Chaos-based true random number generators
- Luis L Bonilla^{1}Email author,
- Mariano Alvaro^{1} and
- Manuel Carretero^{1}
https://doi.org/10.1186/s13362-016-0026-4
© Bonilla et al. 2016
Received: 28 January 2016
Accepted: 20 June 2016
Published: 29 June 2016
Abstract
Random number (bit) generators are crucial to secure communications, data transfer and storage, and electronic transactions, to carry out stochastic simulations and to many other applications. As software generated random sequences are not truly random, fast entropy sources such as quantum systems or classically chaotic systems can be viable alternatives provided they generate high-quality random sequences sufficiently fast. The discovery of spontaneous chaos in semiconductor superlattices at room temperature has produced a valuable nanotechnology option. Here we explain a mathematical model to describe spontaneous chaos in semiconductor superlattices at room temperature, solve it numerically to reveal the origin and characteristics of chaotic oscillations, and discuss the limitations of the model in view of known experiments. We also explain how to extract verified random bits from the analog chaotic signal produced by the superlattice.
Keywords
random bit generator semiconductor superlattice deterministic and stochastic chaos1 Background
Generation of random numbers at high speed is at the core of many activities of economic importance. Online gambling, finance, computer telecommunications, online commerce and data encryption systems, [1–3], stochastic modeling [4], and Monte Carlo simulations [5] among many others, rely on fast random number generators (RNGs). We also talk about random bit generators (RBGs) when emphasizing that binary numbers are produced. Usually, these generators are based on numerical algorithms that produce seemingly unpredictable number sequences. The generator is a function whose input is a short random seed, and whose output is a long stream which is indistinguishable from truly random bits. Such numerical strings yield the keys for secure storage and transmission of data. This conventional approach is cheap and fast, as it is limited only by the processor speed. However, the number sequences thus produced are only pseudorandom, as two identical programs that begin at the same state will produce the same sequence. Then vulnerability in the pseudorandom number generator (PRNG) may follow, as it famously was the case for Microsoft Windows operating system secure encryption several years ago [6].
To get cryptographically secure PRNGs, it is convenient to have generated truly random numbers that may be obtained ideally from inherently random or unpredictable processes. Deterministic processes that are difficult to predict have been used in gambling since antiquity. For instance, the mechanics of coin tossing shows that small uncertainties in the initial condition ensure equal probability of heads and tails provided some parameter (e.g., initial velocity) is large enough [7, 8]. Similar analyses apply to the case of rolling dice, card shuffling or spinning a roulette wheel. An obvious drawback of these mechanical methods is that they are too slow for practical use. Other physical sources of entropy are too sensitive to external influences and lack robustness, for example, thermal noise or electrical noise in diodes and resistors. These physical processes yield a low analog signal and are easily affected by disturbances including temperature fluctuations. More robust systems are based on quantum mechanical uncertainty, e.g., on whether a photon is detected, but they are limited to relatively low rates of number generation (tens of Mb/s) [9, 10]. Recently, fast generation of truly random numbers (tens or hundreds of Gb/s) has been achieved using chaotic semiconductor lasers [11–14] and superlattices [15]. In both cases, quantum fluctuations are amplified by chaotic dynamics to a macroscopic fluctuating signal. This signal can be detected by using conventional electronics that is much faster than optical photon counting detectors. While semiconductor lasers require a mixture of optical and electronic components, semiconductor superlattices are all electronic submicron devices that can be integrated in more complex circuits. As of now, these two types of devices have been shown to reliably produce truly random sequences of numbers at fast rates in laboratory experiments. If they show to be scalable, these devices could be vastly useful, as the performance and reliability of our digital networked society relies on the ability to generate fast and cheaply large quantities of random numbers.
In this paper, we comment the possible use of spontaneously chaotic semiconductor superlattices (SLs) as true random number generators. In Section 2, we discuss the mathematical model for a single SL under voltage bias. The model consists of a number of coupled stochastic differential equations together with algebraic boundary and voltage bias conditions. In Section 3.1, numerical solutions of the model equations show that the thermal and shot noises existing in the SL enhance stable spontaneous chaos in voltage intervals where the corresponding deterministic model exhibits chaos. The noises also induce chaos in nearby voltage intervals where the deterministic system had periodic oscillations. We also discuss the relation of our results to experiments and which features of the model need to be revised in order to optimize the chaotic oscillations. In Section 3.2 and following Ref. [15], we explain how to obtain a high-speed true random bit generator by processing the chaotic current oscillations provided by the device. Section 4 summarizes our findings and perspectives for fast random bit generators based on semiconductor superlattices. Two Appendices provide details on the derivation of the model equations.
