon_s}{{|{E(t )} |}^2}}} - \frac{1}{{{\tau_p}}}} \right]E(t )+ {\kappa _{inj}}{A_s}\textrm{exp}[{i({2\pi \mathrm{\Delta }ft - {\omega_1}{\tau_{inj}}} )} ]$$

(2)$$\frac{{dN(t )}}{{dt}} = {J_2} - \frac{{N(t )}}{{{\tau _s}}} - \frac{{{G_N}({N(t )- {N_0}} )}}{{1 + {\varepsilon _s}{{|{E(t )} |}^2}}}{|{E(t )} |^2}$$

where E(t) is the complex electric field and N(t) is the carrier density. The optical intensity I(t) and the optical phase ϕ(t) are calculated from the complex electric field E(t) = E_Re(t) + i E_Im(t), where I(t) = E_Re² + E_Im² and ϕ(t) = arctan(E_Im / E_Re). G_N is the gain coefficient, N₀ is the carrier density at transparency, ɛ_s is the gain saturation coefficient, τ_p is the photon lifetime, and τ_s is the carrier lifetime.

The parameter values are set as follows: The injection currents of Lasers 1 and 2 are j₁ = 1.11 and j₂ = 1.36, respectively, where j_i = J_i/J_th,i, J_i is the injection current and J_th,i is the lasing threshold for Laser i. The distance between Lasers 1 and 2 is 1.2 m. The propagation delay time of light from Lasers 1 to 2 is τ_inj = 4.003 ns. The injection strength from Laser 1 to 2 is κ_inj = 10.0 ns^–1 (variable). The optical wavelength and angular optical frequency of Laser 1 are λ₁ = 1537 nm and ω₁ = 1.226 ×10¹⁵ s^–1, respectively. The detuning of the initial optical frequencies between Lasers 1 and 2 is Δf = f₁ – f₂ = 1.3 GHz. The linewidth enhancement factor is set to α = 3.0. Other parameter values are set to be the same as in [54,55].

The second right-hand side term in Eq. (1) represents the optical injection from Lasers 1 to 2. The output of Laser 1 is in a steady state and the electric field amplitude A_s of Laser 1 is expressed as follows [39]:

(3)$${A_s} = \sqrt {\frac{{{G_N}{\tau _p}{N_{th}}({{j_1} - 1} )}}{{{G_N}{\tau _s} + {\varepsilon _s}}}}$$

where N_th is the carrier density at the lasing threshold [54,55].

We numerically calculate the temporal waveforms of the Laser 2 outputs using the Lang–Kobayashi equations. Figure 2 shows the numerical results of the temporal waveforms of the optical intensity I(t) calculated from the numerical model. We vary the optical injection strength κ_inj, while the optical frequency detuning is fixed at Δf = 1.3 GHz. A quasiperiodic oscillation is observed for κ_inj = 9.0 ns^–1, as in Fig. 2(a). As κ_inj increases, chaotic temporal waveforms are obtained, as shown in Figs. 2(b)–2(d). Here, more irregular oscillations can be observed for larger κ_inj. Chaotic oscillations are obtained in the range 9.2 ns^–1 ≤ κ_inj ≤ 13.5 ns^–1. We focus on the chaotic oscillations in this range and evaluate the (ɛ, τ) entropy (see Section 2.4).

Fig. 2. Numerical results of the temporal waveforms of the optical intensity for different optical injection strengths κ_inj: (a) κ_inj = 9.0 ns^–1, (b) 10.0 ns^–1, (c) 11.5 ns^–1, and (d) 13.5 ns^–1. The optical frequency detuning is fixed at Δf = 1.3 GHz.

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2.2 Evaluation method of (ɛ, τ) entropy

In this subsection, we evaluate the (ɛ, τ) entropy in an optically injected laser system. We optimize the vertical resolution ɛ and sampling time τ. The evaluation method of the (ɛ, τ) entropy can be explained as follows:

We calculate the (ɛ, τ) entropy introduced by Cohen and Procassia [49,50]. A chaotic temporal waveform sampled at the sampling time τ as well as a vector of length d of the sampled temporal waveform is considered as follows [50]:

(4)$${{\boldsymbol x}_i} = [{x({i\tau } ),\; x({({i + 1} )\tau } ),x({({i + 2} )\tau } ), \cdots ,\; x({({i + d - 1} )\tau } )} ]\; \; \; ({1 \le i \le L} )$$

where L is the total number of vectors. The distance between the two vectors is defined as follows:

(5)$$dis{t_d}[{{{\boldsymbol x}_i},{{\boldsymbol x}_j}} ]= \mathop {\max }\limits_{} [{|{x({i\tau } )- x({j\tau } )} |,\; \cdots ,\; |{x({({i + d - 1} )\tau } )- x({({j + d - 1} )\tau } )} |} ]$$

where max[x_i] returns the maximum value of the elements in the vector x_i. We randomly select a reference vector x_r (r ${\in} $ {r₁,…, r_R}) from all vectors x_i. In addition, we calculate the number of vectors L’, whose distance is less than ɛ from the reference x_r.

(6)$$L^{\prime} = Number({dis{t_d}[{{{\boldsymbol x}_r},{{\boldsymbol x}_i}} ]< \varepsilon } )\; \; \; \; \; \; \; ({1 \le i \le L} )$$

More specifically, we count the number of vectors x_i within an ɛ-sized d-dimensional hypercube from the reference vector x_r. Here, we include the reference vector in L’ to avoid L’ = 0. We calculate the ratio of L’ to the total number of vectors L.

