Ultrahigh-speed random number generation based on a chaotic semiconductor laser

Scalable parallel ultrafast optical random bit generation based on a single chaotic microcomb

Random bit generators are critical for information security, cryptography, stochastic modeling, and simulations. Speed and scalability are key challenges faced by current physical random bit generation. Herein, we propose a massively parallel scheme for ultrafast random bit generation towards rates of order 100 terabit per second based on a single micro-ring resonator. A modulation-instability-driven chaotic comb in a micro-ring resonator enables the simultaneous generation of hundreds of independent and unbiased random bit streams. A proof-of-concept experiment demonstrates that using our method, random bit streams beyond 2 terabit per second can be successfully generated with only 7 comb lines. This bit rate can be easily enhanced by further increasing the number of comb lines used. Our approach provides a chip-scale solution to random bit generation for secure communication and high-performance computation, and offers superhigh speed and large scalability.

Feedback insensitivity in a self-chaotic microcavity laser

Insensitivity to external optical feedback is experimentally demonstrated in a self-chaotic deformed square microcavity laser for the first time, to the best of our knowledge. Both the optical and radio frequency (RF) spectra of the microlaser remain unaffected for external optical feedback with feedback strength as high as 9.9 dB. In addition, the autocorrelation function curve exhibits no time-delayed peaks. The insensitivity makes the self-chaotic microcavity laser promising for applications in feedback-insensitive optical sources.

Massive and parallel 10 Tbit/s physical random bit generation with chaotic microcomb

Ultrafast physical random bit (PRB) generators and integrated schemes have proven to be valuable in a broad range of scientific and technological applications. In this study, we experimentally demonstrated a PRB scheme with a chaotic microcomb using a chip-scale integrated resonator. A microcomb contained hundreds of chaotic channels, and each comb tooth functioned as an entropy source for the PRB. First, a 12 Gbits/s PRB signal was obtained for each tooth channel with proper post-processing and passed the NIST Special Publication 800-22 statistical tests. The chaotic microcomb covered a wavelength range from 1430 to 1675 nm with a free spectral range (FSR) of 100 GHz. Consequently, the combined random bit sequence could achieve an ultra-high rate of about 4 Tbits/s (12 Gbits/s × 294 = 3.528 Tbits/s), with 294 teeth in the experimental microcomb. Additionally, denser microcombs were experimentally realized using an integrated resonator with 33.6 GHz FSR. A total of 805 chaotic comb teeth were observed and covered the wavelength range from 1430 to 1670 nm. In each tooth channel, 12 Gbits/s random sequences was generated, which passed the NIST test. Consequently, the total rate of the PRB was approximately 10 Tbits/s (12 Gbits/s × 805 = 9.66 Tbits/s). These results could offer potential chip solutions of Pbits/s PRB with the features of low cost and a high degree of parallelism.

Harnessing microcomb-based parallel chaos for random number generation and optical decision making

Optical chaos is vital for various applications such as private communication, encryption, anti-interference sensing, and reinforcement learning. Chaotic microcombs have emerged as promising sources for generating massive optical chaos. However, their inter-channel correlation behavior remains elusive, limiting their potential for on-chip parallel chaotic systems with high throughput. In this study, we present massively parallel chaos based on chaotic microcombs and high-nonlinearity AlGaAsOI platforms. We demonstrate the feasibility of generating parallel chaotic signals with inter-channel correlation <0.04 and a high random number generation rate of 3.84 Tbps. We further show the application of our approach by demonstrating a 15-channel integrated random bit generator with a 20 Gbps channel rate using silicon photonic chips. Additionally, we achieved a scalable decision-making accelerator for up to 256-armed bandit problems. Our work opens new possibilities for chaos-based information processing systems using integrated photonics, and potentially can revolutionize the current architecture of communication, sensing and computations.

Learned pseudo-random number generator: WGAN-GP for generating statistically robust random numbers

Pseudo-random number generators (PRNGs) are software algorithms generating a sequence of numbers approximating the properties of random numbers. They are critical components in many information systems that require unpredictable and nonarbitrary behaviors, such as parameter configuration in machine learning, gaming, cryptography, and simulation. A PRNG is commonly validated through a statistical test suite, such as NIST SP 800-22rev1a (NIST test suite), to evaluate its robustness and the randomness of the numbers. In this paper, we propose a Wasserstein distance-based generative adversarial network (WGAN) approach to generating PRNGs that fully satisfy the NIST test suite. In this approach, the existing Mersenne Twister (MT) PRNG is learned without implementing any mathematical programming code. We remove the dropout layers from the conventional WGAN network to learn random numbers distributed in the entire feature space because the nearly infinite amount of data can suppress the overfitting problems that occur without dropout layers. We conduct experimental studies to evaluate our learned pseudo-random number generator (LPRNG) by adopting cosine-function-based numbers with poor random number properties according to the NIST test suite as seed numbers. The experimental results show that our LPRNG successfully converted the sequence of seed numbers to random numbers that fully satisfy the NIST test suite. This study opens the way for the "democratization" of PRNGs through the end-to-end learning of conventional PRNGs, which means that PRNGs can be generated without deep mathematical know-how. Such tailor-made PRNGs will effectively enhance the unpredictability and nonarbitrariness of a wide range of information systems, even if the seed numbers can be revealed by reverse engineering. The experimental results also show that overfitting was observed after about 450,000 trials of learning, suggesting that there is an upper limit to the number of learning counts for a fixed-size neural network, even when learning with unlimited data.

