The odd degree Chebyshev polynomials are known to be permutation polynomials on the residue class rings modulo 2^w, so that applying the Chebyshev polynomials provide closed orbits with finite periods. We investigate the regularity of the periods for polynomials which are defined by the odd degree Chebyshev polynomials. It is shown that, if the degree of the polynomials divided by 8 leaves a remainder of 3 or 5, there is one orbit whose period is the size of the ring. It is shown that, if the degree of Chebyshev polynomials divided by 8 leaves a remainder of 1 or 7, there are orbits whose periods depend on a remainder which the degree divided by 4 times the size of the ring leave. After that, we propose a public key cryptosystem which use the polynomials, and investigate the relation between the security and the period. It is shown that, if the period is too small, the security is weak.