(7)$$P({{{\boldsymbol x}_r};\varepsilon ,\tau ,d} )= \frac{{L^{\prime}}}{L}$$

P(x_r; ɛ, τ, d) represents the probability of the existence of two vectors within the distance ɛ. We repeat this procedure for the R reference vectors. We then calculate the correlation integral H(ɛ, τ, d) as the logarithm of the averaged P(x_r; ɛ, τ, d) over R.

(8)$$H({\varepsilon ,\tau ,d} )={-} \frac{1}{R}\mathop \sum \limits_{r = 1}^R lo{g_2}\; P({{{\boldsymbol x}_r}:\varepsilon ,\tau ,d} )$$

We use the binary logarithm (base-2 logarithm) to calculate the entropy rate in units of bits per second for the entropy of random number generation.

The entropy rate h_CP(ɛ, τ) can be calculated from the difference between the correlation integrals between d and d + 1.

(9)$${h_{CP}}({\varepsilon ,\; \tau } )= \frac{1}{\tau }\; \mathop {\lim }\limits_{d \to \infty } \mathop {\lim }\limits_{R,L^{\prime} \to \infty } [{H({\varepsilon ,\tau ,d + 1} )- H({\varepsilon ,\tau ,d} )} ]$$

The entropy rate h_CP(ɛ, τ) (entropy per time unit) is calculated by multiplying the entropy with the sampling frequency 1/τ.

Equation (9) indicates that entropy can be generated if neighboring vectors within the distance ɛ exist at d but vanish at d + 1. If the number of neighboring vectors within the distance ɛ remains constant between d and d + 1, no entropy is generated. Therefore, entropy represents the ratio of the loss of neighboring vectors as d increases (i.e., as the temporal waveform becomes longer).

We introduce the normalized vertical resolution ɛ_N instead of ɛ as follows:

(10)$$\varepsilon = \sigma \times {2^{{\varepsilon _N}}}$$

where σ is the standard deviation of the chaotic temporal waveform used for the entropy evaluation.

To verify the correctness of the evaluation of the (ɛ, τ) entropy, we also calculate the KS entropy h_KS from the sum of positive Lyapunov exponents [20,54].

(11)$${h_{KS}} = \mathop \sum \limits_{i|\lambda \rangle 0} {\lambda _i}$$

Several Lyapunov exponents exist in multidimensional nonlinear dynamical systems, which are called the Lyapunov spectrum. We calculate the Lyapunov spectrum using sets of linearized equations derived from Eqs. (1) and (2) [20,54]. The linearized variables can be considered as a vector for each linearized equation; then, the vectors for the sets of linearized equations are orthogonalized. The Lyapunov spectrum can be obtained by calculating the time average of the logarithm of the norm of the orthogonal vectors [20,54]. We consider the KS entropy as a reference to justify the evaluation of the (ɛ, τ) entropy. We also use the binary logarithm for the KS entropy for comparison with the (ɛ, τ) entropy (i.e., bits per second).

2.3 Numerical results of evaluation of (ɛ, τ) entropy

First, we evaluate the (ɛ, τ) entropy of the chaotic temporal waveform at κ_inj = 10.0 ns^–1, as shown in Fig. 2(b). We investigate the influence of the sampling resolution ɛ_N and temporal waveform length d on the (ɛ, τ) entropy. We use one million data points (one-mega sample) for evaluation of the (ɛ, τ) entropy. Figure 3(a) shows the (ɛ, τ) entropy as ɛ_N varies for different d. For a smaller ɛ_N, the entropy rate saturates at a single value, except for d = 10. Thus, ɛ_N must be small to obtain saturation of the entropy rate. In addition, the entropy rate approaches the KS entropy for larger values of d (denoted by the dotted line in Fig. 3(a)). This result indicates that d must be sufficiently large, and a longer time series is required to obtain the correct value of the (ɛ, τ) entropy.

Fig. 3. Numerical results of the (ɛ, τ) entropy as (a) ɛ_N is varied for different d, and (b) d is varied for different ɛ_N. The dotted line indicates the KS entropy h_KS.

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Figure 3(b) shows the (ɛ, τ) entropy as d is varied for different values of ɛ_N. As d increases, the value of the (ɛ, τ) entropy decreases gradually, approaching the value of the KS entropy for larger values of d and smaller values of ɛ_N. Therefore, a large d and small ɛ_N are suitable for evaluating the (ɛ, τ) entropy. In fact, an excessively small ɛ_N results in a decrease in the entropy rate owing to the lack of sampled data, as shown in Fig. 3(a). Therefore, an intermediate value of ɛ_N is suitable for evaluating the (ɛ, τ) entropy. We use ɛ_N = 0 (i.e., ɛ = σ, the standard deviation of the temporal waveform), and d = 100 for the following calculations.

Next, we investigate the influence of the sampling time τ on the (ɛ, τ) entropy. Figure 4(a) shows the (ɛ, τ) entropy as d is varied for different 1/τ values (denoted as the sampling frequency in Giga-Samples/second (GS/s)). As d increases, the entropy rate decreases monotonically. For small 1/τ, the entropy approaches zero with increasing d because of the lack of sampled data in the ɛ-sized hypercube. By contrast, for a large 1/τ, the entropy rate decreases, although the value does not saturate at d = 100. We speculate that longer temporal waveforms are required to achieve saturation of the entropy rate. We select 1/τ = 50 GS/s (τ = 20 ps) to obtain the optimal value of the entropy rate for d = 100, which is close to the KS entropy (denoted by the dotted line in Fig. 4(a)).

