The synchronization process of two mutually delayed coupled deterministic chaotic maps is demonstrated both analytically and numerically. The synchronization is preserved when the mutually transmitted signal is concealed by two commutative private fi
lters that are placed on each end of the communication channel. We demonstrate that when the transmitted signal is a convolution of the truncated time delayed output signals or some powers of the delayed output signals synchronization is still maintained. The task of a passive attacker is mapped onto Hilberts tenth problem, solving a set of nonlinear Diophantine equations, which was proven to be in the class of NP-Complete problems. This bridge between two different disciplines, synchronization in nonlinear dynamical processes and the realm of the NPC problems, opens a horizon for a new type of secure public-channel protocols.
Small networks of chaotic units which are coupled by their time-delayed variables, are investigated. In spite of the time delay, the units can synchronize isochronally, i.e. without time shift. Moreover, networks can not only synchronize completely,
but can also split into different synchronized sublattices. These synchronization patterns are stable attractors of the network dynamics. Different networks with their associated behaviors and synchronization patterns are presented. In particular, we investigate sublattice synchronization, symmetry breaking, spreading chaotic motifs, synchronization by restoring symmetry and cooperative pairwise synchronization of a bipartite tree.
Two chaotic systems which interact by mutually exchanging a signal built from their delayed internal variables, can synchronize. A third unit may be able to record and to manipulate the exchanged signal. Can the third unit synchronize to the common c
haotic trajectory, as well? If all parameters of the system are public, a proof is given that the recording system can synchronize as well. However, if the two interacting systems use private commutative filters to generate the exchanged signal, a driven system cannot synchronize. It is shown that with dynamic private filters the chaotic trajectory even cannot be calculated. Hence two way (interaction) is more than one way (drive). The implication of this general result to secret communication with chaos synchronization is discussed.
