Network Analysis of Chebyshev Polynomial in a Fixed-precision Digital Domain

Recently, the dynamical systems generated by iterating a polynomial on a finite field are widely used in cryptography, physics and so on. Especially, Chebyshev polynomial plays a significant role in such application. This paper depicts the variation rules of Chebyshev integer sequence for every initial value in a ring. As for most of the initial states, the period of the corresponding sequence increase with the implementation precision. In addition, from the perspective of associate state-mapping networks, the circle structure of the Chebyshev polynomial implemented in a fixed-point arithmetic domain is investigated. It is found that the number of circles of different length remains unchanged with respect to the precision. The results can be used to understand the dynamic characteristics of Chebyshev polynomial and evaluate its cryptographic applications.