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Volume 2017, Article ID 8692046, 21 pages
Research Article

Finite Precision Logistic Map between Computational Efficiency and Accuracy with Encryption Applications

1Engineering Mathematics and Physics Department, Faculty of Engineering, Cairo University, Giza 12613, Egypt
2Nanoelectronics Integrated Systems Center, Nile University, Cairo 12588, Egypt
3Electronics and Communication Engineering Department, Faculty of Engineering, Cairo University, Giza 12613, Egypt

Correspondence should be addressed to Wafaa S. Sayed; ge.ude.uc.gne@deyas.s.aafaw

Received 30 July 2016; Revised 8 October 2016; Accepted 4 December 2016; Published 12 February 2017

Academic Editor: Alicia Cordero

Copyright © 2017 Wafaa S. Sayed et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Chaotic systems appear in many applications such as pseudo-random number generation, text encryption, and secure image transfer. Numerical solutions of these systems using digital software or hardware inevitably deviate from the expected analytical solutions. Chaotic orbits produced using finite precision systems do not exhibit the infinite period expected under the assumptions of infinite simulation time and precision. In this paper, digital implementation of the generalized logistic map with signed parameter is considered. We present a fixed-point hardware realization of a Pseudo-Random Number Generator using the logistic map that experiences a trade-off between computational efficiency and accuracy. Several introduced factors such as the used precision, the order of execution of the operations, parameter, and initial point values affect the properties of the finite precision map. For positive and negative parameter cases, the studied properties include bifurcation points, output range, maximum Lyapunov exponent, and period length. The performance of the finite precision logistic map is compared in the two cases. A basic stream cipher system is realized to evaluate the system performance for encryption applications for different bus sizes regarding the encryption key size, hardware requirements, maximum clock frequency, NIST and correlation, histogram, entropy, and Mean Absolute Error analyses of encrypted images.