Chaotic Communication via Temporal Transfer Entropy

We propose a new perspective on communication using chaos. A binary message is encoded into the temporally causal relations on a coupled maps ring of $N$ chaotic nodes. From the analysis of temporal transfer entropy, the masked information can be recovered from transmitted signals at the receiver. The communication scheme has been demonstrated to be robust against external noise and some traditional attacks.

Figure 1

(a) Communication scheme based on temporal causality of the coupled maps ring. (b) Encoding process of the binary information.

Figure 2

Temporal transfer entropies ${\mathrm{TE}}_{S_1,S_2}^t$ and ${\mathrm{TE}}_{S_1,S_1}^t$ as functions of time for a CMR of Ulam maps. At $t={10}^4$, the coupling direction is reversed. The two qualities of TTE clearly reflect the variation of the causality.

Figure 3

(a) Relaxation times $T_R$ as functions of $\Delta p$. The slope of the reference lines is $-1.1$. (b) The dependence of TTE on $\Delta p$. With the increase of $\Delta p$, the distinction between the predominant state (empty circles) and the inferior state (solid circles) becomes smaller.

Figure 4

(a) A sequence of binary messages ${\mu }^k$ with the length of each information bit $T_{\mu }=200$. (b) The evolution of TTE in the directions of $S_1\to S_2$ (blue solid line) and $S_2\to S_1$ (red dotted line). (c) The sequence of ${\sigma }^k$: ${\sigma }^k>0$ indicates ${\mu }^k=1$, while ${\sigma }^k<0$ indicates ${\mu }^k=0$.

Figure 5

The error probability $P_e$ versus the signal-to-noise-ratio (SNR). In each simulation, ${10}^5$ randomly generated symbols are transmitted.