Query complexity of generalized Simon's problem

Simon's problem plays an important role in the history of quantum algorithms, as it inspired Shor to discover the celebrated quantum algorithm solving integer factorization in polynomial time. Besides, the quantum algorithm for Simon's problem has been recently applied to break symmetric cryptosystems. Generalized Simon's problem, denoted by $\mathsf{GSP}(p,n,k)$, is a natural extension of Simon's problem. In this paper we consider the query complexity of $\mathsf{GSP}(p,n,k)$. First, it is not difficult to design a quantum algorithm solving the above problem with query complexity of $O(n-k)$. However, so far it is not clear what is the classical query complexity of the problem, and revealing this complexity is necessary for clarifying the computational power gap between quantum and classical computing on the problem. To tackle this problem, we prove that any classical (deterministic or randomized) algorithm for $\mathsf{GSP}(p,n,k)$ has to query at least $\Omega\left(\max\{k, \sqrt{p^{n-k}}\}\right)$ values and any classical nonadaptive deterministic algorithm for $\mathsf{GSP}(p,n,k)$ has to query at least $\Omega\left(\max\{k, \sqrt{k \cdot p^{n-k}}\}\right)$ values. Hence, we clearly show the classical computing model is less powerful than the quantum counterpart, in terms of query complexity for the generalized Simon's problem. Moreover, we obtain an upper bound $O\left(\max\{k, \sqrt{k \cdot p^{n-k}}\}\right)$ on the classical deterministic query complexity of $\mathsf{GSP}(p,n,k)$, by devising a subtle classical algorithm based on group theory and the divide-and-conquer approach. Therefore, we have an almost full characterization of the classical deterministic query complexity of the generalized Simon\u2019s problem.