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This paper addresses issues surrounding the concept of fractional quantum mechanics, related to lights propagation in inhomogeneous nonlinear media, specifically restricted to a so-called gravitational optics. Besides Schrödinger–Newton equation, we have also concerned with linear and nonlinear Airy beam accelerations in flat and curved spaces and fractal photonics, related to nonlinear Schrödinger equation, where impact of the fractional Laplacian is discussed. Another important feature of the gravitational optics' implementation is its geometry with the paraxial approximation, when quantum mechanics, in particular, fractional quantum mechanics, is an effective description of optical effects. In this case, fractional-time differentiation reflexes this geometry effect as well. [ABSTRACT FROM AUTHOR]

The inherent random-like behavior and one-way property of iteration in chaotic systems provide a good basis for designing Hash function. In the era of big data, due to the increasing data capacity in applications, fast Hash functions with parallel mode are highly desirable when authenticating data integrity. We analyze the issue of how to parallelize Hash function with iterative structure. Some security requirements on parallel Hash function are presented. In addition, using chaotic map and block cipher, we construct a keyed parallel Hash function. The message blocks are firstly processed in parallel by a DM-like structure. Furthermore, a tree mode with chaotic map is utilized to combine the outputs of the hash round function in parallel. The proposed Hash function is analyzed by theory and tested by computer simulations. The test results show that the proposed scheme can resist the various common attacks against Hash functions. It satisfies the secure performance requirements of Hash function. Owing to the usage of the parallel mode to process messages, the proposed chaos-based Hash function possess high efficiency and has high potential in applications to guarantee data integrity on a parallel computing platform. [ABSTRACT FROM AUTHOR]