Pseudorandom number generator based on the Bernoulli map on cubic algebraic integers

We develop a pseudorandom bit generator using chaotic true orbits of the Bernoulli map on real cubic algebraic integers having complex conjugates. Although this generator has a high computational cost, it exactly simulates the Bernoulli map that can generate ideal random binary sequences. In particular, we clarify a seed selection method that can select initial points (i.e., seeds) without bias and can avoid overlaps in latter parts of the pseudorandom binary sequences derived from them. Moreover, in order to evaluate the memory usage of our generator, we give upper bounds concerning the growth of the representation of points on a true orbit. We also report results of a large variety of tests indicating that the generated pseudorandom sequences have good statistical properties as well as an advantage over one of the most popular generators at this point, the Mersenne Twister MT19937.