Fig. 4. Numerical results of the (ɛ, τ) entropy as (a) d is varied for different 1/τ (sampling frequency in Giga-Samples/second, GS/s), and (b) d is varied for different data length. The dotted line indicates the KS entropy h_KS.

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We also vary the length of the chaotic time series used for the entropy evaluation. Figure 4(b) shows the entropy rate as d is varied for different data lengths. The entropy rate decreases monotonically and saturates to the value of the KS entropy for longer data lengths. Therefore, a longer data length is preferable, although entropy evaluation is time-consuming. We select a data length of one-mega point (L = 10⁶).

From these results, the parameter values used for the evaluation of the (ɛ, τ) entropy can be summarized as follows: ɛ_N = 0 (ɛ = σ), τ = 20 ps, d = 100, R = 5000, and L = 10⁶.

We also calculate the (ɛ, τ) entropy under different conditions. All the variables of the dynamical model (e.g., the optical intensity I(t), optical phase ϕ(t), and carrier density N(t) in Eqs. (1) and (2)) are used to evaluate the (ɛ, τ) entropy. However, it may be useful to evaluate the entropy with respect to only one variable used for random number generation when the entropy of random number generation is estimated. Therefore, we compare the three cases of the evaluation of the (ɛ, τ) entropy using (i) the three variables (I(t), ϕ(t), and N(t)) in the phase space, (ii) attractor reconstruction using time-delayed embedding of one variable I(t), and (iii) one variable I(t).

Figure 5 shows the results of the comparison of the (ɛ, τ) entropy calculated from these three cases. The (ɛ, τ) entropy obtained from the three variables shows the maximum value for all d because the complexity of the three variables is included. By contrast, the (ɛ, τ) entropy obtained from the attractor reconstruction and that from one variable appear very similar because only the dynamics of one variable are used. More interestingly, all three values converge to the KS entropy for d = 100. We speculate that the optically injected laser system only has one positive Lyapunov exponent, and the entropy is approximately the same between the cases of all the variables and one variable. From these results, we consider only one variable (the optical intensity I(t)) for the evaluation of the (ɛ, τ) entropy in the following sections. This evaluation method of the (ɛ, τ) entropy matches the scheme for random number generation, where one variable (the optical intensity) is used to create random bit sequences.

Fig. 5. Comparison of the (ɛ, τ) entropy calculated from the attractors obtained from the three variables (black curve), time-delayed embedding from one variable I(t) (red curve), and the time series from one variable I(t) (blue curve). The dotted line indicates the KS entropy h_KS.

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2.4 Parameter dependence of (ɛ, τ) entropy

In this subsection, we evaluate the (ɛ, τ) entropy of chaos for different laser parameter values. We vary the optical injection strength κ_inj of the semiconductor laser to obtain different chaotic waveforms and evaluate the (ɛ, τ) entropy for these chaotic temporal waveforms. For comparison, we calculate the KS entropy from the Lyapunov spectrum, as obtained from the linearized forms of the original rate equations [20,54].

Figure 6(a) shows the (ɛ, τ) entropy for the chaotic temporal waveforms of the laser outputs with different κ_inj values. The entropy rate increases with an increase in κ_inj, and exhibits the maximum entropy rate of 3.8 Gb/s at κ_inj = 11.4 ns^–1. Meanwhile, the entropy rate becomes almost zero at κ_inj = 10.1 and 12.4 ns^–1, because of the existence of periodic windows in the bifurcation diagram. For comparison, the black dotted curve in Fig. 6(a) represents the KS entropy, and the values of the (ɛ, τ) entropy agree well with those of the KS entropy. Figure 6(b) shows the correlation plot between the (ɛ, τ) entropy and KS entropy, as obtained from the data in Fig. 6(a). A very high correlation can be observed, with a correlation value of 0.983. From these results, we confirm that the (ɛ, τ) entropy can match the KS entropy; the (ɛ, τ) entropy is a good metric for the entropy of chaotic temporal waveforms.

Fig. 6. (a) Numerical results of the (ɛ, τ) entropy (red solid curve) and the KS entropy (black dotted curve) as a function of the optical injection strength κ_inj. (b) Correlation plot between the (ɛ, τ) entropy and the KS entropy obtained from (a).

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Next, we investigate the influence of spontaneous emission noise in chaotic temporal waveforms on the (ɛ, τ) entropy for evaluation of the experimental data. We add white Gaussian noise ξ(t) to the electric field amplitude of Eq. (1). ξ(t) is white Gaussian noise with the properties < ξ(t) > = 0 and < ξ(t₀) ξ(t) > = 2Dδ(t – t₀), where < > denotes the ensemble average and δ(t) is the Dirac delta function. The amount of noise can be controlled by changing the noise strength D. We measure the signal-to-noise ratio (SNR) to determine the amount of noise.

