This paper presents a new generator of chaotic bit sequences with mixed-mode (continuous and discrete) inputs. The generator has an improved level of chaotic properties in comparison with the existing single source (input) digital chaotic bit generators. The 0-1 test is used to show the improved chaotic behavior of our generator having a chaotic continuous input (Chua, Rössler or Lorenz system) intermingled with a discrete input (logistic, Tinkerbell or Henon map) with various parameters. The obtained sequences of chaotic bits show some features of random processes with increased entropy levels, even in the cases of small numbers of bit representations. The properties of the new generator and its binary sequences compare well with those obtained from a truly random binary reference quantum generator, as evidenced by the results of the $ent$ tests.
Recently two encryption schemes were proposed by combining circular bit shift and XOR operations, under the control of a pseudorandom bit sequence (PRBS) generated from a chaotic system. This paper studies the security of these two encryption schemes and reports the following findings: 1) there exist some security defects in both schemes; 2) the underlying chaotic PRBS can be reconstructed as an equivalent key by using only two chosen plaintexts; 3) most elements in the underlying chaotic PRBS can be obtained by a differential known-plaintext attack using only two known plaintexts. Experimenta…
We deal with randomness-quantifiers and concentrate on their ability do discern the hallmark of chaos in time-series used in connection with pseudo random number generators (PRNG). Workers in the field are motivated to use chaotic maps for generating PRNGs because of the simplicity of their implementation. Although there exist very efficient general-purpose benchmarks for testing PRNGs, we feel that the analysis provided here sheds additional didactic light on the importance of the main statistical characteristics of a chaotic map, namely, i) its invariant measure and ii) the mixing constant. …
We propose generalized bit cumulants for chaotic systems, within nonextensive thermodynamic approach. In this work, we apply the first and second generalized cumulants to one dimensional logistic and logistic-like family of maps.
In this paper, a new chaotic pseudo-random number generator (PRNG) is proposed. It combines the well-known ISAAC and XORshift generators with chaotic iterations. This PRNG possesses important properties of topological chaos and can successfully pass NIST and TestU01 batteries of tests. This makes our generator suitable for information security applications like cryptography. As an illustrative example, an application in the field of watermarking is presented.
In this paper, a new pseudo-random number generator (PRNG) based on chaotic iterations is proposed. This method also combines the digits of two XORshifts PRNGs. The statistical properties of this new generator are improved: the generated sequences can pass all the DieHARD statistical test suite. In addition, this generator behaves chaotically, as defined by Devaney. This makes our generator suitable for cryptographic applications. An illustration in the field of data hiding is presented and the robustness of the obtained data hiding algorithm against attacks is evaluated.
Pseudo-Random Bit Generation (PRBG) is required in many aspects of cryptography as well as in other applications of modern security engineering. In this work, PRBG based on 2D symmetrical chaotic mappings of logistic type is considered. The sequences generated with a chaotic PRBG of this type, are statistically tested and the computational effectiveness of the generators is estimated. Considering this PRBG valid for cryptography, the size of the available key space is also calculated. Different cryptographic applications can be suitable to this PRBG, being a stream cipher probably the most imm…
A scheme is suggested of the parametric generator of chaotic oscillations with attractor represented by a kind of Smale-Williams solenoid that operates under a periodic sequence of pump pulses at two different frequencies. Simulation of chaotic dynamics using the software product Multisim is provided.
We demonstrate the physical generation of random bits at high bit rates (> 100 Gb/s) using optical chaos from a solitary laser diode and therefore without the complex addition of either external optical feedback or injection. This striking result is obtained despite the low dimension and relatively small bandwidth of the laser chaos, i.e. two characteristics that have been so far considered as limiting the performances of optical chaos-based applications. We unambiguously attribute the successful randomness at high speed to the physics of the laser chaotic polarization dynamics and the resulti…
Internet communication systems involving cryptography and data hiding often require billions of random numbers. In addition to the speed of the algorithm, the quality of the pseudo-random number generator and the ease of its implementation are common practical aspects. In this work we will discuss how to improve the quality of random numbers independently from their generation algorithm. We propose an additional implementation technique in order to take advantage of some chaotic properties. The statistical quality of our solution stems from some well-defined discrete chaotic iterations that sa…
Agent-based models have demonstrated their power and flexibility in Econophysics. However their major challenge is still to devise more realistic simulation scenarios. The complexity of Economy makes appealing the idea of introducing chaotic number generators as simulation engines in these models. Chaos based number generators are easy to use and highly configurable. This makes them just perfect for this application.
