Abstract
This paper discusses the auto-correlation functions of m-bit random numbers obtained from m chaotic binary sequences generated by one-dimensional nonlinear maps. First, we provide the theoretical auto-correlation function of an m-bit sequence obtained by m binary sequences that are assumed to be uncorrelated to each other. The auto-correlation function is expressed by a simple form using the auto-correlation functions of the binary sequences. This implies that the auto-correlation properties of the m-bit sequences can be easily controlled by the auto-correlation functions of the original binary sequences. In numerical experiments using a computer, we generated m-bit random sequences using some chaotic binary sequences with prescribed auto-correlations generated by one-dimensional chaotic maps. The numerical experiments show that the numerical auto-correlation values are almost equal to the corresponding theoretical ones, and we can generate m-bit sequences with a variety of auto-correlation properties. Furthermore, we also show that the distributions of the generated m-bit sequences are uniform if all of the original binary sequences are balanced (i.e., the probability of 1 (or 0) is equal to ) and independent of one another.
Highlights
Chaos-based random number generation has been widely studied [1,2,3,4,5,6,7,8,9,10,11,12,13,14]
We introduce the following two chaotic maps with I = [0, 1] and f ∗ ( x ) = 1, which will be used in numerical experiments later
The auto-correlation function (ACF) of the generated binary sequence depends on the combination of the chaotic map and the binary function
Summary
Chaos-based random number generation has been widely studied [1,2,3,4,5,6,7,8,9,10,11,12,13,14]. Using one-dimensional (1-D) chaotic maps, we can design various kinds of chaotic sequences including uncorrelated and correlated ones [1,8,9,11,16], which can be adapted according to the desired properties depending on the applications. The statistical properties of the m-bit random numbers depend on the original m binary sequences. The main purpose of this paper is to discuss the controllability of the auto-correlations of the m-bit random numbers but not to generate only purely random numbers. We provided the theoretical auto-correlation function of the m-bit sequence generated by m binary sequences.
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