Figure 7 shows the (ɛ, τ) entropy as the injection strength is varied. For an SNR of 30 dB, the values of the (ɛ, τ) entropy increase compared with the case without noise (green curve). However, the tendency of the (ɛ, τ) entropy is similar in the cases with and without noise, except in the periodic windows because the periodic oscillations become irregular owing to the noise. As the amount of noise increases (as the SNR decreases), the (ɛ, τ) entropy increases, and the curve in Fig. 7 flattens. This result indicates that the dynamics tend to be governed by noise, and less parameter dependence is found. From these results, we can estimate the (ɛ, τ) entropy from chaotic temporal waveforms even in the presence of noise, which affects its values.

Fig. 7. Numerical results of the (ɛ, τ) entropy as a function of the optical injection strength κ_inj in the presence of noise. The curves represent the cases of no noise (green) and SNR values of 30 dB (black), 25 dB (red), and 20 dB (blue).

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3. Experimental results

3.1 Experimental setup

In this section, we evaluate the (ɛ, τ) entropy from the experimentally obtained chaotic temporal waveforms. Figure 8 shows the experimental setup for the optically injected semiconductor laser. Two distributed-feedback (DFB) semiconductor lasers with fiber pigtails are used (NTT Electronics, KELD1C5GAAA, 1548-nm wavelength), referred to as Lasers 1 and 2. The stable output of Laser 1 is unidirectionally injected into Laser 2 through an optical isolator and optical attenuator to generate chaotic outputs. The optical injection strength from Lasers 1 to 2 is measured using an optical power meter at one of the fiber pigtails of the fiber coupler in front of Laser 2. The output of Laser 2 is divided into two outputs via the fiber coupler, whereas one of the outputs is measured using a photodetector (New Focus, 1554-B, 12 GHz bandwidth). The converted electronic signals at the photodetector are sent to a digital oscilloscope (Tektronix, DPO73304D, 33-GHz bandwidth, 100 GS/s) and a radio-frequency (RF) spectrum analyzer (Agilent, N9010A-544, 44-GHz bandwidth) to measure the temporal waveforms and the RF spectra of the laser outputs, respectively. The other output is sent to an optical spectrum analyzer (Yokogawa, AQ6370D) to measure the optical spectra of the laser outputs. We use polarization-maintaining (PM) fiber components to match the polarization direction of light in our experiment.

Fig. 8. Experimental setup for optically injected chaotic semiconductor laser.

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The injection currents of Laser 1 and 2 are set to 13.51 mA (1.50 I_th,1) and 13.34 mA (1.55 I_th,2), respectively, where I_th,1 = 9.0 mA and I_th,2 = 8.6 mA are the lasing thresholds of Lasers 1 and 2, respectively. The center wavelengths of the outputs of Lasers 1 and 2 are set to 1548.126 and 1548.107 nm, respectively. The wavelength detuning Δλ = λ₂ − λ₁ between the two lasers is –0.019 nm, corresponding to the frequency detuning Δf = f₁ – f₂ of –2.4 GHz, which is defined such that the signs of Δλ and Δf match. The maximum optical injection strength from Lasers 1 to 2 is 400 nW. Injection locking does not occur under this condition, and periodic or chaotic oscillations of the output of Laser 2 are observed.

3.2 Experimental results of evaluation of (ɛ, τ) entropy

We experimentally observe the temporal waveforms of the output of Laser 2 for different parameter values. Figure 9 shows the temporal waveforms of the output of Laser 2 for different injection strengths. For a small injection strength, chaotic oscillations with slight irregularities appear, as shown in Fig. 9(a). As the injection strength increases, more irregular chaotic oscillations are obtained, as shown in Figs. 9(b) and 9(c). For the maximum injection strength in Fig. 9(d), the temporal waveform consists of chaotic oscillations and a stable output with noisy oscillations, indicating intermittent dynamics. We speculate that this intermittency results from unstable injection locking between Lasers 1 and 2. We can experimentally observe chaotic oscillations for a wide range of optical injection strengths.

Fig. 9. Experimental results of chaotic temporal waveforms at different injection strengths: (a) injection strength of 100 nW, (b) 200 nW, (c) 300 nW, and (d) 400 nW.

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We evaluate the (ɛ, τ) entropy from these experimental data for different injection strengths. We set the parameter values of ɛ_N = 0, τ = 20 ps, and d = 100 to evaluate the (ɛ, τ) entropy. Figure 10 shows the experimental results of the (ɛ, τ) entropy for different injection strengths. The entropy increases gradually as the injection strength is increased, and the maximum value of the entropy rate of 6.2 Gb/s is obtained at an injection strength of 300 nW. As the injection strength increases further, the entropy decreases owing to the intermittent dynamics. The experimental results of the evaluation of the (ɛ, τ) entropy are compared with those obtained from the numerical results in the presence of noise (SNR = 25 dB), indicated by the red curve in Fig. 7. The tendency of the entropy rate appears to be similar for the experimental and numerical results. Thus, the (ɛ, τ) entropy can be adequately estimated from the experimentally obtained chaotic temporal waveforms.

Fig. 10. Experimental result of the (ɛ, τ) entropy for different injection strengths.

