Research of Lyapunov exponent of S-boxes

Abstract

In the design of cryptographic algorithms, S-boxes provide the cryptosystems with the information confusion function. The traditional cryptography indexes of the S-boxes generally include linear deviation, differential characteristics, algebraic immunity, fixed point mumber, snowslide effect, and so on. In 2006, Kocarev et al. (Kocarev L, Szczepanski J, Amigo J M and Tomovski I 2006 IEEE Transactions on Circuits and Systems-I: regular papers 53 6 1300) set up a discrete chaos theory based on the finite set. In light of the theory in this paper, we introduce the definition of the Lyapunov exponent with Hamming distance, calculate and compare the Lyapunov exponent values of the S-boxes in several cryptographic algorithms. In this paper we prove that a map defined on the Euclidean distance has a maximal Lyapunov exponent value of 0. In this paper it is shown that the relationship between the Lyapunov exponent and the snowslide effect of the S-box is the relationship between the butterfly effect in chaos theory and the snowslide effect in cryptography. The definition of the Lyapunov exponent of the proposed S-boxes may be complementary to the traditional cryptography indexes of the S-box.