A chaos-based approach to the design of cryptographically secure substitutions

We show that chaotic maps may be used for designing so-called substitution boxes for ciphers resistant to linear and differential cryptanalysis, providing an alternative to the algebraic methods. Our approach is based on the approximation of mixing maps by periodic transformations. The expectation behind is, of course, that the nice chaotic properties of such maps will be inherited by their approximations, at least if the convergence rate is appropriate and the associated partitions are sufficiently fine. We show that this is indeed the case and that, in principle, substitutions with close-to-optimal immunity to linear and differential cryptanalysis can be designed along these guidelines. We provide also practical examples and numerical evidence for this approximation philosophy.