Periodic Properties of Chebyshev Polynomial Sequences Over the Residue Ring $\mathbb{Z}/2^{k}\mathbb{Z}$

This brief analyzes in detail the relation of the period, the initial value, and the degree of Chebyshev polynomials over the residue ring Z/2
^k
Z. In particular, it is demonstrated that a recently proposed public-key cryptosystem based on Chebyshev polynomials over Z/2
^k
Z is not secure.