  Odd degree Chebyshev polynomials over a ring of modulo $2^w$ have two kinds
of period. One is an "orbital period". Odd degree Chebyshev polynomials are
bijection over the ring. Therefore, when an odd degree Chebyshev polynomial
iterate affecting a factor of the ring, we can observe an orbit over the ring.
The "orbital period" is a period of the orbit. The other is a "degree period".
It is observed when changing the degree of Chebyshev polynomials with a fixed
argument of polynomials. Both kinds of period have not been completely studied.
In this paper, we clarify completely both of them. The knowledge about them
enables us to efficiently solve degree decision problem of Chebyshev polynomial
over the ring, and so a key-exchange protocol with Chebyshev polynomial over
the ring is not secure. In addition, we generalize the discussion and show that
a key-exchange protocol with more generalized permutation polynomials which
belong to a certain class is not secure.