2 Mathematical model and methods for a single superlattice
SLs were proposed in 1970 by Esaki and Tsu to develop a device that exhibits Bloch oscillations [18]. Although such oscillations were observed in experiments years later [19], no practical device using them has been so far developed. Instead, SLs have been used to build gigahertz oscillators for communications purposes, detectors for terahertz signals or infrared radiation and quantum-cascade lasers commercially used for a variety of purposes, such as environmental sensing and pollution controlling, industrial processes control (e.g., combustion control, converter diagnosis, collision avoiding radar in automotive industry), medical applications (breath analysis, early detection of ulcers), etc. [17]. In 2012, experiments demonstrated the existence of spontaneously chaotic oscillations of the current through a SL at room temperature under voltage bias, as in the sketch of Figures 1(a) and (b) [20]. This paved the way to using SLs as true random number generators [15].
Modeling electron transport in a SL is a bit more complicated than modeling mass transport in a fluid. One first thought could be following the route from Boltzmann equation to Navier-Stokes equations, as done in the kinetic theory of gases [21], and advocated in the mathematical literature on semiconductors [22, 23]. This has achieved some success in the case of strongly coupled SLs [24, 25], but none so far for weakly coupled SLs, as those displaying spontaneous chaos in experiments [15, 20]. The main reason for this failure is that Boltzmann-type equations for SLs are based on electrons populating minibands at zero electric field, and such a picture is far from reality in the presence of electric fields that are sufficiently strong: \(eFl>\Delta\), where \(-e<0\) is the electron charge, −F the electric field, l the SL period and Δ is the miniband width [17].
2.1 Model
The mathematical model we use has some limitations that may have to be overcome in more precise future studies. It corresponds to an idealized SL in which all periods have identical values of \(d_{W}\), \(d_{B}\), \(N_{D}\), and \(V_{B}\). The effective mass \(m^{*}\) and permittivity ε are the same irrespective on whether they correspond to a barrier or a well. In addition, the model does not include 2D effects due to imperfect growth: barrier and well widths may vary in the SL cross section perpendicular to the growth direction. Some of these effects were addressed in [37].
2.2 Methods
We have solved the stochastic model given by Eqs. (1)-(14) with \(\eta(t)=0\) (internal noise only) for the SL of Refs. [15, 20, 31] at 300 K using a standard stochastic Euler-Maruyama method (explicit Euler method corresponding to Ito integration) [38]. Coding of the numerical method follows the indications in Ref. [39]. To calculate the largest Lyapunov exponent (LLE), we have simultaneously integrated all perturbed and unperturbed trajectories during 10,000 ns and used the Benettin et al. algorithm [40] with a renormalization period of 1 ns. LLE calculations with the Gao et al. algorithm [41] give similar results.
3 Results and discussion
3.1 Spontaneous chaotic oscillations
Spontaneous chaos at room temperature has been observed in quite recent experiments with voltage biased, doped, weakly coupled SLs [15, 20]. Before the 2012 experiments, spontaneous chaos was observed only at very low temperatures (from 4 to 77 K) [42]. The new key modification that allows observing oscillations of the current at room temperature is adding 55% of gallium in the barriers. The technical reasons are discussed in [20] and references cited therein. Early theoretical explanations of spontaneous chaotic oscillations at low temperature are based on complex dynamics of wave front solutions [43] when applied to discrete model equations such as those of Section 2; see Ref. [44]. At room temperature, wave fronts are not sharp, and spontaneous chaos arises due to other reasons, as explained in Ref. [27] and in what follows.
3.1.1 Internal noise induces and enhances chaos
We have found self-sustained oscillations in two voltage intervals that appear as plateaus in the SL current-voltage characteristics, for \(V<(\mathcal{E}_{C2}-\mathcal{E}_{C1})/e\) and for \(V>(\mathcal {E}_{C2}-\mathcal{E}_{C1})/e\). At the left end of both intervals, small-amplitude current oscillations appear as supercritical Hopf bifurcations from the stationary state. They are caused by the repeated creation of field pulses that dissolve before arriving at the collector. The range of voltages for which this behavior is observed is much narrower on the first voltage interval than on the second one. On both voltage intervals, oscillations die via supercritical Hopf bifurcations. The reverse tunneling current \(J_{i\to i+1}^{-}\) given by (5) is much larger for the smaller fields at the first voltage plateau than for the larger fields at the second plateau. The internal noise is correspondingly larger compared with the mean current at the first plateau. Spontaneous chaotic oscillations in the first voltage interval were discussed in Ref. [27]. Here we shall present similar results obtained for voltages on the second plateau.