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4. Entropy evaluation for random number generation

In this section, we investigate the relationship between the entropy rate and random number generation to guarantee the uncertainty of the physical random bits. We use the chaotic temporal waveforms generated experimentally, as described in Section 3.2. Random numbers can be generated by discretizing the amplitude of chaotic temporal waveforms. We match the size ɛ to the vertical resolution of the random number generation (i.e., the vertical resolution of the 8-bit digital oscilloscope). For example, the entropy of the most significant bit (MSB) can be estimated by setting ɛ = 128 (the vertical range is [–128:128]), and the entropy of the second MSB is calculated by setting ɛ = 64, and so on. We denote MSB n as the bits from the most to the n-th significant bit (e.g., MSB 3 includes the most, second-most, and third-most significant bits). As the number of MSB n increases, a smaller value of ɛ is used.

Figure 11 shows the (ɛ, τ) entropy and the (ɛ, τ) entropy per sample as d is varied for different MSB n. The entropy decreases as d increases, particularly for larger values of n. The (ɛ, τ) entropy can be estimated correctly from the values of the plateaus of the curves, which can be found for the curves of MSB 1, 2, and 3. The lack of plateaus may result from the small amount of sampled data because more data are required for larger n and smaller ɛ. From these results, we set d = 30 (dotted line in Fig. 11) and evaluate the (ɛ, τ) entropy from MSB 1–3 in the following calculations.

Fig. 11. (ɛ, τ) entropy and (ɛ, τ) entropy per sample as d is varied for different most significant bit (MSB) n. The sampling time is set to τ = 20 ps (sampling frequency of 50 GS/s).

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We evaluate the (ɛ, τ) entropy of MSB 1, 2, and 3 with respect to the sampling frequency 1/τ. Figure 12(a) shows the (ɛ, τ) entropy as the sampling frequency is varied for MSB 1, 2, and 3. The entropy increases as the sampling frequency increases, and saturates at over 20 GS/s for MSB 1 and 2. This result indicates that excessively fast sampling does not generate entropy owing to the correlation among neighboring sampled bits. By contrast, the entropy increases slightly as the sampling frequency increases for MSB 3.

Fig. 12. (a) (ɛ, τ) entropy rate and (b) (ɛ, τ) entropy per sample as the sampling frequency 1/τ is varied for MSB 1, 2, and 3.

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Figure 12(b) shows the (ɛ, τ) entropy per sample with respect to the sampling frequency. This value is estimated from the (ɛ, τ) entropy in Fig. 12(a) divided by the sampling frequency. The entropy per sample increases as the sampling frequency decreases, as shown in Fig. 12(b). We found that the entropy per sample exceeds one bit (indicated by the dotted line) for MSB 3 when the sampling frequency is set to 10 GS/s or less. This result indicates that the entropy of one bit per sample is guaranteed for MSB 3 sampled at 10 GS/s.

We consider how to extract physical random bits from the sampled chaotic data while guaranteeing the entropy of one bit per sample. We use MSB 3 data sampled at 10 GS/s to evaluate physical random number generation. First, we evaluate the entropy per sample for the most, second-most, and third-most significant bits, which are denoted as bits 1, 2, and 3, respectively.

Figure 13(a) shows the entropy per sample for each significant bit as d is varied. The entropy per sample can be estimated from the plateaus of these curves. The entropy per sample is almost constant and close to one (∼ 0.99) for bit 3 at the plateau (blue curve in Fig. 13(a)). By contrast, bits 1 and 2 have an entropy of less than one at the plateaus. Therefore, physical random bits with an entropy of one bit per sample can be generated using the third-most significant bit (bit 3) of this data.

Fig. 13. (a) (ɛ, τ) entropy per sample as d is varied for the most (bit 1), second-most (bit 2), and third-most (bit 3) significant bits. (b) (ɛ, τ) entropy per sample for the bits generated by the exclusive-OR (XOR) operation among bits 1, 2, and 3 at the same sampling point.

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For comparison, we evaluate the effect of postprocessing of random number generation in terms of entropy. Figure 13(b) shows the entropy per sample for the bits generated by the exclusive-OR (XOR) operation among bits 1, 2, and 3 at the same sampling point. The entropy per sample presented in Fig. 13(b) resembles the average entropy for the three significant bits, as shown in Fig. 13(a). Therefore, the XOR operation does not increase the entropy per sample. From these results, direct use of the third-most significant bit in the 8-bit data is recommended for physical random number generation, which guarantees the entropy of one bit per sample.

5. Discussion

We evaluated the (ɛ, τ) entropy of chaotic temporal waveforms generated from both numerical simulations and experiments. We also estimated the (ɛ, τ) entropy per sample for random bit sequences generated from the chaotic temporal waveforms obtained experimentally. We found that the entropy of one bit per sample can be guaranteed using the third-most significant bit of chaotic temporal waveforms sampled at 8-bit vertical resolution and 10 GS/s. These quantitative values may vary for different chaotic data. However, we confirmed that the middle significant bits can be useful for generating physical random bits with uncertainty. In fact, higher significant bits (e.g., MSB) are correlated among neighboring sampled data, which degrades the entropy rate. By contrast, lower significant bits (e.g., the least significant bit, LSB) are dominated by the effect of spontaneous emission noise and detection noise. The value of the entropy strongly depends on the laser device and measurement equipment [46]. Therefore, we recommend the use of one of the middle significant bits between MSB and LSB as a physical random bit that carries the entropy of one bit per sample.

In addition, the sampling frequency is a crucial parameter, which we set as 10 GS/s to guarantee the entropy of one bit per sample, according to our analysis. A faster sampling frequency cannot enhance the entropy; however, faster random number generation is required for engineering applications. Therefore, we must determine the optimal sampling frequency, which is the fastest sampling frequency under the condition of an entropy of one bit per sample.