Photonic elementary cellular automata for simulation of complex phenomena

Cellular automata are a class of computational models based on simple rules and algorithms that can simulate a wide range of complex phenomena. However, when using conventional computers, these 'simple' rules are only encapsulated at the level of software. This can be taken one step further by simplifying the underlying physical hardware. Here, we propose and implement a simple photonic hardware platform for simulating complex phenomena based on cellular automata. Using this special-purpose computer, we experimentally demonstrate complex phenomena, including fractals, chaos, and solitons, which are typically associated with much more complex physical systems. The flexibility and programmability of our photonic computer present new opportunities to simulate and harness complexity for efficient, robust, and decentralized information processing using light.

Source-independent quantum random number generator against tailored detector blinding attacks

Randomness, mainly in the form of random numbers, is the fundamental prerequisite for the security of many cryptographic tasks. Quantum randomness can be extracted even if adversaries are fully aware of the protocol and even control the randomness source. However, an adversary can further manipulate the randomness via tailored detector blinding attacks, which are hacking attacks suffered by protocols with trusted detectors. Here, by treating no-click events as valid events, we propose a quantum random number generation protocol that can simultaneously address source vulnerability and ferocious tailored detector blinding attacks. The method can be extended to high-dimensional random number generation. We experimentally demonstrate the ability of our protocol to generate random numbers for two-dimensional measurement with a generation speed of 0.1 bit per pulse.

Chaos generated in a semiconductor microcavity

The dynamics of chaos in quantum systems has attracted much interest in connection with the fundamental aspects of quantum mechanics. We study the chaotic dynamics of both the excitonic mode and the cavity mode in a microcavity containing a quantum well driven by an external field. We investigate how the chaotic dynamics is influenced by the frequencies of the exciton and the cavity, the coupling constant between the exciton and cavity, the Coulomb interaction between excitons, and the response of the exciton to the cavity and the external field. We show that chaos can be generated synchronously in both the cavity and the excitonic mode by choosing appropriate parameters. Moreover, this kind of chaos can be controlled by the coupling constant, the strength of the interaction between excitons, the external field, the response of the excitons to the cavity, and the detuning between the cavity field and the excitonic field. The present study may have applications in chaos-based neural networks and extreme event statistics.

Multiplexing quantum tunneling diodes for random number generation

Random numbers are indispensable resources for application in modern science and technology. Therefore, a dedicated entropy source is essential, particularly for cryptographic tasks and modern applications. In this work, we experimentally demonstrated a scheme to generate random numbers by multiplexing eight tunnel diodes onto a single circuit. As a result, the data rate of random number generation was significantly enhanced eightfold. In comparison to the original scheme that employed one diode, this multiplexing scheme produced data with higher entropy. These data were then post-processed with the Toeplitz-hashing extractor, yielding final outputs that achieved almost full entropy and passed the U.S. National Institute of Standards and Technology (NIST) Special Publication 800-90B validation. These data also passed the NIST Special Publication 800-22 statistical randomness examination and had no sign of patterns detected from an autocorrelation analysis.

Laminar chaos in systems with quasiperiodic delay

A type of chaos called laminar chaos was found in singularly perturbed dynamical systems with periodic time-varying delay [Phys. Rev. Lett. 120, 084102 (2018)]0031-900710.1103/PhysRevLett.120.084102. It is characterized by nearly constant laminar phases, which are periodically interrupted by irregular bursts, where the intensity levels of the laminar phases vary chaotically from phase to phase. In this paper, we demonstrate that laminar chaos can also be observed in systems with quasiperiodic delay, where we generalize the concept of conservative and dissipative delays to such systems. It turns out that the durations of the laminar phases vary quasiperiodically and follow the dynamics of a torus map in contrast to the periodic variation observed for periodic delay. Theoretical and numerical results indicate that introducing a quasiperiodic delay modulation into a time-delay system can lead to a giant reduction of the dimension of the chaotic attractors. By varying the mean delay and keeping other parameters fixed, we found that the Kaplan-Yorke dimension is modulated quasiperiodically over several orders of magnitudes, where the dynamics switches quasiperiodically between different types of high- and low-dimensional types of chaos.

Pseudolaminar chaos from on-off intermittency

In finite-dimensional, chaotic, Lorenz-like wave-particle dynamical systems one can find diffusive trajectories, which share their appearance with that of laminar chaotic diffusion [Phys. Rev. Lett. 128, 074101 (2022)0031-900710.1103/PhysRevLett.128.074101] known from delay systems with lag-time modulation. Applying, however, to such systems a test for laminar chaos, as proposed in [Phys. Rev. E 101, 032213 (2020)2470-004510.1103/PhysRevE.101.032213], these signals fail such a test, thus leading to the notion of pseudolaminar chaos. The latter can be interpreted as integrated periodically driven on-off intermittency. We demonstrate that, on a signal level, true laminar and pseudolaminar chaos are hardly distinguishable in systems with and without dynamical noise. However, very pronounced differences become apparent when correlations of signals and increments are considered. We compare and contrast these properties of pseudolaminar chaos with true laminar chaos.

