Complexity of Solving Hash Puzzles Using Revised Boyer Quantum Algorithm

This study investigates the computational complexity of solving hash puzzles using quantum algorithms. A hash puzzle is formalized as a problem for finding a pre-image whose hash value falls within a specified target set. It is known that Grover quantum search algorithm and Boyer algorithm, which is a Bayesian utilization of Grover algorithm, can compute the preimage of a given hash value more efficiently than non-quantum algorithms. However, those quantum algorithms use implicit assumptions and approximations that are not compatible with a general hash puzzle. This study revises Boyer algorithm by excluding those assumptions and approximations, and clarifies the relation between the size of the target set of a hash puzzle and the complexity necessary for solving the puzzle. It is shown that, in the quantum framework, a hash puzzle with fewer targets can be easier to solve than a hash puzzle with more targets, which contradicts to the classical understanding where puzzles with fewer targets are more difficult to solve.