Batch and stream entropy with fixed partitions for chaos-based random bit generators

Measures are proposed for reliably estimating the entropy of bits produced in an entropy source using a chaotic physical system. The measures are reliable with respect to a “guessing” attack and depend on the end-to-end method of transfer of entropy from the chaotic physical system to the bit entropy source. Fixed partitions are considered to correspond with practical methods for fast digital sampling of analog signals. We propose two different measures corresponding to the batch and streaming modes of entropy transfer. Numerical examples are provided to demonstrate features of dependence of the batch and stream entropy on fixed partitions with uniform or nonuniform types of chaos.

Figure 1

Sampling interval $\tau$ dependence of the batch entropy (left) and stream entropy (right) for the Bernoulli map (top) and logistic map (bottom). $\epsilon =2^{-3}$ and various values of the offset $\delta$ are shown. Noise $\sigma =0$. String length $d=4$.

Figure 2

$\epsilon$ dependence of the batch entropy (left) and stream entropy (right) for the Bernoulli map (top) and logistic map (bottom) with a fixed sampling interval $\tau =4$ and various values of the offset $\delta$. Noise $\sigma =0$. String length $d=4$.

Figure 3

$\epsilon$ dependence of the batch entropy (left) and stream entropy (right) for the logistic map, with a fixed sampling interval $\tau =4$, various values of the offset $\delta$, and noise $\sigma ={10}^{-4}$ (top) and $\sigma ={10}^{-1}$ (bottom). String length $d=4$.