Noise robustness of unpredictability in a chaotic laser system: Toward reliable physical random bit generation

We introduce a concept of noise robustness in dynamical systems with noise and argue that this concept is essential to guarantee the reliability of physical random bit generators (RBGs). As an example of promising physical RBGs we consider a chaotic laser system and show that it has the property of noise robustness with respect to changes in the temporal correlation of the noise source. Moreover, employing a simple model of tangent space dynamics, we give a theoretical interpretation of the numerical results and in particular show that the Lyapunov exponent determines a theoretical boundary of a noise-robust region in parameter space. These theoretical results are expected to be significant not only for chaotic lasers, but also for a broad class of chaotic dynamical systems with correlated noise.

Figure 1

(a) Schematic illustration of noise parameters $p$ and properties of random bits $Q\left(p\right)$. (b) Schematic illustration of noise controllable region $C\left(p^{☆}\right)$ (a yellow region surrounded by a dotted line frame) and noise-robust region $R\left(p^{☆}\right)$ (a blue region surrounded by a solid line frame) in the case of $m=2$.

Figure 2

Illustration of solution orbits $x_i={\phi }_{\gamma ,W}^{T_s}\left(x_{i-1}\right)$ in state space. The solid line arrows denote the operation of ${\phi }_{\gamma ,W}^{T_s}$ and the dashed line arrows denote the operation of ${\phi }_{\gamma ,W^{\prime }}^{T_s}$.

Figure 3

Correlation coefficient $C(T_{\gamma },T_s)$. The white dashed line is $T_s=f_{\epsilon }\left(T_{\gamma }\right)$ ($\epsilon =0.1$). The green solid curve is a theoretical curve $T_s=f_{\text{th}}\left(T_{\gamma }\right)$ derived in Sec. 5 by using a simple model of tangent space dynamics.

Figure 4

(a) Typical time series of the intensity $\left|E\right(t\left)\right|$. (b) Power spectrum of the intensity corresponding to (a).

Figure 5

Correlation coefficient in the case of $\xi \left(t\right)$ being the white Gaussian noise $C(0,T_s)$. The error bars indicate the standard deviation of an ensemble of 30 time series.