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International audience; In this paper, we investigate the influence of different numerical solutions methods on the chaoticity of fractional chaotic systems. In particular, fractional Chen system is discussed and approximated from both analytical and numerical points of view applying different calculation methods. The impacts of the chosen methods to the chaotic behavior of the fractional system are compared and analyzed.

International audience; Nonlinear dynamic systems and chaotic systems have been quite exhaustively researched in the domain of cryptography. However, the possibility of using the fractional chaotic system in the cryptosystem design has been much less explored while it bears advantages such as enlarged keyspace, compared to classical nonlinear systems. This paper, therefore, proposes a novel structure for the pseudo-random number generator based on fractional chaotic systems which consists of 3 different fractional chaotic systems, namely fractional Chen's system, Lu's system, and fractional generalized double-humped logistic map(FGDHL). Then, the outputs of this fractional chaotic pseudo-random number generator(FCPRNG) are used as a keystream for an image encryption scheme. The confusion layer of the scheme is conducted by a dynamic DNA encoding and decoding method combined with a 2D cat map for the permutation in the DNA-bases level. The diffusion layer is performed through the adoption of a 32 bits discrete logistic map. The performance and security analysis have been conducted for the above-designed cryptosystem, proving that the proposed cryptosystem is practical and secure in image encryption.

Two fractional chaotic maps, Lu and Chen fractional chaotic maps, have been analyzed using an innovative predictor-corrector method based on a non-uniform grid. An original chaotic law to control the grid size at each iteration has been introduced. The choice of the non-uniform grid was shown to play a fundamental role to obtain chaotic behavior for a larger fractional parameter range, while decreasing the computational time. The obtained results show that the complexity of the resulting fractional chaotic system in terms of Lyapunov exponents, Fourier transforms and chaos robustness is sensitive to the choice of the grid and can be considerably improved by the proposed method.

For both chaos-based stream ciphers and chaos-based block ciphers, key streams have a crucial influence on their security. A well designed pseudo-chaotic number generator (PCNG) that exhibits both chaotic properties and pseudo-randomness is a good candidate for creating the cryptographic key stream for encryption purposes. PCNGs are based on multiple chaotic maps. Since the majority of the chaotic maps are defined using real numbers, most of the proposed PCNGs use floating-point notations. However, this data type, especially the double-precision notation, has disadvantages of high computation cost and inefficient resource utilization. Also, the quantification errors may undermine the reliability of the produced key stream. To overcome these drawbacks, a key stream generation algorithm using a PCNG scheme is proposed in this paper. The PCNG is based on reformulated skew tent maps over a 32-bit integer field. It not only reduces the resource utilization from the hardware perspective, but also ensures the key stream performance over various operation platforms. Furthermore, the proposed PCNG uses a parameter changeable strategy, which can help to expand the key space, and thus increases the immunity against the brute-force attack. The quality of the key stream produced by the PCNG has been tested in a stream cipher. The analysis and the obtained test results have demonstrated that the proposed PCNG is secure and reliable to generate cryptographic key streams for encryption purposes.

This paper is devoted to the analysis of calculation methods for solving fractional chaotic systems and the impact of these different approaches on the behavior of the fractional chaotic system. Two widely used time domain fractional differential equations solving approaches are discussed, the fractional ABM correctorpredictor method based on Caputo fractional derivative definition, and the long memory calculation approach based on Grunwald fractional derivative. These numerical solutions calculation methods are employed to depict the phase portrait of a class of commensurate fractional chaotic systems. The Lyapunov exponent and bifurcation diagrams of the systems over various fractional orders and parameters are illustrated to detect the impact on the dynamics of the chaotic system applying different calculation approaches.

International audience; In this paper, we design a pseudo-chaotic random number generator using fractional chaotic systems. A non-uniform grid calculation method is proposed and employed to numerically solve the fractional systems by introducing a skew-tent map to vary the step size of the grid. Greater chaoticity in terms of Lyapunov exponent has been achieved by applying the proposed approach to the solution calculation of the fractional chaotic Chen's and Lu's systems. By adopting the piecewise constant argument method, one 1D fractional generalized double-humped logistic system (FGDHL) is discretized. A fractional pseudo-chaotic random number generator (FPCRNG) has been proposed by performing XOR (exclusive-or) operations to the states of the fractional Chen's system, fractional Lu's system, and the FGDHL systems. The security analysis of the generator and the statistical experiment of a stream cipher implementing the proposed FPCRNG prove that the proposed structure is efficient and can be used in the cryptosystem design.

