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Quantum Information Processing

Quantum algorithms for learning Walsh spectra of multi-output Boolean functions

Cui Jingyi, Jiansheng Guo, Linhong Xu, Mingming Li

Published May 2, 2019• 2 citations
Publisher
DOI
10.1007/s11128-019-2303-9

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Boolean functions are important building blocks in cryptography for their wide application in both stream and block cipher systems. For cryptanalysis of such systems one tries to find out linear functions that are correlated to the Boolean functions used in the crypto system. Let fff be an nnn-variable Boolean function and its Walsh spectra is denoted by Wf(ω)W_f(ω)Wf​(ω) at the point ω∈{0,1}nω\in \{0, 1\}^nω∈{0,1}n. The Boolean function is available in the form of an oracle. We like to find an ωωω such that Wf(ω)≠0W_f(ω) \neq 0Wf​(ω)=0 as this will provide one of the linear functions which are correlated to fff. We show that the quantum algorithm proposed by Deutsch and Jozsa (1992) solves the above mentioned problem in constant time. However, the best known classical algorithm to solve this problem requires exponential time in nnn. We also analyse certain classes of cryptographically significant Boolean functions and highlight how the basic Deutsch-Jozsa algorithm performs on them.
Boolean functions are important building blocks in cryptography for their wide application in both stream and block cipher systems. For cryptanalysis of such systems, one tries to find out linear functions that are correlated to the Boolean functions used in the crypto system. Let f be an n-variable Boolean function and its Walsh spectra is denoted by Wf(ω) at the point ω ∈ {0, 1}n. The Boolean function is available in the form of an oracle. We like to find a ω such that Wf(ω) ≠ 0 as this will provide one of the linear functions which are correlated to f. We show that the quantum algorithm proposed by Deutsch and Jozsa7 solves this problem in constant time. However, the best known classical algorithm to solve the problem requires exponential time in n. We also analyze certain classes of cryptographically significant Boolean functions and highlight how the basic Deutsch–Jozsa algorithm performs on them.