The query complexity of order-finding

Abstract

We consider the problem where P is an unknown permutation on {0,1,...,2^n - 1}, y is an element of {0,1,...,2^n - 1}, and the goal is to determine the minimum r > 0 such that P^r(y) = y (where P^r is P composed with itself r times). Information about P is available only via queries that yield P^x(y) from any x in {0,1,...,2^m - 1} and y in {0,1,...,2^n - 1} (where m is polynomial in n). The main resource under consideration is the number of these queries. 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.