Properties making a chaotic system a good pseudo random number generator

We discuss the properties making a deterministic algorithm suitable to generate a pseudo random sequence of numbers: high value of Kolmogorov-Sinai entropy, high dimensionality of the parent dynamical system, and very large period of the generated sequence. We propose the multidimensional Anosov symplectic (cat) map as a pseudo random number generator. We show what chaotic features of this map are useful for generating pseudo random numbers and investigate numerically which of them survive in the discrete state version of the map. Testing and comparisons with other generators are performed.

Figure 1

The $ϵ$ block entropy for the Fibonacci map with ${\tau }_1=2,{\tau }_2=5,a=b=1$, and different values of $ϵ$. The change of the slope from $\ln (1∕ϵ)$ to $h_{KS}$ is clearly visible for $m={\tau }_2=5$.

Figure 2

$H_4\left(ϵ\right)-H_3\left(ϵ\right)$ for the 2D cat map of Eq. (26) with $M=2^{20}$ and different values of $a,b$ as a function of $ϵ$. The horizontal lines indicate the $h_{KS}$ values.

Figure 3

The period of the orbits for the multidimensional cat map, with different values of $N$, as a function of $M^N$ (only the prime values of $M$ are considered). The straight lines refer to the probabilistic argument $\mathcal{T}\simeq M^N$.