We devise a pseudorandom number generator that exactly computes chaotic true orbits of the Bernoulli map on quadratic algebraic integers. Moreover, we describe a way to select the initial points (seeds) for generating multiple pseudorandom binary sequences. This selection method distributes the initial points almost uniformly (equidistantly) in the unit interval, and latter parts of the generated sequences are guaranteed not to coincide. We also demonstrate through statistical testing that the generated sequences possess good randomness properties.
References
These lengths have been approximately evaluated through calculation of their average values7 and through numerical simulations.8
Normality is a necessary but not a sufficient condition for a sequence to be random. For example, the base-2 Champernowne number is normal in base 2, but its binary expansion is nonrandom not only in the sense of algorithmic randomness16 but also because it has statistical properties different from those of random sequences.17 In fact, we tested its first 1012 binary digits using the method described in Sec. IV, and we found that only one of the 162 tests was passed. (Incidentally, the passed test was Linear Complexity Test.).
Also, in the case of the tent map MT, we can generate pseudorandom binary sequences based on true orbits using a transformation similar to Eq. (1). Because MT on S is not injective, we cannot guarantee that latter parts of binary sequences will not coincide, in the same way as for MB. However, by taking an initial point set I, whose elements each belong to a different quadratic field, such as Ib, we can guarantee for MT not only that latter parts of the binary sequences derived from I do not coincide but also strong difference among the derived sequences.