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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Published in:	Quantum information processing Vol. 18; no. 6; pp. 1 - 31
Main Authors:	Cui, Jingyi, Guo, Jiansheng, Xu, Linhong, Li, Mingming
Format:	Journal Article
Language:	English
Published:	New York Springer US 01.06.2019 Springer Nature B.V
Subjects:	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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References	Wocjan, Abeyesinghe (CR33) 2008; 78 Bravyi, Harrow, Hassidim (CR30) 2011; 57 Xie, Qiu, Cai (CR17) 2018; 17 Wocjan, Chiang, Nagaj (CR35) 2009; 80 Brassard, Peter, Mosca (CR24) 2002; 305 Adcock, Cleve (CR28) 2002 Gregor, Alexander (CR4) 2017; 2017 Golomb (CR37) 1959; 6 CR38 CR15 Montanaro, de Wolf (CR16) 2016; 7 CR12 CR34 CR11 Jerrum, Valiant, Vazirani (CR32) 1986; 43 CR10 Lanyon, Barbieri, Almeida (CR39) 2009; 5 Li, Yang (CR14) 2015 Dong, Li, Wang (CR7) 2019; 62 O’Donnell (CR19) 2014 CR2 Wu, Feng (CR18) 2016 Kushilevitz, Mansour (CR44) 1993; 22 Chailloux, Naya-Plasencia, Schrottenloher (CR3) 2017; 2017 Goldreich, Goldwasser, Ron (CR20) 1998; 45 CR6 CR8 Montanaro (CR31) 2015; 471 CR29 CR27 Barenco, Bennett, Cleve (CR36) 1995; 52 CR25 Simon (CR5) 1997; 26 CR46 Buhrman, Cleve, Watrous, Dewolf (CR26) 2001; 87 Gangopadhyay, Behera, Panigrahi (CR22) 2018; 17 CR43 CR42 Zhou, Ralph, Kalasuwan (CR41) 2011; 2 CR40 Dong, Wang (CR9) 2018; 61 Maitra, Mukhopadhyay (CR23) 2005; 03 Shor (CR1) 1999; 41 Zhou, Lu, Zhang (CR13) 2015; 14 Deutsch, Jozsa (CR21) 1992; 439 Daemen, Rijmen (CR45) 2007; 1 Mark Adcock (2303_CR28) 2002 2303_CR40 X-Y Dong (2303_CR7) 2019; 62 S Maitra (2303_CR23) 2005; 03 P Wocjan (2303_CR35) 2009; 80 C-G Wu (2303_CR18) 2016 A Montanaro (2303_CR31) 2015; 471 2303_CR27 2303_CR46 2303_CR25 2303_CR42 BP Lanyon (2303_CR39) 2009; 5 2303_CR43 2303_CR6 PW Shor (2303_CR1) 1999; 41 2303_CR8 Q Zhou (2303_CR13) 2015; 14 2303_CR2 R O’Donnell (2303_CR19) 2014 A Barenco (2303_CR36) 1995; 52 X-Q Zhou (2303_CR41) 2011; 2 2303_CR29 P Wocjan (2303_CR33) 2008; 78 S Bravyi (2303_CR30) 2011; 57 O Goldreich (2303_CR20) 1998; 45 L Gregor (2303_CR4) 2017; 2017 X-Y Dong (2303_CR9) 2018; 61 M Jerrum (2303_CR32) 1986; 43 E Kushilevitz (2303_CR44) 1993; 22 S Golomb (2303_CR37) 1959; 6 2303_CR15 DR Simon (2303_CR5) 1997; 26 D Deutsch (2303_CR21) 1992; 439 2303_CR38 2303_CR11 2303_CR12 A Montanaro (2303_CR16) 2016; 7 2303_CR34 2303_CR10 Hongwei Li (2303_CR14) 2015 A Chailloux (2303_CR3) 2017; 2017 G Brassard (2303_CR24) 2002; 305 H Buhrman (2303_CR26) 2001; 87 Z Xie (2303_CR17) 2018; 17 S Gangopadhyay (2303_CR22) 2018; 17 J Daemen (2303_CR45) 2007; 1
References_xml	– volume: 7 start-page: 1 year: 2016 end-page: 81 ident: CR16 article-title: A survey of quantum property testing publication-title: Theory Comput. Libr. Gr. Surv. – volume: 439 start-page: 553 issue: 1907 year: 1992 end-page: 558 ident: CR21 article-title: Rapid solution of problems by quantum computation publication-title: Proc. R. Soc. Lond. A (Math. Phys. Eng. Sci.) doi: 10.1098/rspa.1992.0167 – volume: 22 start-page: 1331 issue: 6 year: 1993 end-page: 1348 ident: CR44 article-title: Learning decision trees using the Fourier spectrum publication-title: SIAM J. Comput. doi: 10.1137/0222080 – ident: CR43 – volume: 5 start-page: 134 year: 2009 end-page: 140 ident: CR39 article-title: Simplifying quantum logic using higher-dimensional Hilbert spaces publication-title: Nat. Phys. doi: 10.1038/nphys1150 – volume: 17 start-page: 160 year: 2018 