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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 1/2) and independent of one another.

Various Auto-Correlation Functions of m-Bit Random Numbers Generated from Chaotic Binary Sequences