Abstract
By applying the structure of the group of reduced residue classes of residue ring R=Z/2^wZ, we obtain the following results. We prove that the equations of degree two can be solved in at most polynomial time and their solutions have many branches generally. We can decode Diffie-Hellman type of key exchange algorithm given by substituting of Chebyshev polynomials in at most polynomial time, in the case that the generators of the key are even numbers. Moreover we show the characteristic of quantity of computation of the map obtained by the discretized chaotic map using Chebyshev polynomials.