The measurement of the entropy rate is extremely important to guarantee the quality of physical random number generators. Furthermore, the separation of entropy sources between chaotic waveforms and external noise is crucial for evaluating the origin of entropy production. There are different metrics of entropy evaluation, and the development of entropy measurements is strongly encouraged for the quantitative evaluation of entropy for physical random number generators.

6. Conclusions

We evaluated the (ɛ, τ) entropy of a chaotic time series in an optically injected semiconductor laser. We numerically optimized the vertical resolution ɛ and sampling time τ by comparing the (ɛ, τ) entropy with the KS entropy. We estimated the (ɛ, τ) entropy when the optical injection strength was varied. We also evaluated the (ɛ, τ) entropy from the experimentally obtained chaotic temporal waveforms. The conditions of random number generation that satisfied the entropy of one bit per sample were investigated. We found that the extraction of the third-most significant bit from 8-bit chaotic data resulted in an entropy of one bit per sample, which certified physical random number generation.

We evaluated the physical entropy source of a chaotic semiconductor laser as a single-bit random number generator because the entropy of one bit per sample is satisfied. The extraction of one of the middle significant bits can be useful for certified physical random number generation. The evaluation of the entropy rate is highly crucial for certified physical random number generators used for applications in information security, which requires the unpredictability of generated random bits. Further investigation of entropy evaluation is encouraged for the development of high-quality certified physical random number generators.

Funding

Japan Society for the Promotion of Science (JP19H00868, JP20K15185); Core Research for Evolutional Science and Technology (JPMJCR17N2); Telecommunications Advancement Foundation.

Disclosures

The authors declare no conflicts of interest.

References

1. A. Uchida, K. Amano, M. Inoue, K. Hirano, S. Naito, H. Someya, I. Oowada, T. Kurashige, M. Shiki, S. Yoshimori, K. Yoshimura, and P. Davis, “Fast physical random bit generation with chaotic semiconductor lasers,” Nat. Photonics 2(12), 728–732 (2008). [CrossRef]

2. I. Reidler, Y. Aviad, M. Rosenbluh, and I. Kanter, “Ultrahigh-speed random number generation based on a chaotic semiconductor laser,” Phys. Rev. Lett. 103(2), 024102 (2009). [CrossRef]

3. I. Kanter, Y. Aviad, I. Reidler, E. Cohen, and M. Rosenbluh, “An optical ultrafast random bit generator,” Nat. Photonics 4(1), 58–61 (2010). [CrossRef]

4. J. Zhang, Y. Wang, M. Liu, L. Xue, P. Li, A. Wang, and M. Zhang, “A robust random number generator based on differential comparison of chaotic laser signals,” Opt. Express 20(7), 7496–7506 (2012). [CrossRef]

5. R. M. Nguimdo, G. Verschaffelt, J. Danckaert, X. Leijtens, J. Bolk, and G. Van der Sande, “Fast random bits generation based on a single chaotic semiconductor ring laser,” Opt. Express 20(27), 28603–28613 (2012). [CrossRef]

6. A. Wang, P. Li, J. Zhang, J. Zhang, L. Li, and Y. Wang, “4.5 Gbps high-speed real-time physical random bit generator,” Opt. Express 21(17), 20452–20462 (2013). [CrossRef]

7. N. Oliver, M. C. Soriano, D. W. Sukow, and I. Fischer, “Fast random bit generation using a chaotic laser: approaching the information theoretic limit,” IEEE J. Quantum Electron. 49(11), 910–918 (2013). [CrossRef]

8. M. Virte, E. Mercier, H. Thienpont, K. Panajotov, and M. Sciamanna, “Physical random bit generation from chaotic solitary laser diode,” Opt. Express 22(14), 17271–17280 (2014). [CrossRef]

9. P. Li, Y. Sun, X. Liu, X. Yi, J. Zhang, X. Guo, Y. Guo, and Y. Wang, “Fully photonics-based physical random bit generator,” Opt. Lett. 41(14), 3347–3350 (2016). [CrossRef]

10. K. Hirano, T. Yamazaki, S. Morikatsu, H. Okumura, H. Aida, A. Uchida, S. Yoshimori, K. Yoshimura, T. Harayama, and P. Davis, “Fast random bit generation with bandwidth-enhanced chaos in semiconductor lasers,” Opt. Express 18(6), 5512–5524 (2010). [CrossRef]

11. Y. Akizawa, T. Yamazaki, A. Uchida, T. Harayama, S. Sunada, K. Arai, K. Yoshimura, and P. Davis, “Fast random number generation with bandwidth-enhanced chaotic semiconductor lasers at 8 × 50 Gb/s,” IEEE Photonics Technol. Lett. 24(12), 1042–1044 (2012). [CrossRef]

12. X.-Z. Li and S.-C. Chan, “Random bit generation using an optically injected semiconductor laser in chaos with oversampling,” Opt. Lett. 37(11), 2163–2165 (2012). [CrossRef]

13. X.-Z. Li and S.-C. Chan, “Heterodyne random bit generation using an optically injected semiconductor laser in chaos,” IEEE J. Quantum Electron. 49(10), 829–838 (2013). [CrossRef]