ident: CR22 article-title: Generalization and demonstration of an entanglement-based Deutsch-Jozsa-like algorithm using a 5-qubit quantum computer publication-title: Quantum Inf. Proc. doi: 10.1007/s11128-018-1932-8 – volume: 52 start-page: 3457 year: 1995 ident: CR36 article-title: Elementary gates for quantum computation publication-title: Phys. Rev. A doi: 10.1103/PhysRevA.52.3457 – ident: CR2 – volume: 87 start-page: 167902 year: 2001 ident: CR26 article-title: Quantum fingerprinting publication-title: Phys. Rev. Lett. doi: 10.1103/PhysRevLett.87.167902 – volume: 03 start-page: 359 issue: 02 year: 2005 end-page: 370 ident: CR23 article-title: The Deutsch-Jozsa alogrithm revisited in the domain of cryptographically significant Boolean funcations publication-title: Int. J. Quantum Inf. doi: 10.1142/S0219749905000980 – ident: CR12 – volume: 1 start-page: 221 issue: 3 year: 2007 end-page: 242 ident: CR45 article-title: Probability distributions of correlation and differentials in block ciphers publication-title: J. Math. Cryptol. doi: 10.1515/JMC.2007.011 – volume: 2017 start-page: 211 year: 2017 end-page: 240 ident: CR3 article-title: An efficient quantum collision search algorithm and implications on symmetric cryptography publication-title: ASIACRYPT – volume: 41 start-page: 303 issue: 2 year: 1999 end-page: 332 ident: CR1 article-title: Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer publication-title: SIAM Rev. doi: 10.1137/S0036144598347011 – ident: CR10 – volume: 2 start-page: 413 year: 2011 ident: CR41 article-title: Adding control to arbitrary unknown quantum operations publication-title: Nat. Commun. doi: 10.1038/ncomms1392 – ident: CR6 – ident: CR29 – volume: 6 start-page: 176 issue: 5 year: 1959 end-page: 186 ident: CR37 article-title: On the classification of Boolean functions publication-title: IRE Trans. Circuit Theory doi: 10.1109/TCT.1959.1086595 – volume: 305 start-page: 53 year: 2002 end-page: 74 ident: CR24 article-title: Quantum amplitude amplification and estimation publication-title: Contemp. Math. doi: 10.1090/conm/305/05215 – ident: CR8 – ident: CR40 – volume: 17 start-page: 139 year: 2018 ident: CR17 article-title: Quantum algorithms on Walsh transform and Hamming distance for Boolean functions publication-title: Quantum Inf. Proc. doi: 10.1007/s11128-018-1885-y – ident: CR25 – volume: 62 start-page: 22501 year: 2019 ident: CR7 article-title: Quantum cryptanalysis on some generalized feistel schemes publication-title: Sci. China Inf. Sci. doi: 10.1007/s11432-017-9436-7 – ident: CR27 – ident: CR42 – start-page: 323 year: 2002 end-page: 334 ident: CR28 article-title: A Quantum Goldreich-Levin Theorem with Cryptographic Applications publication-title: STACS 2002 doi: 10.1007/3-540-45841-7_26 – volume: 57 start-page: 3971 issue: 6 year: 2011 end-page: 3981 ident: CR30 article-title: Quantum algorithms for testing properties of distributions publication-title: IEEE Trans. Inf. Theory doi: 10.1109/TIT.2011.2134250 – ident: CR46 – volume: 78 start-page: 042336 year: 2008 ident: CR33 article-title: Speedup via quantum sampling publication-title: Phys. Rev. A doi: 10.1103/PhysRevA.78.042336 – volume: 45 start-page: 653 issue: 4 year: 1998 end-page: 750 ident: CR20 article-title: Property testing and its connection to learning and approximation publication-title: J. ACM doi: 10.1145/285055.285060 – ident: CR15 – ident: CR38 – volume: 61 start-page: 102501 year: 2018 ident: CR9 article-title: Quantum Key-recovery attack on feistel