In the literature, there are many image encryption algorithms that have been constructed based on different chaotic maps. However, those algorithms do well in the cryptographic process, but still, some developments need to be made in order to enhance the security level supported by them. This paper introduces a new cryptographic algorithm that depends on a logistic and two-dimensional chaotic economic map. The robustness of the introduced algorithm is shown by implementing it on several types of images. The implementation of the algorithm and its security are partially analyzed using some statistical analyses such as sensitivity to the key space, pixels correlation, the entropy process, and contrast analysis. The results given in this paper and the comparisons performed have led us to decide that the introduced algorithm is characterized by a large space of key security, sensitivity to the secret key, few coefficients of correlation, a high contrast, and accepted information of entropy. In addition, the results obtained in experiments show that our proposed algorithm resists statistical, differential, brute-force, and noise attacks.
Many researchers have used quadratic utility function to study its influences on economic games with product differentiation. Such games include Cournot, Bertrand, and a mixed-type game called Cournot-Bertrand. Within this paper, a cubic utility function that is derived from a constant elasticity of substitution production function (CES) is introduced. This cubic function is more desirable than the quadratic one besides its amenability to efficiency analysis. Based on that utility a two-dimensional Cournot duopoly game with horizontal product differentiation is modeled using a discrete time scale. Two different types of games are studied in this paper. In the first game, firms are updating their output production using the traditional bounded rationality approach. In the second game, firms adopt Puu's mechanism to update their productions. Puu's mechanism does not require any information about the profit function; instead it needs both firms to know their production and their profits in the past time periods. In both scenarios, an explicit form for the Nash equilibrium point is obtained under certain conditions. The stability analysis of Nash point is considered. Furthermore, some numerical simulations are carried out to confirm the chaotic behavior of Nash equilibrium point. This analysis includes bifurcation, attractor, maximum Lyapunov exponent, and sensitivity to initial conditions.
Securing transmission of information between legitimate transmitter and receiver sides is a great challenge for mathematicians, computer scientists and engineers in recent years. This paper aims at achieving three goals. The first of them is to introduce a novel fractional order two dimensional (2D) map having very complex chaotic behavior and distinct large positive values of Lyapunov exponents over wide range of parameters, compared with other 2D maps in literature. Secondly, a new reliable secure encryption scheme combining the associated chaotic pseudo-orbits of the proposed map with the advantages of elliptic curves in public key cryptography is suggested, for first time, and applied to colored images. The hybrid scheme is capable to confirm reliable secret keys exchange in addition to highly obscure and hide transmitted information messages. Finally, a thorough mathematical analysis of security performance and evaluation of encryption scheme immunity against all possible attacks are carried out and proved its efficiency and robustness. INDEX TERMS Chaos-based cryptography, chaotic maps, discrete fractional calculus, elliptic curves, pseudo-orbits.
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