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中文摘要：
在密码算法的设计中，S-盒有着信息混淆的重要功能．传统的S．盒的密码学指标一般包括线性偏差、差分特征、代数免疫度、不动点个数、雪崩效应等．2006年，Kocarev给出了有限集合上的离散混沌理论．本文借鉴该理论，在汉明距离的基础上给出了S-盒的Lyapunov指数的定义，利用该定义计算了几个密码算法中的S-盒的Lyapunov指数值，并进行了比较．证明了在欧氏距离上定义的Lyapunov指数最大的映射，按本文提出的S-盒的Lyapunov指数的定义其Lyapunov指数为0；讨论了S．盒的Lyapunov指数与S。盒的雪崩效应之间的关系，该关系实际上是混沌理论中的蝴蝶效应与密码学中的雪崩效应之间的关系．本文提出的S-盒的Lyapunov指数的定义可视为对传统的S．盒的密码学指标的补充．
英文摘要：
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.
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