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