Pseudo-chaotic number generators (PCNGs), designed based on multiple chaotic maps, have a significant influence on the security of chaos-based cryptosystems. Since the majority of the chaotic maps are based on real numbers, most of the proposed PCNGs use floating-point notations. However, this data type, especially the double precision notation, has disadvantages of high computation cost and inefficient resource utilization. Also, due to the high sensitivity of the chaotic systems and the finite precision nature of software and hardware platforms, the truncation or round-off errors may cause a big difference in the generated pseudo-chaotic numbers, which may undermine the PCNG's security reliability. To overcome these drawbacks, a new secure PCNG scheme based on the reformulated skew tent maps over a 32-bit integer field is proposed in this paper. It not only reduces the resource utilization from the hardware perspective, but also ensures the PCNG's performance over various operation platforms. Furthermore, the proposed PCNG uses a parameter changeable strategy, which expands the key space of the cryptosystem that adopts the PCNG, and thus increases the immunity against the brute-force attack. The security and statistical test results have demonstrated that the proposed PCNG has reliable security properties with good randomness and it is suitable for cryptographic applications.

In this paper, we focus on the design of fractional pseudo-chaotic random number generator (FPCRNG) based on the coupling of fractional chaotic systems. The proposed FPCRNG is composed of 3 fractional chaotic systems, including one fractional generalized double-humped logistic system, two 3D fractional systems Chen's system and Lu's system, and one classical skew-tent map. A non-uniform grid calculation method is employed by introducing the skew-tent map into the numerical calculation of the states of the Chen's system and Lu's system to obtain greater chaoticity in terms of Lyapunov exponent. The XOR (exclusive-or) operations are applied to the fractional systems to obtain the final pseudo-chaotic outputs. The security analysis and statistical experiment of a stream cipher implementing the FPCRNG prove that the proposed structure is effective and can be used in the cryptosystem.

In this paper, we design, realize and evaluate a new secure cryptosystem based on a Pseudo-Chaotic Number Generator(PCNG), a global diffusion and a block cipher in Cipher Block Chaining(CBC) mode. The proposed PCNG, defined over a finite field, is based on a discrete Chebyshev map coupled with a Pseudo-Random Number Generator(PRNG) and a discrete skew tent map. Our PCNG can remove a hidden danger of deteriorated security caused by the dynamical degradation encountered in numerical implementation of chaotic maps defined on real numbers. The global diffusion uses a Horizontal Addition Diffusion(HAD) and a Vertical Addition Diffusion(VAD) followed by a modified 2-D cat map. The block cipher in CBC mode is based on a confusion layer using the AES S-Box, followed by a diffusion layer built on the modified 2-D cat map and a key addition operation. The global diffusion enhances the diffusion process of the block cipher. Conducted security analysis and the obtained results have proven that the proposed cryptosystem design is robust with high security and that it can be applied to practical implementations.

International audience

In this paper an improved Lozi system speed performance is achieved in application to chaotic switched model. Generally speaking, the main advantage of the switched encryption model is its robustness to the noise, while the main drawback is the slow processing speed. In this article we propose to gain model productivity by adjusting parameter in Lozi chaotic generator. Chaos generator is sensitive to any structure, therefore the solution shouldn’t influence the pseudo randomness which is required for encryption. We provide the results of switched chaotic model based on Lozi chaotic generator with changeable parameter studied for chaoticity and pseudo-randomness with NIST, largest Lyapunov exponent, auto-correlation, cross-correlation and cumulative distribution.

This paper deals with chaotic encryption and observer synthesis for coherent detection using a new ring-coupled map firstly introduced by Lozi. The specificity of the switched map under investigation is that it exhibits excellent properties for chaotic encryption, but has no standard solution for the observer design. Unlike the previous works, here we propose to design a modified output injection observer, which shows to be more efficient than the extended Luenberger observer. The noise in the transmission channel is dealt with by implementing a non-coherent detection scheme.

The increasing number of e-transactions requires more secure and innovative schemes for secure information storage and transmission. Since the encryption should be unique for each transaction, there is a big necessity of new generators of very huge numbers of encryption keys, and chaotic random number generators seem to be perfectly suitable for this application. In this paper a new robust, gigaperiodic and simple in implementation chaotic generator is proposed. The generator construction is based on the principle of ring-coupling. The proposed chaotic generator successfully passed statistical and analytical tests: NIST, largest Lyapunov exponent, autocorrelation, cross-correlation, uniform distribution. For the best precision of uniform distribution, approximate density function has been applied, the distribution errors are analyzed by written software. The resulting chaotic system promises the designed robust implementation to cryptosystems.

