Summary: Cryptography by Cellular Automata
How Fast Can Complexity Emerge in Nature?
Benny Applebaum # Yuval Ishai + Eyal Kushilevitz +
Computation in the physical world is restricted by the following spatial locality constraint: In a single
unit of time, information can only travel a bounded distance in space. A simple computational model
which captures this constraint is a cellular automaton: A discrete dynamical system in which cells are
placed on a grid and the state of each cell is updated via a local deterministic rule that depends only on
the few cells within its close neighborhood. Cellular automata are commonly used to model real world
systems in nature and society.
Cellular automata were shown to be capable of a highly complex behavior. However, it is not clear
how fast this complexity can evolve and how common it is with respect to all possible initial config
urations. We examine this question from a computational perspective, identifying ``complexity'' with
computational intractability. More concretely, we consider an ncell automaton with a random initial
configuration, and study the minimal number of computation steps t = t(n) after which the following
problems can become computationally hard:
. The inversion problem. Given the configuration y at time t, find an initial configuration x which
leads to y in t steps.
. The prediction problem. Given an arbitrary sequence of # > n intermediate values of cells in the