Hyperchaos, Intermittency, Noise and Disorder in Modified Semiconductor Superlattices

Weakly coupled semiconductor superlattices under DC voltage bias are nonlinear systems with many degrees of freedom whose nonlinearity is due to sequential tunneling of electrons. They may exhibit spontaneous chaos at room temperature and act as fast physical random number generator devices. Here we present a general sequential transport model with different voltage drops at quantum wells and barriers that includes noise and fluctuations due to the superlattice epitaxial growth. Excitability and oscillations of the current in superlattices with identical periods are due to nucleation and motion of charge dipole waves that form at the emitter contact when the current drops below a critical value. Insertion of wider wells increases superlattice excitability by allowing wave nucleation at the modified wells and more complex dynamics. Then hyperchaos and different types of intermittent chaos are possible on extended DC voltage ranges. Intrinsic shot and thermal noises and external noises produce minor effects on chaotic attractors. However, random disorder due to growth fluctuations may suppress any regular or chaotic current oscillations. Numerical simulations show that more than 70% of samples remain chaotic when the standard deviation of their fluctuations due to epitaxial growth is below 0.024 nm (10% of a single monolayer) whereas for 0.015 nm disorder suppresses chaos.

Fast physical random bit generation based on a chaotic optical injection system with multi-path optical feedback

Based on the chaotic signal provided by a simple chaotic system, a random bit sequence with a rate of 640 Gb/s is generated through adopting the circulating exclusive-or (CXOR) post-processing method. Such a simple chaotic system is built via a slave semiconductor laser subject to optical injection of a chaotic signal originated from a master semiconductor laser under multi-path optical feedback. First, through inspecting the dependences of the time-delay-signature (TDS) and bandwidth of the chaotic signal on some key operation parameters, optimized parameters are determined for generating a high-quality chaotic signal with a large bandwidth and low TDS. Second, the high-quality chaotic signal is converted to an 8-bit digital signal by sampling with a digital oscilloscope at 80 GSa/s. Next, through adopting the CXOR post-processing method, a bit sequence with a rate of 640 Gb/s is obtained. Finally, the randomness is estimated by the National Institute of Standard Technology (NIST) Special Publication 800-22 statistical tests, and the results demonstrate that the obtained random bit sequence can pass all the NIST tests.

Chaotic microlasers caused by internal mode interaction for random number generation

Chaotic semiconductor lasers have been widely investigated for generating unpredictable random numbers, especially for lasers with external optical feedback. Nevertheless, chaotic lasers under external feedback are hindered by external feedback loop time, which causes correlation peaks for chaotic output. Here, we demonstrate the first self-chaotic microlaser based on internal mode interaction for a dual-mode microcavity laser, and realize random number generation using the self-chaotic laser output. By adjusting mode frequency interval close to the intrinsic relaxation oscillation frequency, nonlinear dynamics including self-chaos and period-oscillations are predicted and realized numerically and experimentally due to internal mode interaction. The internal mode interaction and corresponding carrier spatial oscillations pave the way of mode engineering for nonlinear dynamics in a solitary laser. Our findings provide a novel and easy method to create controllable and robust optical chaos for high-speed random number generation.

Electro-optic chaotic system based on time delay feature hiding and key space enhancement based on chaotic post-processing

To improve the output performance of the classical all-optical chaotic system and solve the security problems of its key exposure and small key space, a new chaotic system, to the best of our knowledge, based on logistic map post-processing is proposed. In terms of the general output performance of the system, the spectrum of the proposed system is flatter than the classical system. Through a bifurcation diagram and permutation entropy analysis, it is found that the output of the system is extremely complex. In terms of security, the simulation results show that, with a reasonable selection of system parameters, key hiding can be achieved under a large parameter range. Moreover, through the sensitivity analysis of logistic parameters, it can be seen that the introduction of logistic parameters can improve the key space of the system and further improve the security of the system.

Nonlinear Dynamics of Mid-Infrared Interband Cascade Lasers Subject to Variable-Aperture Optical Feedback

In this work, we experimentally investigate the nonlinear dynamics of an interband cascade laser (ICL) under variable-aperture optical feedback implemented by a gold mirror combining with a ring-actuated iris diaphragm (RAID). By continuously varying the diameter of RAID (DR), the evolution of the dynamical state of ICL with the aperture of the optical feedback can be inspected. The characteristics of each dynamical state are characterized by time series, power spectra, phase portraits, and Lyapunov exponents. The results show that, with the decrease of DR, the dynamical state of the ICL under variable-aperture optical feedback presents an evolution from complex, simple to stable. Diverse dynamical states including period one state (P1), period two state (P2), multi-period state (MP), quasi-period state (QP), low-frequency fluctuation (LFF), chaotic state (C), and hyperchaos have been observed. Through mapping the evolution of dynamical states with DR for the ICL biased at different currents, different evolved routes of the dynamical states are revealed.

Wideband chaotic tri-mode microlasers with optical feedback

A tri-mode micro-square laser under optical feedback is proposed and demonstrated to generate chaos with the broadband flat microwave spectrum. By adjusting lasing mode intensities, frequency intervals, and optical feedback strength, we can enhance the chaotic bandwidth significantly. The existence of two mode-beating peaks makes the flat bandwidth much larger than the relaxation oscillation frequency. Effective bandwidth of 35.3 GHz is experimentally achieved with the flatness of 8.3 dB from the chaotic output spectrum of the tri-mode mode laser under optical feedback.

