The query complexity of order-finding

We consider the problem where π is an unknown permutation on (0, 1,..., 2ⁿ-1), γ₀∈(0, 1,..., 2ⁿ -1), and the goal is to determine the minimum r>0 such that π^r(y₀)=y₀. Information about π is available only via queries that yield π^x(y) from any x∈(0, 1,..., 2ⁿ-1) and γ_∈(0, 1,..., 2ⁿ-1) (where m is polynomial in n). The resource under consideration is the number of these queries (hence our model of computation is the decision tree). We show that the number of queries necessary to solve the problem in the classical probabilistic bounded error model is exponential in n. This contrasts sharply with the quantum bounded-error model, where a constant number of queries suffices