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

Polynomial-time quantum algorithms for finding the linear structures of Boolean function 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 $$\alpha$$ α such that $$f(x)=f(x\oplus \alpha )$$ f ( x ) = f ( x ⊕ α ) . We do not know the n-bit string $$\alpha$$ α , excepting the Hamming weight $$W(\alpha )=m, 1\le m\le n$$ W ( α ) = m , 1 ≤ m ≤ n . Find the string $$\alpha$$ α .” In case $$W(\alpha )=1$$ W ( α ) = 1 , we present an efficient quantum algorithm to solve this linear construction for the general $$n$$ n . In case $$W(\alpha )>1$$ W ( α ) > 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 ( n 2 ) . From this view, the quantum algorithm is more efficient than any classical algorithm. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Quantum Information Processing Springer Journals

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

Quantum Information Processing, Volume 14 (4) – Feb 4, 2015
12 pages

1

/lp/springer_journal/polynomial-time-quantum-algorithms-for-finding-the-linear-structures-zIpSdan0mu
Publisher
Springer Journals
Subject
Physics; Quantum Information Technology, Spintronics; Quantum Computing; Data Structures, Cryptology and Information Theory; Quantum Physics; Mathematical Physics
ISSN
1570-0755
eISSN
1573-1332
D.O.I.
10.1007/s11128-015-0940-1
Publisher site
See Article on Publisher Site

### 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 $$\alpha$$ α such that $$f(x)=f(x\oplus \alpha )$$ f ( x ) = f ( x ⊕ α ) . We do not know the n-bit string $$\alpha$$ α , excepting the Hamming weight $$W(\alpha )=m, 1\le m\le n$$ W ( α ) = m , 1 ≤ m ≤ n . Find the string $$\alpha$$ α .” In case $$W(\alpha )=1$$ W ( α ) = 1 , we present an efficient quantum algorithm to solve this linear construction for the general $$n$$ n . In case $$W(\alpha )>1$$ W ( α ) > 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 ( n 2 ) . From this view, the quantum algorithm is more efficient than any classical algorithm.

### Journal

Quantum Information ProcessingSpringer Journals

Published: Feb 4, 2015

### References

• On linear structures of Boolean functions
Wu, WL; Xiao, GZ
• Linear symmetries of Boolean functions
Xiao, W

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