The linear response of synchronized chaotic units with delayed couplings and feedback to small external perturbations is investigated in the context of communication with chaos synchronization. For iterated chaotic maps, the distribution of distances is calculated numerically and, for some special cases, analytically as well. Depending on model parameters, this distribution has power law tails leading to diverging moments of distances in the region of synchronization. The corresponding linear equations have multiplicative and additive noise due to perturbations and chaos. The response to small…
The linear response of synchronized time-delayed chaotic systems to small external perturbations, i.e., the phenomenon of chaos pass filter, is investigated for iterated maps. The distribution of distances, i.e., the deviations between two synchronized chaotic units due to external perturbations on the transfered signal, is used as a measure of the linear response. It is calculated numerically and, for some special cases, analytically. Depending on the model parameters this distribution has power law tails in the region of synchronization leading to diverging moments of distances. This is a co…
Utilization of chaotic signals for covert communications remains a very promising practical application. Multiple studies indicated that the major shortcoming of recently proposed chaos-based communication schemes is their susceptibility to noise and distortions in communication channels. In this talk we discuss a new approach to communication with chaotic signals, which demonstrates good performance in the presence of channel distortions. This communication scheme is based upon chaotic signals in the form of pulse trains where intervals between the pulses are determined by chaotic dynamics of…
New chaos-based communication schemes for transmission of analog and digital information are suggested. The carrier signal is produced by chaotic generator having well-defined oscillation phases at least on short time intervals. For data extraction, a special procedure of phase detection for chaotic signals is developed. An efficiency of the scheme is demonstrated by examples of coupled generators of structurally stable (robust) chaos with hyperbolic attractors of Smale-Williams and Arnold's "cat map" types. Robustness of the transmitter and the receiver makes it possible to transmit informati…
In this work, Pseudo-Random Bit Generation (PRBG) based on 2D chaotic mappings of logistic type is considered. The sequences generated with two Pseudorandom Bit Generators (PRBGs) of this type are statistically tested and the computational effectiveness of the generators is estimated. The role played by the symmetry and the geometrical properties of the underlying chaotic attractors is also explored. Considering these PRBGs valid for cryptography, the size of the available key spaces are calculated. Additionally, a novel mechanism called 'symmetry-swap' is introduced in order to enhance the PR…
Departing from the geodesic flow on a surface of negative curvature as a classic example of the hyperbolic chaotic dynamics, we propose an electronic circuit operating as a generator of rough chaos. Circuit simulation in NI Multisim software package and numerical integration of the model equations are provided. Results of computations (phase trajectories, time dependences of variables, Lyapunov exponents and Fourier spectra) show good correspondence between the chaotic dynamics on the attractor of the proposed system and of the Anosov dynamics for the original geodesic flow.
We discuss two properties making a deterministic algorithm suitable to generate a pseudo random sequence of numbers: high value of Kolmogorov-Sinai entropy and high-dimensionality. We propose the multi dimensional Anosov symplectic (cat) map as a Pseudo Random Number Generator. We show what chaotic features of this map are useful for generating Pseudo Random Numbers and investigate numerically which of them survive in the discrete version of the map. Testing and comparisons with other generators are performed.
Lots of researches indicate that the inefficient generation of random numbers is a significant bottleneck for information communication applications. Therefore, Field Programmable Gate Array (FPGA) is developed to process a scalable fixed-point method for random streams generation. In our previous researches, we have proposed a technique by applying some well-defined discrete chaotic iterations that satisfy the reputed Devaney's definition of chaos, namely chaotic iterations (CI). We have formerly proven that the generator with CI can provide qualified chaotic random numbers. In this paper, th…
Mean value and cross-covariance function of chaotic binary sequences are evaluated for chaotic maps with specific properties. We also take into account the effect of fixed- and floating-point representations on statistical properties of chaotic generators. Thus, one is able to obtain possible candidates for pseudo-random binary sequences generation. Results of statistical tests applied to chaotic binary sequences are presented. The chaotic binary sequences thus obtained could be used for security improvement in IEEE 802.11 standard.
With the constant increase of the number of autonomous vehicles and connected objects, tools to understand and reproduce their mobility models are required. We focus on chaotic dynamics and review their applications in the design of mobility models. We also provide a review of the nonlinear tools used to characterize mobility models, as it can be found in the literature. Finally, we propose a method to generate traces for a given scenario involving moving people, using tools from the nonlinear analysis domain usually dedicated to topological analysis of chaotic attractors.