This paper is devoted to the design of new chaotic Pseudo Random Number Generator (CPRNG). Exploring several topologies of network of 1-D coupled chaotic mapping, we focus first on two dimensional networks. Two coupled maps are studied: TTL^{RC} non-alternate, and TTL^{SC} alternate. The primary idea of the novel maps has been based on an original coupling of the tent and logistic maps to achieve excellent random properties and homogeneous /uniform/ density in the phase plane, thus guaranteeing maximum security when used for chaos base cryptography. In this aim a new nonlinear CPRNG: MTTL_{2}^{SC} is proposed. In addition, we explore higher dimension and the proposed ring coupling with injection mechanism enables us to achieve the strongest security requirements.

In order to improve the communication security, two novel chaotic generators are proposed. The first one is a single order chaotic generator using two cascading stages, and the second one is a second order chaotic generator with single and cascaded layer. The goal is to generate binary pseudo random chaotic signal with high degree of randomness. We analyze their global dynamics properties using system and signal processing tools (Lyapunov exponents, bifurcation diagrams, distribution, pseudo phase, autocorrelation, cross correlation, NIST test). A complete analysis is provided. Experimental and theoretical analyses show that the proposed generators have good cryptographic properties.

International audience; This paper is devoted to the design of new chaotic Pseudo Random Number Generator (CPRNG). Exploring several topologies of network of 1-D coupled chaotic mapping, we focus first on two dimensional networks. Two coupled maps are studied: TTL^{RC} non-alternate, and TTL^{SC} alternate. The primary idea of the novel maps has been based on an original coupling of the tent and logistic maps to achieve excellent random properties and homogeneous /uniform/ density in the phase plane, thus guaranteeing maximum security when used for chaos base cryptography. In this aim a new nonlinear CPRNG: MTTL_{2}^{SC} is proposed. In addition, we explore higher dimension and the proposed ring coupling with injection mechanism enables us to achieve the strongest security requirements.

Editors: Witold Jarczyk, Daniele Fournier-Prunaret, João Manuel Goncalves Cabral; We propose a new mechanism for undersampling chaotic numbers obtained by the ring coupling of one-dimensional maps. In the case of 2 coupled maps this mechanism allows the building of a PRNG which passes all NIST Test. This new geometric undersampling is very effective for generating 2 parallel streams of pseudo-random numbers, as we show, computing carefully their properties, up to sequences of one trillion consecutives iterates of the ring coupled mapping which provides more than 33.5 billions random numbers in very short time.

This work falls within the frame of the research project "Process Control System Integration" (PCSI) which aims to provide automated translation of the languages for plant modelling and simulation, such as Matlab, Simulink, State flow and IEC, in the frame of an object-based infrastructure, the Integrated Design Notation (IDN). The output of the IDN is to be Java code that runs in real time on the plant. The targeted application for the PCSI project used as a case tool is the control of extrusion process, and this paper discusses the modelling and control of a typical 90 mm plastic extruder.

Bandpass sigma-delta modulation is a technique used to convert high-frequency narrowband electronic signals from analog form to digital, or vice versa. It is a fundamentally nonlinear technique, and therefore can be analyzed only approximately by application of linear analysis. The aim of this work is to present certain results concerning the dynamical behavior of bandpass sigma-delta modulators with zero input based on application of methods of nonlinear dynamics and noninvertible map theory. We will consider the effect of certain circuit imperfections on the behavior of the system, and will also consider the effect of operating in a chaotic regime. While many of our results arise from application of standard methods of nonlinear dynamics, we will emphasize in particular those results that can be obtained by application of the critical lines tool of Mira. The paper, therefore, has two aims: To investigate the nonlinear behavior of bandpass sigma-delta modulators and to provide a demonstration of the applicability of the critical lines tool to a system of engineering importance.

International audience; This paper is devoted to the design of new chaotic Pseudo Random Number Generator (CPRNG). Exploring several topologies of network of 1-D coupled chaotic mapping, we focus first on two dimensional networks. Two topologically coupled maps are studied: T T L (RC) non-alternate, and T T L (SC) alternate. The primary idea of the novel maps has been based on an original coupling of the tent and logistic maps to achieve excellent random properties and homogeneous /uniform/ density in the phase plane, thus guaranteeing maximum security when used for chaos base cryptography. In this aim two new nonlinear CPRNG: M T T L (SC 2) and N T T( L2) are proposed. The maps successfully passed numerous statistical, graphical and numerical tests, due to proposed ring coupling and injection mechanisms.