Fast physical random bit generation using a millimeter-wave white noise source

A broadband millimeter-wave (MMW) white noise signal generated by optical heterodyning of two Fabry-Perot laser diodes (FP-LDs) subject to optical feedback is demonstrated and employed for fast physical random bit generation with a simple least significant bits (LSBs) retaining method. Firstly, under suitable feedback conditions, two external-cavity feedback FP-LDs can be easily driven into chaotic states. In this process, the optical spectra of multi-longitudinal modes are significantly broadened. Then, two spectral broadening multi-longitudinal chaotic signals are mixed and converted into an MMW white noise signal through the heterodyne beating technique combined with a fast photodetector. With such an approach, a high dimensional broadband chaos with perfect characteristics of MMW white noise (3-dB bandwidth beyond 50 GHz without any time-delay signature) is experimentally achieved. Finally, taking the generated MMW white noise as the entropy source, 640 Gb/s physical random bit generation is realized by directly selecting 4-LSBs at 160 GS/s sampling rate after an 8-bit analog-digital-convertor.

Batch and stream entropy with fixed partitions for chaos-based random bit generators

Measures are proposed for reliably estimating the entropy of bits produced in an entropy source using a chaotic physical system. The measures are reliable with respect to a "guessing" attack and depend on the end-to-end method of transfer of entropy from the chaotic physical system to the bit entropy source. Fixed partitions are considered to correspond with practical methods for fast digital sampling of analog signals. We propose two different measures corresponding to the batch and streaming modes of entropy transfer. Numerical examples are provided to demonstrate features of dependence of the batch and stream entropy on fixed partitions with uniform or nonuniform types of chaos.

Chaotic time-delay signature suppression using quantum noise

The time-delay signature (TDS) suppression of semiconductor lasers with external optical feedback is necessary to ensure the security of chaos-based secure communications. Here we numerically and experimentally demonstrate a technique to effectively suppress the TDS of chaotic lasers using quantum noise. The TDS and dynamical complexity are quantified using the autocorrelation function and normalized permutation entropy at the feedback delay time, respectively. Quantum noise from quadrature fluctuations of the vacuum state is prepared through balanced homodyne measurement. The effects of strength and bandwidth of quantum noise on chaotic TDS suppression and complexity enhancement are investigated numerically and experimentally. Compared to the original dynamics, the TDS of this quantum noise improved chaos is suppressed up to 94%, and the bandwidth suppression ratio of quantum noise to chaotic laser is 1:25. The experiment agrees well with the theory. The improved chaotic laser is potentially beneficial to chaos-based random number generation and secure communication.

Designing Hyperchaos and Intermittency in Semiconductor Superlattices

Weakly coupled semiconductor superlattices under dc voltage bias are excitable systems with many degrees of freedom that may exhibit spontaneous chaos at room temperature and act as fast physical random number generator devices. Superlattices with identical periods exhibit current self-oscillations due to the dynamics of charge dipole waves but chaotic oscillations exist on narrow voltage intervals. They disappear easily due to variation in structural growth parameters. Based on numerical simulations, we predict that inserting two identical sufficiently separated wider wells increases superlattice excitability by allowing wave nucleation at the modified wells and more complex dynamics. This system exhibits hyperchaos and varieties of intermittent chaos in extended dc voltage ranges. Unlike in ideal superlattices, our chaotic attractors are robust and resilient against noises and against controlled random disorder due to growth fluctuations.

Photonic decision-making for arbitrary-number-armed bandit problem utilizing parallel chaos generation

In this paper, we propose and experimentally demonstrate a novel scheme that helps to solve an any-number-armed bandit problem by utilizing two parallel simultaneously-generated chaotic signals and the epsilon (ɛ)-greedy strategy. In the proposed scheme, two chaotic signals are experimentally generated, and then processed by an 8-bit analog-to-digital conversion (ADC) with 4 least significant bits (LSBs), to generate two amplitude-distribution-uniform sequences for decision-making. The correspondence between these two random sequences and different arms is established by a mapping rule designed in virtue of the ɛ-greedy-strategy. Based on this, decision-making for an exemplary 5-armed bandit problem is successfully performed, and moreover, the influences of the mapping rule and unknown reward probabilities on the correction decision rate (CDR) performance for the 4-armed to 7-armed bandit problems are investigated. This work provides a novel way for solving the arbitrary-number-armed bandit problem.

High-entropy chaos generation using semiconductor lasers subject to intensity-modulated optical injection for certified physical random number generation

This study investigates high-entropy chaos generation using a semiconductor laser subject to intensity-modulated optical injection for certified physical random number generation. Chaos with a continuous spectral profile that is not only widely distributed but also broadly flattened over a bandwidth of 33 GHz is generated. The former suggests that the chaos can be sampled at a high rate while keeping sufficient un-correlation between data samples, and the latter indicates that the chaos possesses high entropy, both of which enhance the generation rate of physical random numbers with guaranteed unpredictability. A minimum entropy value of 2.19 bits/sample is obtained without any post-processing and by excluding the contribution from measurement noise, suggesting that, to the least extent, the chaotic source can be used as a 2-bit physical random number generator at a rate of 160 Gbits/s.

Massively parallel ultrafast random bit generation with a chip-scale laser

Random numbers are widely used for information security, cryptography, stochastic modeling, and quantum simulations. Key technical challenges for physical random number generation are speed and scalability. We demonstrate a method for ultrafast generation of hundreds of random bit streams in parallel with a single laser diode. Spatiotemporal interference of many lasing modes in a specially designed cavity is introduced as a scheme for greatly accelerated random bit generation. Spontaneous emission, caused by quantum fluctuations, produces stochastic noise that makes the bit streams unpredictable. We achieve a total bit rate of 250 terabits per second with off-line postprocessing, which is more than two orders of magnitude higher than the current postprocessing record. Our approach is robust, compact, and energy-efficient, with potential applications in secure communication and high-performance computation.