Excitable media are prevalent models for describing physical, chemical, and biological systems which support wave propagation. In this letter, we show that the time evolution of the medium state at the wave fronts can be determined by complicated chaotic attractors. Wave front dynamics can be controlled by initial data choice. Building on this groundwork, we show that there is a mechano-chemical analog of the Universal Turing machine for morphogenesis problems. Namely, a fixed mechano-chemical system can produce any prescribed cell pattern depending on its input (initial data). This universal …
Central Pattern Generators (CPGs) in invertebrates are comprised of networks of neurons in which every neuron has reciprocal connections to other members of the CPG. This is a ``closed'' network topology. An ``open'' topology, where one or more neurons receives input but does not send output to other member neurons, is not found in these CPGs. In this paper we investigate a possible reason for this topological structure using the ability to perform a biological functional task as a measure of the efficacy of the network. When the CPG is composed of model neurons which exhibit regular membrane …
A nonextensive thermostatic approach to chaotic dynamical systems is developed by expressing generalized Tsallis distribution as escort distribution. We explicitly show the thermodynamic limit and also derive the Legendre Transform structure. As an application, bit variance is calculated for ergodic logistic map. Consistency of the formalism demands a relation between box size ($ε$) and degree of nonextensivity, given as $(1-q)\sim -1/{\rm ln} ε$. This relation is numerically verified for the case of bit variance as well as using basic definition of Tsallis entropy.
Complex spatiotemporal dynamics have been a subject of recent experimental investigations in optical frequency comb microresonators and in driven fiber cavities with a Kerr-type media. We show that this complex behavior has a spatiotemporal chaotic nature. We determine numerically the Lyapunov spectra, allowing to characterize different dynamical behavior occurring in these simple devices. The Yorke-Kaplan dimension is used as an order parameter to characterize the bifurcation diagram. We identify a wide regime of parameters where the system exhibits a coexistence between the spatiotemporal ch…
In this work we show a strategy for synchronization of three optoelectronic oscillators (OEO) operating in chaotic regime. Two applications of synchronized OEOs in secure communications are considered. In the first one the OEO is used to produce a pseudo-random bit sequence. The second application is an optical setup for secure transmission of sampled analog signals. Using numerical simulations, we calculated the bit error rate taking into account parameter mismatch noise and Gaussian noise in the input optical power. The conditions for error rate of up to 15% during key generation are shown.
Designing a pseudorandom number generator (PRNG) is a difficult and complex task. Many recent works have considered chaotic functions as the basis of built PRNGs: the quality of the output would indeed be an obvious consequence of some chaos properties. However, there is no direct reasoning that goes from chaotic functions to uniform distribution of the output. Moreover, embedding such kind of functions into a PRNG does not necessarily allow to get a chaotic output, which could be required for simulating some chaotic behaviors. In a previous work, some of the authors have proposed the idea of …
This study proposes a pseudo random number generator of q-Gaussian random variables for a range of q values, -infinity < q < 3, based on deterministic chaotic map dynamics. Our method consists of chaotic maps on the unit circle and map dynamics based on the piecewise linear map. We perform the q-Gaussian random number generator for several values of q and conduct both Kolmogorov-Smirnov (KS) and Anderson-Darling (AD) tests. The q-Gaussian samples generated by our proposed method pass the KS test at more than 5% significance level for values of q ranging from -1.0 to 2.7, while they pass the AD…
The chaotic systems have been found applications in diverse fields such as pseudo random number generator, coding, cryptography, spread spectrum (SS) communications etc. The inherent capability of generating a large space of PN sequences due to sensitive dependence on initial conditions has been the main reason for exploiting chaos in spread spectrum communication systems. This behaviour suggests that it is straightforward to generate a variety of initial condition induced PN sequences with nice statistical properties by quantising the output of an iterated chaotic map. In the present paper th…
Chaotic systems, presenting complex and non-reproducible dynamics, may be found in nature from the interaction between planets to the evolution of the weather, but can also be tailored using current technologies for advanced signal processing. However, the realization of chaotic signal generators remains challenging, due to the involved dynamics of the underlying physics. In this paper, we experimentally and numerically present a disruptive approach to generate a chaotic signal from a micromechanical resonator. This technique overcomes the long-established complexity of controlling the bucklin…
This paper reports the first finding of cupolets in a chaotic Hindmarsh-Rose neural model. Cupolets (chaotic, unstable, periodic, orbit-lets) are unstable periodic orbits that have been stabilized through a particular control scheme applying a binary control sequence. We demonstrate different neural dynamics (periodic or chaotic) of the Hindmarsh-Rose model through a bifurcation diagram where the external input current, $I$, is the bifurcation parameter. We select a region in the chaotic parameter space and provide the results of numerical simulations. In this chosen parameter space, a control…
