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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.

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.

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.

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 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.

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.

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; In this paper an improved chaotic switching algorithm based on modified Lozi system is applied. Generally speaking, the main advantage of the switched encrypted 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.

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.

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.