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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Vydáno v:	Quantum information processing Ročník 18; číslo 6; s. 1 - 31
Hlavní autoři:	Cui, Jingyi, Guo, Jiansheng, Xu, Linhong, Li, Mingming
Médium:	Journal Article
Jazyk:	angličtina
Vydáno:	New York Springer US 01.06.2019 Springer Nature B.V
Témata:	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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Abstract	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.
AbstractList	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-1logδ-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 O2m+5+n/2τ3log2m+5nδτ2. 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 O2m+9nπτ4log2m+3nδτ2 achieving a further speedup when τ is no less than O(2-n/2+6n). Those two quantum Goldreich–Levin algorithms have their own advantages in implementation and query complexity. 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.
ArticleNumber	180
Author	Cui, Jingyi Guo, Jiansheng Xu, Linhong Li, Mingming
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Cites_doi	10.1098/rspa.1992.0167 10.1137/0222080 10.1038/nphys1150 10.1007/s11128-018-1932-8 10.1103/PhysRevA.52.3457 10.1103/PhysRevLett.87.167902 10.1142/S0219749905000980 10.1515/JMC.2007.011 10.1137/S0036144598347011 10.1038/ncomms1392 10.1109/TCT.1959.1086595 10.1090/conm/305/05215 10.1007/s11128-018-1885-y 10.1007/s11432-017-9436-7 10.1007/3-540-45841-7_26 10.1109/TIT.2011.2134250 10.1103/PhysRevA.78.042336 10.1145/285055.285060 10.1007/s11432-017-9468-y 10.1137/S0097539796298637 10.1007/978-3-662-48683-2_5 10.1103/PhysRevA.80.022340 10.1017/CBO9781139814782 10.1016/0304-3975(86)90174-X 10.1007/978-3-662-48865-2 10.1007/s11128-015-0983-3 10.1145/73007.73010 10.1109/ISIT.2010.5513654 10.1088/1367-2630/16/9/093026 10.46586/tosc.v2016.i1.71-94 10.1007/978-3-662-53008-5_8 10.1103/PhysRevA.89.030303
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Snippet	In classical cryptography, many cryptographic primitives could be treated as multi-output Boolean functions. The analysis of such functions is of great...
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SubjectTerms	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
Title	Quantum algorithms for learning Walsh spectra of multi-output Boolean functions
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