Quantum cryptanalysis of hash and claw-free functions

In this note, we give a quantum algorithm that finds collisions in arbitrary τ-to-one functions after only O ( 3 √ N /τ) expected evaluations of the function. Assuming the function is given by a black box, this is more efficient than the best possible classical algorithm, even allowing probabilism. We also give a similar algorithm for finding claws in pairs of functions. Furthermore, we exhibit a space-time tradeoff for our technique. Our approach uses Grover's quantum searching algorithm in a novel way.