Enhancing unimodal digital chaotic maps through hybridisation
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ORIGINAL PAPER
Enhancing unimodal digital chaotic maps through hybridisation Moatsum Alawida · Azman Samsudin · Je Sen Teh
Received: 10 August 2018 / Accepted: 28 January 2019 © Springer Nature B.V. 2019
Abstract Despite sharing many similar properties with cryptography, digitizing chaotic maps for the purpose of developing chaos-based cryptosystems leads to dynamical degradation, causing many security issues. This paper introduces a hybrid chaotic system that enhances the dynamical behaviour of these maps to overcome this problem. The proposed system uses cascade and combination methods as a nonlinear chaotification function. To depict the capability of the proposed system, we apply it to classical chaotic maps and analyse them using theoretical analysis, conventional, fractal and randomness evaluations. Results show that the enhanced maps have a larger chaotic range, low correlation, uniform data distribution and better chaotic properties. As a proof of concept, simple pseudorandom number generators are then designed based on a classical map and its enhanced variant. Security comparisons between the two generators indicate that the generator based on the enhanced map has better statistical properties as compared to its classical counterpart. This finding showcases the capability of the proposed system in improving the performance of chaos-based algorithms. M. Alawida · A. Samsudin · J. S. Teh (B) School of Computer Sciences, University Sains Malaysia (USM), 11800 Gelugor, Pulau Pinang, Malaysia e-mail: [email protected] M. Alawida e-mail: [email protected] A. Samsudin e-mail: [email protected]
Keywords Chaotic map · Dynamical degradation · Pseudorandom number generator · Unimodal map · Sine map · Logistic map
1 Introduction Many cryptographic algorithms have been proposed based on digital chaos in the past two decades due to the strong relationship between chaos and cryptography. Chaos-based algorithms mostly use the chaotic trajectory as a pseudorandom number generator (PRNG) in their security applications [1–5]. However, digital chaotic maps suffer from dynamical degradation and do not depict ideal chaotic behaviour as compared to their analog counterparts because they are computed using finite precision. They have degraded chaotic characteristics such as shorter cycle lengths, strong correlation characteristics, low complexity and subpar trajectory distribution [6–9]. Therefore, these maps should be enhanced when realised on machines with finite precision in order to provide higher security for its cryptographic algorithms. Digital chaotic maps can be classified into two main categories: high-dimensional (HD) and onedimensional (1D) chaotic maps [10–13]. A HD chaotic map has a complex structure with more than one control parameter and/or system variable. In addition, it has better security features such as uniform data distribution, high randomness and resistance to control parameter or initial condition estimation. However, HD
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chaotic maps are costly to compute due to their c
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