In this paper, we design and implement under Matlab/Simulink some efficient digital chaotic generators for data encryption/decryption process. Some of these generators (logistic, PWLCM, Frey) are known, the others xcos(x), xexp[cos(x)], 2-D Tmap) are proposed. A number of designed generators contain cascaded layer to improve the statistical properties of the generated sequences. We introduce and demonstrate the importance of the perturbing orbit technique to avoid the dynamical degradation caused by the 2N-dimensional finite state space. This technique increases also the orbit cycle length. Finally, to quantify the security level of the proposed generators, we analyze their global dynamical properties using system and signal processing tools (Lyapunov exponents, bifurcation diagrams, distribution, pseudo phase space, autonotcorrelation, cross correlation, NIST tests ldquoNational Institute of Standards and Technology ldquo). Experimental and theoretical analyses show that the proposed generators have good cryptographic properties.

International audience; The modified Lozi system is analyzed as chaotic PRNG and synchronized via observers. The objective of the study is to investigate chaotic-based encryption method that preserves CSK model advantages, but improves the security level. The CSK model have been discussed to message encryption because it implies better resistance against noise, but there are many evidences of the model weaknesses. The investigation provides the original CSK model analyses of secure message transmission over the communication channel by examining identifiability and observability; switched regimes detection; sensitivity to initial conditions and session key; NIST tests of the encrypted signal; correlation between wrong decrypted messages; system ergodicity. The proposed model has a significant effect on the security level of the transmitted signal that successfully passed chaotic and randomness tests. The results suggest that the original CSK model can be used for information security applications.

Many systems in signal processing are of the form of discrete-time feedback loops containing a quantizer, and are modelled by discontinuous, piecewise linear and noninvertible maps. This paper aims to show the applicability of piecewise-continuous models to such systems, taking as an example a system from signal processing, the bandpass sigma-delta modulator, which is analysed in the case of ideal, imperfect and chaotic regimes for zero and sinusoidal inputs. It is shown that the phase plane consists of regions with different numbers of rank one backward iterates, separated by critical lines. Noninvertible map theory is later used to explain aspects of the complex behaviour of the sigma-delta modulator.

Chaotic systems have been employed to design pseudo-random number generators (PRNG) and applied to cryptosystems due to their promising features, such as randomness and sensitivity to initial conditions. The fractional chaotic systems, though muchless discussed than the classical integer order chaotic maps and systems, possess intriguing intricacy which can provide novelty, complexity, and extra secret keys to the Chaotic PRNG (CPRNG) design, which in turn enhance the security of the cryptosystem.This thesis investigated different numerical calculation approaches for fractional chaotic systems. A non-uniform gird calculationmethod with two different grid compositions was proposed to solve the 3D fractional chaotic systems numerically. The FractionalCPRNGs (FCPRNG), which meet the randomness and statistical requirements, were designed for the first time employing threedifferent fractional chaotic systems. In addition, a stream cipher and a block cipher based on DNA encoding and decoding methods were proposed and studied using the designed FCPRNGs. Both ciphers have been verified to be secure and reliable.; Dans cette thèse, nous avons utilisé des systèmes chaotiques pour concevoir des générateurs de nombres pseudoaléatoires(PRNG) et appliqué ces derniers aux cryptosystèmes en raison de leurs caractéristiques prometteuses, telles que le caractèrealéatoire et la sensibilité aux conditions initiales. Les systèmes chaotiques fractionnaires, bien que moins discutés que les carteset systèmes chaotiques classiques d’ordre entier, possèdent une complexité inhérente qui apporte de la nouveauté, de la complexité et des clés secrètes supplémentaires à la conception Chaotic PRNG (CPRNG), qui à son tour améliore la sécurité du cryptosystème. Cette thèse a étudié les différentes approches de calcul numérique pour les systèmes chaotiques fractionnaires. Une méthode de calcul utilisant une grille non uniforme avec deux compositions de grille différentes a été proposée pour résoudre numériquement les systèmes chaotiques fractionnaires 3D. Les CPRNG Fractionnaires (FCPRNG), qui répondent aux exigences aléatoires et statistiques, ont été conçus pour la première fois en utilisant trois systèmes chaotiques fractionnaires différents. De plus, un chiffrement par flux et un chiffrement par blocs basés sur des méthodes de codage et de décodage de l’ADN ont été proposés et étudiés à l’aide des FCPRNG conçus. Les deux schémas de chiffrements ont été vérifiés comme étant sûrs et fiables.