Encryption using cellular automata chain-rules

Chain rules are maximally chaotic CA rules that can be constructed at random to provide a huge number of encryption keys — where the the CA is run backwards to encrypt, forwards to decrypt. The methods are based on the reverse algorithm and the Z-parameter [5]. 1 The CA reverse algorithm and basins of attraction In the simplest cellular automata [4], each cell in a ring of cells updates its value (0,1) as a function of the values of its k neighbours. All cells update synchronously — in parallel, in discrete time-steps, moving through a deterministic forward trajectory. Each “state” of the ring, a bit string, has one successor, but may have multiple or zero predecessors. A book, “The Global Dynamics of Cellular Automata” [5] published in 1992 introduced a reverse algorithm for finding the pre-images (predecessors) of states for any finite 1d binary CA with periodic boundary conditions, which made it possible to reveal the precise topology of “basins of attraction” for the first time — represented by state transition graphs — states linked into trees rooted on attractor cycles, which could be drawn automatically, as in Fig. 1. The software was attached to the book on a floppy disk — the origin of what later became DDLab [12]. As state-space necessarily includes every possible piece of information encoded within the size of its string, including excerpts from Shakespeare, copies of the Mona Lisa, and one’s own thumb print, and given that each unique string is linked somewhere within the graph according to a dynamical rule, this immediately suggested that a string with some relevant information could be recovered from another string linked to it in some remote location in the graph, for example by running backwards from string A (the information) to arrive after a number of time steps at string B (the encryption), then running forwards from B back to A to decrypt (or the method could be reversed) — so here was a new approach to encryption where the rule is the encryption key. Gutowitz patented analogous methods using dynamical systems, CA in particular [2], but these are different from the methods I will describe, where its crucial to distinguish a type of CA rule were the graph linking state-space has the appropriate topology to allows efficient encryption/decryption. Fig. 1. Three basins of attraction with contrasting topology, n=15, k=3. The direction of time flows inward towards the attractor, then clockwise. One complete set of equivalent trees is shown in each case, and just the Garden-ofEden (leaf) states are shown as nodes. Data for each is provided as follows: attractor period=p, volume=v, leaf density=d, longest transient=t, max indegree=Pmax. topleft: rule 250, Zleft=0.5, Zright=0.5, too convergent for encryption, p=1, v=32767, d=0.859, t=14, Pmax=1364,. topright: rule 110, Zleft=0.75, Zright=0.625, too convergent for encryption, p=295, v=10885, d=0.55, t=39, Pmax=30,. bottom: rule 30, a chain rule, Zleft=0.5, Zright=1, OK for encryption, p=1455, v=30375, d=0.042, t=321, Pmax=2,