A quantum algorithm for examining oracles

In this paper, we consider a quantum algorithm for solving the following problem: ``Suppose $f$ is a function given as a black box (that is also called an oracle) and $f$ is invariant under some AND-mask. Examine a property of $f$ by querying the oracle.'' We compare the efficiency of our quantum algorithm with that of classical algorithms by evaluating the expected number of queries for each algorithm. We show that our quantum algorithm is more efficient than any classical algorithm in some cases. However, our quantum algorithm does not exhibit an exponential speedup in the size of an input, compared with the best classical algorithm. Our algorithm extracts a global property of $f$ (that is, invariance of $f$) while it neglects local properties of $f$ (that is, outputs of $f$). We can regard our algorithm as an application of Simon's algorithm.