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

In classical cryptography, many cryptographic primitives could be treated as multi-output Boolean functions. The analysis of such functions is of great interest for cryptologists owing to their wide ranges of applications. Since each multi-output Boolean function can be uniquely determined by its Wa...

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Vydané v:	Quantum information processing Ročník 18; číslo 6; s. 1 - 31
Hlavní autori:	Cui, Jingyi, Guo, Jiansheng, Xu, Linhong, Li, Mingming
Médium:	Journal Article
Jazyk:	English
Vydavateľské údaje:	New York Springer US 01.06.2019 Springer Nature B.V
Predmet:	Algorithms Amplitudes Boolean algebra Boolean functions Boundary conditions Codes Coefficients Complexity Cryptography Data Structures and Information Theory Machine learning Mathematical Physics Monte Carlo simulation Physics Physics and Astronomy Quantum Computing Quantum Information Technology Quantum Physics Queries Spectra Spintronics Walsh transforms Walsh spectrum Quantum computation Goldreich–Levin theorem Quantum learning algorithm Multi-output Boolean function
ISSN:	1570-0755, 1573-1332
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Shrnutí:	In classical cryptography, many cryptographic primitives could be treated as multi-output Boolean functions. The analysis of such functions is of great interest for cryptologists owing to their wide ranges of applications. Since each multi-output Boolean function can be uniquely determined by its Walsh transform, the Walsh spectra could reveal the properties of multi-output Boolean functions. In this paper, several quantum algorithms for learning Walsh spectra of multi-output Boolean functions are proposed. Firstly, with the usage of the amplitude estimation algorithm based on the Monte Carlo method, we present a quantum algorithm that allows one to estimate the Walsh coefficient of a multi-output Boolean function at a specified point with an additive error ϵ and probability at least 1 - δ . The corresponding query complexity is O ( ϵ - 1 log δ - 1 ) . There is an almost quadratic speedup over the classical algorithm. Secondly, we propose a generalized phase kick-back technique for multi-output Boolean functions to encode multiple Walsh coefficients on the amplitudes of states. Based on this generalized technique, a quantum Goldreich–Levin algorithm for arbitrary multi-output Boolean function F : { 0 , 1 } n → { 0 , 1 } m where m , n ∈ Z is proposed to find those Walsh coefficients satisfying the threshold boundary condition τ with probability at least 1 - δ . The whole query complexity is O 2 m + 5 + n / 2 τ 3 log 2 m + 5 n δ τ 2 . Finally, by using the same idea of the swap-test circuit, the query complexity of the modified quantum Goldreich–Levin algorithm could be lowered to O 2 m + 9 n π τ 4 log 2 m + 3 n δ τ 2 achieving a further speedup when τ is no less than O ( 2 - n / 2 + 6 n ) . Those two quantum Goldreich–Levin algorithms have their own advantages in implementation and query complexity.
Bibliografia:	ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14
ISSN:	1570-0755 1573-1332
DOI:	10.1007/s11128-019-2303-9