Real-time all-optical random numbers based on optical Boolean chaos

In this work, a method of generating all-optical random numbers based on optical Boolean chaotic entropy source is proposed. This all-optical random number generation system consists of a Boolean chaotic entropy source and an optical D flip-flop. The Boolean chaotic entropy source is composed of an optical XOR gate and two self-delayed feedback; meanwhile, the optical D flip-flop is composed of two optical AND gates and one SR latch. The optical Boolean chaotic signal possesses the dynamic characteristics of complexity and binarization, so random numbers would be generated only by extracted from chaotic signals with the optical D flip-flop. This all-optical random number generation system achieves the result of 5 Gb/s random numbers that is testable. The whole process of random number generation could be completed in the optical domain without photoelectric conversion, more importantly, the device could be integrated.

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

We evaluate the (ɛ, τ) entropy of chaotic laser outputs generated by an optically injected semiconductor laser for physical random number generation. The vertical resolution ɛ and sampling time τ are numerically optimized by comparing the (ɛ, τ) entropy with the Kolmogorov-Sinai entropy, which is estimated from the Lyapunov exponents using linearized model equations. We then investigate the dependence of the (ɛ, τ) entropy on the optical injection strength of the laser system. In addition, we evaluate the (ɛ, τ) entropy from the experimentally obtained chaotic temporal waveforms in an optically injected semiconductor laser. Random bits with an entropy close to one bit per sampling point are extracted to satisfy the conditions of physical random number generation. We find that the extraction of the third-most significant bit from eight-bit experimental chaotic data results in an entropy of one bit per sample for certified physical random number generation.

Gbps physical random bit generation based on the mesoscopic chaos of a silicon photonics crystal microcavity

We present an experimental and theoretical physical random bit (PRB) generator using the mesoscopic chaos from a photonic-crystal optomechanical microcavity with a size of ∼10µm and very low operating intracavity energy of ∼60 Femto-Joule that was fabricated with CMOS compatible processes. Moreover, two kinds of PRB generation were proposed with rates over gigabits per second (Gbps). The randomness of the large PRB strings was further verified using the NIST Special Publication 800-22. In addition, the Diehard statistical test was also used to confirm the quality of the obtained PRBs. The results of this study can offer a new generation of dedicated PRB solutions that can be integrated on Si substrates, which can speed up systems and eliminate reliance on external mechanisms for randomness collection.

DNA synthesis for true random number generation

The volume of securely encrypted data transmission required by today's network complexity of people, transactions and interactions increases continuously. To guarantee security of encryption and decryption schemes for exchanging sensitive information, large volumes of true random numbers are required. Here we present a method to exploit the stochastic nature of chemistry by synthesizing DNA strands composed of random nucleotides. We compare three commercial random DNA syntheses giving a measure for robustness and synthesis distribution of nucleotides and show that using DNA for random number generation, we can obtain 7 million GB of randomness from one synthesis run, which can be read out using state-of-the-art sequencing technologies at rates of ca. 300 kB/s. Using the von Neumann algorithm for data compression, we remove bias introduced from human or technological sources and assess randomness using NIST's statistical test suite.

Deep Learning-Based Security Verification for a Random Number Generator Using White Chaos

In this paper, a deep learning (DL)-based predictive analysis is proposed to analyze the security of a non-deterministic random number generator (NRNG) using white chaos. In particular, the temporal pattern attention (TPA)-based DL model is employed to learn and analyze the data from both stages of the NRNG: the output data of a chaotic external-cavity semiconductor laser (ECL) and the final output data of the NRNG. For the ECL stage, the results show that the model successfully detects inherent correlations caused by the time-delay signature. After optical heterodyning of two chaotic ECLs and minimal post-processing are introduced, the model detects no patterns among corresponding data. It demonstrates that the NRNG has the strong resistance against the predictive model. Prior to these works, the powerful predictive capability of the model is investigated and demonstrated by applying it to a random number generator (RNG) using linear congruential algorithm. Our research shows that the DL-based predictive model is expected to provide an efficient supplement for evaluating the security and quality of RNGs.

Time-delay signature suppression of polarization-resolved wideband chaos in VCSELs with dual-path chaotic optical injections

The combining investigation on the time-delay signature (TDS) and chaos bandwidth have been theoretically investigated in a vertical-cavity surface-emitting laser (VCSEL) system with dual-path chaotic optical injections. In this scheme, the polarized chaos with the TDS from an external-cavity master VCSEL is routed into two different paths and then unidirectionally injected into another solitary slave VCSEL. With the aid of the autocorrelation function and the effective bandwidth calculation, the TDS and bandwidth of polarized chaos from the chaotic system are quantitatively evaluated. The results show that, in such a dual-path chaotic optical-injection system, the high-quality polarized chaos with the successful TDS suppression and chaotic bandwidth enhancement can be achieved in wider parameter regions in contrast with the case for the single-path chaotic optical injection. Further research also finds that the injected time-delay difference between two injection paths is desired to mismatch the feedback time delay, which is conducive to suppressing TDS and expanding bandwidth of polarized chaos. Besides, the better chaotic quality with low TDS and wide bandwidth can be expected by choosing the appropriate injection strengths of two injection paths.

