Periodicity of odd degree Chebyshev polynomials over a ring of modulo $2^w$

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 the periodicity are expected to be useful for analysis of cryptography and designing a new pseudo random number generator. In addition, we show that degree decision problem of Chebyshev polynomial over the ring can effectively be solved by using the periodicity.