Easy and hard functions for the Boolean hidden shift problem

We study the quantum query complexity of the Boolean hidden shift problem.
Given oracle access to f(x+s) for a known Boolean function f, the task is to
determine the n-bit string s. The quantum query complexity of this problem
depends strongly on f. We demonstrate that the easiest instances of this
problem correspond to bent functions, in the sense that an exact one-query
algorithm exists if and only if the function is bent. We partially characterize
the hardest instances, which include delta functions. Moreover, we show that
the problem is easy for random functions, since two queries suffice. Our
algorithm for random functions is based on performing the pretty good
measurement on several copies of a certain state; its analysis relies on the
Fourier transform. We also use this approach to improve the quantum rejection
sampling approach to the Boolean hidden shift problem.