Reconstruction of parameters and unobserved variables of a semiconductor laser with optical feedback from intensity time series

We propose a method for the reconstruction of time-delayed feedback systems having unobserved variables from scalar time series. The method is based on the modified initial condition approach, which allows one to significantly reduce the number of starting guesses for an unobserved variable with a time delay. The proposed method is applied to the reconstruction of the Lang-Kobayashi equations, which describe the dynamics of a single-mode semiconductor laser with external optical feedback. We consider the case where only the time series of laser intensity is observable and the other two variables of the model are hidden. The dependence of the quality of the system reconstruction on the accuracy of assignment of starting guesses for unobserved variables and unknown laser parameters is studied. The method could be used for testing the security of information transmission in laser-based chaotic communication systems.

Entropy evaluation of white chaos generated by optical heterodyne for certifying physical random number generators

The entropy of white chaos is evaluated to certify physical random number generators. White chaos is generated from the electric subtraction of two optical heterodyne signals of two chaotic outputs in semiconductor lasers with optical feedback. We use the statistical test suites of NIST Special Publication 800-90B for the evaluation of physical entropy sources of white chaos with an eight-bit resolution. The minimum value of entropy is 2.1 for eight most significant bits data. The entropy of white chaos is enhanced from that of the chaotic output of the semiconductor lasers. We evaluate the effect of detection noise and distinguish between the entropy that originates from the white chaos and the detection noise. It is found that the entropy of five most significant bits originates from white chaos. The minimum value of entropy is 1.1 for five most significant bits data, and it is considered that the entropy can be obtained at at least one bit per sample.

Evaluating entropy rate of laser chaos and shot noise

Evaluating entropy rate of high-dimensional chaos and shot noise from analog raw signals remains elusive and important in information security. We experimentally present an accurate assessment of entropy rate for physical process randomness. The entropy generation of optical-feedback laser chaos and physical randomness limit from shot noise are quantified and unambiguously discriminated using the growth rate of average permutation entropy value in memory time. The permutation entropy difference of filtered laser chaos with varying embedding delay time is investigated experimentally and theoretically. High-resolution maps of the entropy difference are observed over the range of the injection-feedback parameter space. We also clarify an inverse relationship between the entropy rate and time delay signature of laser chaos over a wide range of parameters. Compared to the original chaos, the time delay signature is suppressed up to 95% with the minimum of 0.015 via frequency-band extractor, and the experiment agrees well with the theory. Our system provides a commendable entropy evaluation and source for physical random number generation.

Parallel optical random bit generator

We present an optical approach for high-speed parallel random bit generation based on stochastic pulse-to-pulse fluctuation in the supercontinuum (SC). Through spectrally demultiplexing the SC pulse sequence into different wavelength channels, we simultaneously extract multiple independent fast random bit streams from each SC pulse subsequence via associated comparators in parallel. Proof-of-concept experiments demonstrate that using our method, four 10 Gb/s random bit streams are obtained from a SC pulse source with verified randomness. Moreover, this method also provides a promising strategy to fabricate ultrafast random bit generators with Tb/s throughput capacity just by increasing additional wavelength channels.

Tailoring the mode-switching dynamics in quantum-dot micropillar lasers via time-delayed optical feedback

Microlasers are ideal candidates to bring the fascinating variety of nonlinear complex dynamics found in delay-coupled systems to the realm of quantum optics. Particularly attractive is the possibility of tailoring the devices' emission properties via non-invasive delayed optical coupling. However, until now scarce research has been done in this direction. Here, we experimentally and theoretically investigate the effects of delayed optical feedback on the mode-switching dynamics of an electrically driven bimodal quantum-dot micropillar laser, characterizing its impact on the micropillar's output power, optical spectrum and photon statistics. Feedback is found to influence the switching dynamics and its characteristics time scales. In addition, stochastic switching is reduced with the subsequent impact on the microlaser photon statistics. Our results contribute to the comprehension of feedback-induced phenomena in micropillar lasers and pave the way towards the external control and tailoring of the properties of these key systems for the nanophotonics community.

Parameter dependence of high-frequency nonlinear oscillations and intrinsic chaos in short GaAs/(Al, Ga)As superlattices

We explore the design parameter space of short (5-25 period), n-doped, Ga/(Al,Ga)As semiconductor superlattices (SSLs) in the sequential resonant tunneling regime. We consider SSLs at cool (77 K) and warm (295 K) temperatures, simulating the electronic response to variations in (a) the number of SSL periods, (b) the contact conductivity, and (c) the strength of disorder (aperiodicities). Our analysis shows that the chaotic dynamical phases exist on a number of sub-manifolds of codimension zero within the design parameter space. This result provides an encouraging guide towards the experimental observation of high-frequency intrinsic dynamical chaos in shorter SSLs.

16 Gbps random bit generation using chaos in near-symmetric erbium-doped fiber ring laser

We demonstrate a bias-free random bit generator at 16 Gbps using chaos in a near-symmetric erbium-doped fiber (EDF) ring laser. The laser consists of two EDFs, each pumped at 980 nm, two intracavity filters of central wavelength 1549.30 nm, and two 90:10 output couplers. The presence of chaos at the laser output is demonstrated by computing the largest Lyapunov exponent for different embedding dimensions. The laser outputs are photodetected and subtracted to generate an electrical difference signal, which is then sampled at 2 GSa/s and postprocessed to extract random bits at 16 Gbps. The random bits exhibit very low autocorrelation (∼10^-4) and have successfully passed all National Institute of Standards and Technology statistical tests and Diehard battery of tests.

