Suppose that we are given a function f . n n and an initial

Suppose that we are given a function f . ℤ_n → ℤ_n and an initial value x₀ ∈ ℤ_n. Define x_i = f (x_i-1) for i = 1, 2, .... Let t and u > 0 be the smallest values such that x_t+i = x_t+u+i for i = 0, 1, .... In the terminology of Pollard’s rho algorithm, t is the length of the tail and u is the length of the cycle of the rho. Give an efficient algorithm to determine t and u exactly, and analyze its running time.