3.1.2 Comparison with experiments
Experiments [15, 20, 31] show that oscillations appear for voltages on the first plateau and that they have frequencies about 7.5 times larger than those predicted by simulations of our mathematical model [27]. The current spikes observed in experiments are more irregular than those appearing in simulations. These features of oscillations observed in experiments point to the presence of imperfections not taken into account in the model. In earlier work on the role of imperfections [37], numerical simulations of a related discrete model showed that a 3% fluctuation in doping density could increment by a factor of 5 the oscillation frequency. Obvious imperfections that should be taken into account in our model include: (i) fluctuations of the doping density, (ii) fluctuations in \(d_{B}\) and \(d_{W}\), (iii) fluctuations in \(V_{B}\). Once imperfections are included in the mathematical model, we can pose the objective of optimizing chaos, i.e., introducing intentional imperfections so as to widen the voltage intervals for which there are chaotic oscillations, and increase the LLE and the complexity of attractors. These features would increase the usefulness of the device as a true random number generator.
3.2 Random Bit Generation from chaotic oscillations
There are a number of ways to obtain a RBG out of a chaotic signal. In this section, we will explain the methods used by Li et al. [15], using one of the figures they extracted from experimental measurements.
The generation of the random bit stream from the chaotic consists of the following two steps. First, we calculate the nth derivative using \(n+1\) successive values of the recorded signal. Second, we append the m least significant bits (LSBs) of the results of the nth derivative to the bit sequence. Recall that the LSBs are the most sensitive to small fluctuations. Using \(n=4\) and retaining \(m=5\) LSBs out of 8 bits, truly random bits are generated at 6.25 Gbit/s using a sampling rate of 1.25 GHz [15]. For higher sampling rates, larger values of n are needed to achieve verified randomness using the NIST statistical test suite [47]. Then a more efficient method is to use a linear combination of chaotic signals to fill the time interval between large peaks. These signals may come from different SLs or from far segments of the recorded chaotic signal of a single SL, as in [15]. In the latter case, the chaotic nature of the signal ensures lack of correlation between the segments thereof. To minimize the possible emergence of bias in the combined analog signals, each pair of signals is combined by subtraction [4]. Figure 4(c) shows a linear combination SL1 + SL3 − SL2 − SL4 of four uncorrelated traces of the chaotic signal, digitized at 40 GHz. Li et al. obtained a 40 Gbit/s RBG with verified randomness using such a linear combination, a 10 GHz sampling rate and 4 LSBs. They obtained a faster rate verified RBG by combining 6 signals and a faster sampling rate. See [15] for details.
4 Conclusions
The discovery of fast spontaneous chaotic oscillations of the current through semiconductor superlattices at room temperature brings to light their possible applications as true random bit generators [15]. Fast true random bit generators coming from tiny submicron all-electronic devices could be invaluable in secure communications and data storage. In this paper, we have discussed a mathematical model to describe spontaneous chaos in idealized superlattices with identical wells and barriers. Our numerical simulations show that spontaneous chaos possibly may appear directly from a two-frequency quasiperiodic attractor. We have also shown that the unavoidable shot and thermal noises existing in the nanostructure both enhance existing deterministic chaos (increasing its fractal dimension and largest Lyapunov exponent) and induce chaos in nearby voltage intervals. We have discussed that the differences between numerical and experimental results may be due to imperfections in the doping density, the gallium content in the barriers, and the size thereof. A better model needs to be developed to discuss the imperfections and their effect in the chaotic oscillations: ideally we could tune chaos via the introduction of controlled imperfections. We also explain how to extract verified random bit generators from a chaotic signal by digitalization and extraction of least significant bits from high order numerical derivatives, or by combining several chaotic signals coming either several superlattices or from far apart segments of the same long chaotic signal.
Declarations
Acknowledgements
This work has been supported by the Spanish Ministerio de Economía y Competitividad grants FIS2011-28838-C02-01 and MTM2014-56948-C2-2-P.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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