Strain induced polarization chaos in a solitary VCSEL

Physical curiosity at the beginning, optical chaos is now attracting increasing interest in various technological areas such as detection and ranging or secure communications, to name but a few. However, the complexity of optical chaos generators still significantly hinders their development. In this context, the generation of chaotic polarization fluctuations in a single laser diode has proven to be a significant step forward, despite being observed solely for quantum-dot vertical-cavity surface-emitting lasers (VCSELs). Here, we demonstrate experimentally that a similar polarization dynamics can be consistently obtained in quantum-well VCSELs. Indeed, by introducing anisotropic strain in the laser cavity, we successfully triggered the desired chaotic dynamics. The simplicity of the proposed approach, based on low-cost and easily available components including off-the-shelf VCSELs, paves the way to the wide spread use of solitary VCSELs for chaos-based applications.

Experimental demonstration of 12.5 GHz wideband chaos in symmetric dual-port EDFRL

We study the dynamics of chaos in a dual-port erbium-doped fiber ring laser (EDFRL). The laser consists of two erbium-doped fibers, intracavity filters at 1549.32 nm, isolators, and couplers. At both ports, the laser transitions into the chaotic regime for pump currents greater than 100 mA via the period doubling route. We calculate the largest Lyapunov exponent using Rosenstein's algorithm. We obtain positive values for the largest Lyapunov exponent (≈0.2) for embedding dimensions 5, 7, 9, and 11 indicating chaos. We compute the power spectral density of the photocurrents at the output ports of the laser. We observe a bandwidth of 12.5 GHz at both ports. This ultra-wideband nature of chaos obtained has potential applications in high-speed random number generation and communication.

Solution-Processed Carbon Nanotube True Random Number Generator

With the growing adoption of interconnected electronic devices in consumer and industrial applications, there is an increasing demand for robust security protocols when transmitting and receiving sensitive data. Toward this end, hardware true random number generators (TRNGs), commonly used to create encryption keys, offer significant advantages over software pseudorandom number generators. However, the vast network of devices and sensors envisioned for the "Internet of Things" will require small, low-cost, and mechanically flexible TRNGs with low computational complexity. These rigorous constraints position solution-processed semiconducting single-walled carbon nanotubes (SWCNTs) as leading candidates for next-generation security devices. Here, we demonstrate the first TRNG using static random access memory (SRAM) cells based on solution-processed SWCNTs that digitize thermal noise to generate random bits. This bit generation strategy can be readily implemented in hardware with minimal transistor and computational overhead, resulting in an output stream that passes standardized statistical tests for randomness. By using solution-processed semiconducting SWCNTs in a low-power, complementary architecture to achieve TRNG, we demonstrate a promising approach for improving the security of printable and flexible electronics.

Real-time online photonic random number generation

We present a real-time scheme for ultrafast random number (RN) extraction from a broadband photonic entropy source. Ultralow jitter mode-locked pulses are used to sample the stochastic intensity fluctuations of the entropy source in the optical domain. A discrete self-delay comparison technology is exploited to quantize the sampled pulses into continuous RN streams directly. This scheme is bias free, eliminates the electronic jitter bottleneck confronted by currently available physical RN generators, and has no need for threshold tuning and post-processing. To demonstrate its feasibility, we perform a proof-of-principle experiment using an optically injected chaotic laser diode. RN streams at up to 7  Gb/s with verified randomness were thereby successfully extracted in real time. With the provision of a photonic entropy source with sufficient bandwidth, the present approach is expected to provide RN generation rates of several tens of gigabits per second.

Generation of flat wideband chaos with suppressed time delay signature by using optical time lens

We propose a flat wideband chaos generation scheme that shows excellent time delay signature suppression effect, by injecting the chaotic output of general external cavity semiconductor laser into an optical time lens module composed of a phase modulator and two dispersive units. The numerical results demonstrate that by properly setting the parameters of the driving signal of phase modulator and the accumulated dispersion of dispersive units, the relaxation oscillation in chaos can be eliminated, wideband chaos generation with an efficient bandwidth up to several tens of GHz can be achieved, and the RF spectrum of generated chaotic signal is nearly as flat as uniform distribution. Moreover, the periodicity of chaos induced by the external cavity modes can be simultaneously destructed by the optical time lens module, based on this the time delay signature can be completely suppressed.

640-Gbit/s fast physical random number generation using a broadband chaotic semiconductor laser

An ultra-fast physical random number generator is demonstrated utilizing a photonic integrated device based broadband chaotic source with a simple post data processing method. The compact chaotic source is implemented by using a monolithic integrated dual-mode amplified feedback laser (AFL) with self-injection, where a robust chaotic signal with RF frequency coverage of above 50 GHz and flatness of ±3.6 dB is generated. By using 4-least significant bits (LSBs) retaining from the 8-bit digitization of the chaotic waveform, random sequences with a bit-rate up to 640 Gbit/s (160 GS/s × 4 bits) are realized. The generated random bits have passed each of the fifteen NIST statistics tests (NIST SP800-22), indicating its randomness for practical applications.

