Polynomial-time quantum algorithms for finding the linear structures of Boolean function

Abstract

In this paper, we present quantum algorithms to solve the linear structures of Boolean functions. "Suppose Boolean function $$f$$f: $$\{0, 1\}^{n}\rightarrow \{0, 1\}$${0,1}n?{0,1} is given as a black box. There exists an unknown n-bit string $$\alpha $$? such that $$f(x)=f(x\oplus \alpha )$$f(x)=f(x??). We do not know the n-bit string $$\alpha $$?, excepting the Hamming weight $$W(\alpha )=m, 1\le m\le n$$W(?)=m,1≤m≤n. Find the string $$\alpha $$?." In case $$W(\alpha )=1$$W(?)=1, we present an efficient quantum algorithm to solve this linear construction for the general $$n$$n. In case $$W(\alpha )>1$$W(?)>1, we present an efficient quantum algorithm to solve it for most cases. So, we show that the problem can be "solved nearly" in quantum polynomial times $$O(n^{2})$$O(n2). From this view, the quantum algorithm is more efficient than any classical algorithm.