14. X. Tang, G.-Q. Xia, E. Jayaprasath, T. Deng, X.-D. Lin, L. Fan, Z.-Y. Gao, and Z.-M. Wu, “Multi-channel physical random bit generation using a vertical-cavity surface-emitting laser under chaotic optical injection,” IEEE Access 6, 3565–3572 (2018). [CrossRef]

15. Y. Wang, S. Xiang, B. Wang, X. Cao, A. Wen, and Y. Hao, “Time-delay signature concealment and physical random bits generation in mutually coupled semiconductor lasers with FBG filtered injection,” Opt. Express 27(6), 8446–8455 (2019). [CrossRef]

16. R. Sakuraba, K. Iwakawa, K. Kanno, and A. Uchida, “Tb/s physical random bit generation with bandwidth-enhanced chaos in three-cascaded semiconductor lasers,” Opt. Express 23(2), 1470–1490 (2015). [CrossRef]

17. X. Tang, Z.-M. Wu, J.-G. Wu, T. Deng, J.-J. Chen, L. Fan, Z.-Q. Zhong, and G.-Q. Xia, “Tbits/s physical random bit generation based on mutually coupled semiconductor laser chaotic entropy source,” Opt. Express 23(26), 33130–33141 (2015). [CrossRef]

18. N. Li, B. Kim, V. N. Chizhevsky, A. Locquet, M. Bloch, D. S. Citrin, and W. Pan, “Two approaches for ultrafast random bit generation based on the chaotic dynamics of a semiconductor laser,” Opt. Express 22(6), 6634–6646 (2014). [CrossRef]

19. T. Butler, C. Durkan, D. Goulding, S. Slepneva, B. Kelleher, S. P. Hegarty, and G. Huyet, “Optical ultrafast random number generation at 1 Tb/s using a turbulent semiconductor ring cavity laser,” Opt. Lett. 41(2), 388–391 (2016). [CrossRef]

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

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

22. J. F. Dynes, Z. L. Yuan, A. W. Sharpe, and A. J. Shields, “A high speed, post-processing free, quantum random number generator,” Appl. Phys. Lett. 93(3), 031109 (2008). [CrossRef]

23. C. Gabriel, C. Wittmann, D. Sych, R. Dong, W. Mauerer, U. L. Andersen, C. Marquardt, and G. Leuchs, “A generator for unique quantum random numbers based on vacuum states,” Nat. Photonics 4(10), 711–715 (2010). [CrossRef]

24. Y.-Q. Nie, L. Huang, Y. Liu, F. Payne, J. Zhang, and J.-W. Pan, “The generation of 68 Gbps quantum random number by measuring laser phase fluctuations,” Rev. Sci. Instrum. 86(6), 063105 (2015). [CrossRef]

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

26. C. R. S. Williams, J. C. Salevan, X. Li, R. Roy, and T. E. Murphy, “Fast physical random number generator using amplified spontaneous emission,” Opt. Express 18(23), 23584–23597 (2010). [CrossRef]

27. X. Li, A. B. Cohen, T. E. Murphy, and R. Roy, “Scalable parallel physical random number generator based on a superluminescent LED,” Opt. Lett. 36(6), 1020–1022 (2011). [CrossRef]

28. A. Argyris, E. Pikasis, S. Deligiannidis, and D. Syvridis, “Sub-Tb/s physical random bit generators based on direct detection of amplified spontaneous emission signals,” J. Lightwave Technol. 30(9), 1329–1334 (2012). [CrossRef]

29. T. Yamazaki and A. Uchida, “Performance of random number generators using noise-based superluminescent diode and chaos-based semiconductor lasers,” IEEE J. Sel. Top. Quantum Electron. 19(4), 0600309 (2013). [CrossRef]

30. M. Huang, A. Wang, P. Li, H. Xu, and Y. Wang, “Real-time 3 Gbit/s true random bit generator based on a super-luminescent diode,” Opt. Commun. 325, 165–169 (2014). [CrossRef]

31. A. Argyris, S. Deligiannidis, E. Pikasis, A. Bogris, and D. Syvridis, “Implementation of 140 Gb/s true random bit generator based on a chaotic photonic integrated circuit,” Opt. Express 18(18), 18763–18768 (2010). [CrossRef]

32. T. Harayama, S. Sunada, K. Yoshimura, P. Davis, K. Tsuzuki, and A. Uchida, “Fast nondeterministic random-bit generation using on-chip chaos lasers,” Phys. Rev. A 83(3), 031803 (2011). [CrossRef]

33. S. Sunada, T. Harayama, K. Arai, K. Yoshimura, P. Davis, K. Tsuzuki, and A. Uchida, “Chaos laser chips with delayed optical feedback using a passive ring waveguide,” Opt. Express 19(7), 5713–5724 (2011). [CrossRef]

34. R. Takahashi, Y. Akizawa, A. Uchida, T. Harayama, K. Tsuzuki, S. Sunada, K. Arai, K. Yoshimura, and P. Davis, “Fast physical random bit generation with photonic integrated circuits with different external cavity lengths for chaos generation,” Opt. Express 22(10), 11727–11740 (2014). [CrossRef]

35. A. Karsaklian Dal Bosco, N. Sato, Y. Terashima, S. Ohara, A. Uchida, T. Harayama, and M. Inubushi, “Random number generation from intermittent optical chaos,” IEEE J. Sel. Top. Quantum Electron. 23(6), 1801208 (2017). [CrossRef]