Real-time fast physical random number generator with a photonic integrated circuit

Random number generators are essential for applications in information security and numerical simulations. Most optical-chaos-based random number generators produce random bit sequences by offline post-processing with large optical components. We demonstrate a real-time hardware implementation of a fast physical random number generator with a photonic integrated circuit and a field programmable gate array (FPGA) electronic board. We generate 1-Tbit random bit sequences and evaluate their statistical randomness using NIST Special Publication 800-22 and TestU01. All of the BigCrush tests in TestU01 are passed using 410-Gbit random bit sequences. A maximum real-time generation rate of 21.1 Gb/s is achieved for random bit sequences in binary format stored in a computer, which can be directly used for applications involving secret keys in cryptography and random seeds in large-scale numerical simulations.

Chaotic laser based physical random bit streaming system with a computer application interface

We demonstrate a random bit streaming system that uses a chaotic laser as its physical entropy source. By performing real-time bit manipulation for bias reduction, we were able to provide the memory of a personal computer with a constant supply of ready-to-use physical random bits at a throughput of up to 4 Gbps. We pay special attention to the end-to-end entropy source model describing how the entropy from physical sources is converted into bit entropy. We confirmed the statistical quality of the generated random bits by revealing the pass rate of the NIST SP800-22 test suite to be 65 % to 75 %, which is commonly considered acceptable for a reliable random bit generator. We also confirmed the stable operation of our random bit steaming system with long-term bias monitoring.

Minimal-post-processing 320-Gbps true random bit generation using physical white chaos

Chaotic external-cavity semiconductor laser (ECL) is a promising entropy source for generation of high-speed physical random bits or digital keys. The rate and randomness is unfortunately limited by laser relaxation oscillation and external-cavity resonance, and is usually improved by complicated post processing. Here, we propose using a physical broadband white chaos generated by optical heterodyning of two ECLs as entropy source to construct high-speed random bit generation (RBG) with minimal post processing. The optical heterodyne chaos not only has a white spectrum without signature of relaxation oscillation and external-cavity resonance but also has a symmetric amplitude distribution. Thus, after quantization with a multi-bit analog-digital-convertor (ADC), random bits can be obtained by extracting several least significant bits (LSBs) without any other processing. In experiments, a white chaos with a 3-dB bandwidth of 16.7 GHz is generated. Its entropy rate is estimated as 16 Gbps by single-bit quantization which means a spectrum efficiency of 96%. With quantization using an 8-bit ADC, 320-Gbps physical RBG is achieved by directly extracting 4 LSBs at 80-GHz sampling rate.

Compressive Sensing with Optical Chaos

Compressive sensing (CS) is a technique to sample a sparse signal below the Nyquist-Shannon limit, yet still enabling its reconstruction. As such, CS permits an extremely parsimonious way to store and transmit large and important classes of signals and images that would be far more data intensive should they be sampled following the prescription of the Nyquist-Shannon theorem. CS has found applications as diverse as seismology and biomedical imaging. In this work, we use actual optical signals generated from temporal intensity chaos from external-cavity semiconductor lasers (ECSL) to construct the sensing matrix that is employed to compress a sparse signal. The chaotic time series produced having their relevant dynamics on the 100 ps timescale, our results open the way to ultrahigh-speed compression of sparse signals.

Enhanced complexity of optical chaos in a laser diode with phase-conjugate feedback

We demonstrate numerically that a semiconductor laser subjected to phase-conjugate feedback (PCF) can exhibit an enhancement in the complexity of chaos by comparison to conventional optical feedback. Using quantifiers from spectral analysis and information theory, we demonstrate that under similar parametric conditions, PCF exhibits a larger chaotic bandwidth and higher spectral flatness and statistical complexity. These properties are of utmost importance for applications in secure communications and random number generation.

Randomness evaluation for an optically injected chaotic semiconductor laser by attractor reconstruction

State-space reconstruction is investigated for evaluating the randomness generated by an optically injected semiconductor laser in chaos. The reconstruction of the attractor requires only the emission intensity time series, allowing both experimental and numerical evaluations with good qualitative agreement. The randomness generation is evaluated by the divergence of neighboring states, which is quantified by the time-dependent exponents (TDEs) as well as the associated entropies. Averaged over the entire attractor, the mean TDE is observed to be positive as it increases with the evolution time through chaotic mixing. At a constant laser noise strength, the mean TDE for chaos is observed to be greater than that for periodic dynamics, as attributed to the effect of noise amplification by chaos. After discretization, the Shannon entropies continually generated by the laser for the output bits are estimated in providing a fundamental basis for random bit generation, where a combined output bit rate reaching 200 Gb/s is illustrated using practical tests. Overall, based on the reconstructed states, the TDEs and entropies offer a direct experimental verification of the randomness generated in the chaotic laser.

Fully photonics-based physical random bit generator

We propose a fully photonics-based approach for ultrafast physical random bit generation. This approach exploits a compact nonlinear loop mirror (called a terahertz optical asymmetric demultiplexer, TOAD) to sample the chaotic optical waveform in an all-optical domain and then generate random bit streams through further comparison with a threshold level. This method can efficiently overcome the electronic jitter bottleneck confronted by existing RBGs in practice. A proof-of-concept experiment demonstrates that this method can continuously extract 5 Gb/s random bit streams from the chaotic output of a distributed feedback laser diode (DFB-LD) with optical feedback. This limited generation rate is caused by the bandwidth of the used optical chaos.