Discrete Lyapunov Exponent and Differential Cryptanalysis

Partly motivated by the developments in chaos-based block cipher design, a definition of the discrete Lyapunov exponent for an arbitrary permutation of a finite lattice was recently proposed. We explore the relation between the discrete Lyapunov exponent and the maximum differential probability of a bijective mapping (i.e., an S-box or the mapping defined by a block cipher). Our analysis shows that "good" encryption transformations have discrete Lyapunov exponents close to the discrete Lyapunov exponent of a mapping that has a perfect nonlinearity. The converse does not hold.