Binary Chaos Synchronization in Elementary Cellular Automata

Abstract

It is shown that 8 of all 256 elementary cellular automata have the property of chaos synchronization using a binary driving sequence. The binary synchronization signal ensures good noise robustness. A simple measure for characterizing long periodic cycles as being "chaotic" was introduced. It is also shown that certain relations between the profile of attractors shaped by the local rule in the finite state space and the property of chaos synchronization exists. A simple modulation–demodulation scheme is proposed allowing to build spread spectrum communication links at low implementation costs while the cellular automata state space (number of cells) can be taken as large as required by cryptographic reasons.

FPGA implementation of chaotic cellular automaton with binary synchronization property

“…As shown in [3][6] and confirmed by simulations of our FPGA implementation, synchronization between Rx (receiver) and Tx (transmitter) counters occurs in a [mite number of cycles and is independent on the two different starting states of the two counters, provided that a clock recovery scheme is used in the receiver. Various schemes for digital clock recovery may be used, i.e.…”

“…Another direction of our research detailed in [3] was to provide some measure of chaos (randomness) in finite automata. In [3] such a measure was proposed, observing that in a "chaotic counter", unlike in a "normal counter" the jumps (in terms of Hamming distance) between consecutive binary vector states becomes nl2 on average instead of 1.…”

“…In [3] such a measure was proposed, observing that in a "chaotic counter", unlike in a "normal counter" the jumps (in terms of Hamming distance) between consecutive binary vector states becomes nl2 on average instead of 1. Therefore for any arbitrary counting cycle Cj of length Lj a scattering coefficient S j is defined by averaging the Hamming distances between all consecutive binary vector states in that cycle: (2) where k is the time index of consecutive states in the cycle ''j'' A degree of chaos Aj is defined such that it becomes maximum if S j = 0.5 and zero for the extreme, non-chaotic cases of both fixed points and period 2 cycles (with S j = 0 and S j = 1 respectively):…”

“…This counter is a binary automaton providing addresses in the range [0, 2n -1], where n is the number of cells. A detailed description of the theory behind binary synchronization in a certain class of HCA is provided in [3]. There we identified the conservative elementary CA with rule ID=45 (as well as its "twins" ID=75, ID=89 and ID=IOI) as the best CA among all elementary CA with 3 neighbors exhibiting both chaotic behavior (in fact pseudo-random sequence generator) and binary synchronization (one bit of the suffices to recover the entire state in the receiver).…”

An Information Theoretic Analysis of Random Number Generator based on Cellular Automaton

“…It is believed that in spite their simplicity, Elementary Cellular Automata (ECA) show all types of behaviors which can be observed in Cellular Automata in higher dimensions and with more complex local functions. Since the number of possible Elementary Cellular Automata is limited to , these systems have been examined from different perspectives [71], [72].…”