36. K. Ugajin, Y. Terashima, K. Iwakawa, A. Uchida, T. Harayama, K. Yoshimura, and M. Inubushi, “Real-time fast physical random number generator with a photonic integrated circuit,” Opt. Express 25(6), 6511–6523 (2017). [CrossRef]

37. X.-Z. Li, J.-P. Zhuang, S.-S. Li, J.-B. Gao, and S.-C. Chan, “Randomness evaluation for an optically injected chaotic semiconductor laser by attractor reconstruction,” Phys. Rev. E 94, 042214 (2016). [CrossRef]

38. R. Vicente, J. Daudén, P. Colet, and R. Toral, “Analysis and characterization of the hyperchaos generated by a semiconductor laser subject to a delayed feedback loop,” IEEE J. Quantum Electron. 41(4), 541–548 (2005). [CrossRef]

39. K. Kanno, A. Uchida, and M. Bunsen, “Complexity and bandwidth enhancement in unidirectionally coupled semiconductor lasers with time-delayed optical feedback,” Phys. Rev. E 93, 032206 (2016). [CrossRef]

40. M. C. Soriano, L. Zunino, O. A. Rosso, I. Fischer, and C. R. Mirasso, “Time scales of a chaotic semiconductor laser with optical feedback under the lens of a permutation information analysis,” IEEE J. Quantum Electron. 47(2), 252–261 (2011). [CrossRef]

41. J. P. Toomey and D. M. Kane, “Mapping the dynamic complexity of a semiconductor laser with optical feedback using permutation entropy,” Opt. Express 22(2), 1713–1725 (2014). [CrossRef]

42. P. Mu, W. Pan, and N. Li, “Analysis and characterization of chaos generated by free-running and optically injected VCSELs,” Opt. Express 26(12), 15642–15655 (2018). [CrossRef]

43. X. Guo, T. Liu, L. Wang, X. Fang, T. Zhao, M. Virte, and Y. Guo, “Evaluating entropy rate of laser chaos and shot noise,” Opt. Express 28(2), 1238–1248 (2020). [CrossRef]

44. N. J. Corron, R. M. Cooper, and J. N. Blakely, “Evaluating entropy rate of laser chaos and shot noise,” Physica D 332, 34–40 (2016). [CrossRef]

45. E. Barker and J. Kelsey, “Recommendation for the entropy sources used for random bit generation,” NIST Draft Special Publication 800-90B, second draft (2016).

46. K. Yoshiya, Y. Terashima, K. Kanno, and A. Uchida, “Entropy evaluation of white chaos generated by optical heterodyne for certifying physical random number generators,” Opt. Express 28(3), 3686–3698 (2020). [CrossRef]

47. A. M. Hagerstrom, T. E. Murphy, and R. Roy, “Harvesting entropy and quantifying the transition from noise to chaos in a photon-counting feedback loop,” Proc. Natl. Acad. Sci. U. S. A. 112(30), 9258–9263 (2015). [CrossRef]

48. J. D. Hart, Y. Terashima, A. Uchida, G. B. Baumgartner, T. E. Murphy, and R. Roy, “Recommendations and illustrations for the evaluation of photonic random number generators,” APL Photonics 2(9), 090901 (2017). [CrossRef]

49. A. Cohen and I. Procaccia, “Computing the Kolmogorov entropy from time signals of dissipative and conservative dynamical systems,” Phys. Rev. A 31(3), 1872–1882 (1985). [CrossRef]

50. P. Gaspard and X.-J. Wang, “Noise, chaos, and (ɛ, τ)-entropy per unit time,” Phys. Rep. 235(6), 291–343 (1993). [CrossRef]

51. J. S. Richman and J. R. Moorman, “Physiological time-series analysis using approximate entropy and sample entropy,” Am. J. Physiol. Heart Circ. Physiol. 278(6), H2039–H2049 (2000). [CrossRef]

52. J. M. Yentes, N. Hunt, K. K. Schmid, J. P. Kaipust, D. McGrath, and N. Stergiou, “The appropriate use of approximate entropy and sample entropy with short data sets,” Ann. Biomed. Eng. 41(2), 349–365 (2013). [CrossRef]

53. R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. 16(3), 347–355 (1980). [CrossRef]

54. K. Kanno and A. Uchida, “Consistency and complexity in coupled semiconductor lasers with time-delayed optical feedback,” Phys. Rev. E 86, 066202 (2012). [CrossRef]

55. T. Mihana, Y. Mitsui, M. Takabayashi, K. Kanno, S. Sunada, M. Naruse, and A. Uchida, “Decision making for the multi-armed bandit problem using lag synchronization of chaos in mutually coupled semiconductor lasers,” Opt. Express 27(19), 26989–27008 (2019). [CrossRef]

Entropy rate of chaos in an optically injected semiconductor laser for physical random number generation

Abstract

1. Introduction

2. Numerical results

2.1 Numerical model and chaotic temporal waveforms

2.2 Evaluation method of (ɛ, τ) entropy

2.3 Numerical results of evaluation of (ɛ, τ) entropy

2.4 Parameter dependence of (ɛ, τ) entropy

3. Experimental results

3.1 Experimental setup

3.2 Experimental results of evaluation of (ɛ, τ) entropy

4. Entropy evaluation for random number generation

5. Discussion

6. Conclusions

Funding

Disclosures

References

Cited By

Figures (13)

Equations (11)

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