structures publication-title: Sci. China Inf. Sci. doi: 10.1007/s11432-017-9468-y – volume: 471 start-page: 1 issue: 2181 year: 2015 end-page: 20 ident: CR31 article-title: Quantum speedup of Monte Carlo methods publication-title: Proc. R. Soc. Lond. A (Math. Phys. Eng. Sci.) – volume: 26 start-page: 1474 issue: 5 year: 1997 end-page: 1483 ident: CR5 article-title: On the power of quantum computation publication-title: SIAM J. Comput. doi: 10.1137/S0097539796298637 – ident: CR11 – start-page: 44 year: 2015 end-page: 51 ident: CR14 article-title: Quantum Differential Cryptanalysis to the Block Ciphers publication-title: Applications and Techniques in Information Security doi: 10.1007/978-3-662-48683-2_5 – volume: 80 start-page: 022340 year: 2009 ident: CR35 article-title: Quantum algorithm for approximating partition functions publication-title: Phys. Rev. A doi: 10.1103/PhysRevA.80.022340 – ident: CR34 – year: 2014 ident: CR19 publication-title: Analysis of boolean functions doi: 10.1017/CBO9781139814782 – volume: 43 start-page: 169 year: 1986 end-page: 188 ident: CR32 article-title: Random generation of combinatorial structures from a uniform distribution (extended abstract) publication-title: Theor. Comput. Sci. doi: 10.1016/0304-3975(86)90174-X – year: 2016 ident: CR18 publication-title: Boolean Functions and Their Applications in Cryptography doi: 10.1007/978-3-662-48865-2 – volume: 2017 start-page: 161 year: 2017 end-page: 178 ident: CR4 article-title: Grover meets Simon–quantum attacking the FX-construction publication-title: ASIACRYPT – volume: 14 start-page: 2101 issue: 6 year: 2015 end-page: 2109 ident: CR13 publication-title: Quantum differential cryptanalysis. Quantum Inf. Proc. doi: 10.1007/s11128-015-0983-3 – start-page: 44 volume-title: Applications and Techniques in Information Security year: 2015 ident: 2303_CR14 doi: 10.1007/978-3-662-48683-2_5 – ident: 2303_CR29 – volume: 03 start-page: 359 issue: 02 year: 2005 ident: 2303_CR23 publication-title: Int. J. Quantum Inf. doi: 10.1142/S0219749905000980 – volume: 17 start-page: 139 year: 2018 ident: 2303_CR17 publication-title: Quantum Inf. Proc. doi: 10.1007/s11128-018-1885-y – volume: 80 start-page: 022340 year: 2009 ident: 2303_CR35 publication-title: Phys. Rev. A doi: 10.1103/PhysRevA.80.022340 – volume: 1 start-page: 221 issue: 3 year: 2007 ident: 2303_CR45 publication-title: J. Math. Cryptol. doi: 10.1515/JMC.2007.011 – volume: 14 start-page: 2101 issue: 6 year: 2015 ident: 2303_CR13 publication-title: Quantum differential cryptanalysis. Quantum Inf. Proc. doi: 10.1007/s11128-015-0983-3 – volume: 5 start-page: 134 year: 2009 ident: 2303_CR39 publication-title: Nat. Phys. doi: 10.1038/nphys1150 – ident: 2303_CR46 – volume: 439 start-page: 553 issue: 1907 year: 1992 ident: 2303_CR21 publication-title: Proc. R. Soc. Lond. A (Math. Phys. Eng. Sci.) doi: 10.1098/rspa.1992.0167 – ident: 2303_CR42 – ident: 2303_CR25 – volume: 87 start-page: 167902 year: 2001 ident: 2303_CR26 publication-title: Phys. Rev. Lett. doi: 10.1103/PhysRevLett.87.167902 – ident: 2303_CR27 doi: 10.1145/73007.73010 – volume: 22 start-page: 1331 issue: 6 year: 1993 ident: 2303_CR44 publication-title: SIAM J. Comput. doi: 10.1137/0222080 – ident: 2303_CR6 doi: 10.1109/ISIT.2010.5513654 – volume: 61 start-page: 102501 year: 2018 ident: 2303_CR9 publication-title: Sci. China Inf. Sci. doi: 10.1007/s11432-017-9468-y – volume: 2 start-page: 413 year: 2011 ident: 2303_CR41 publication-title: Nat. Commun. doi: 10.1038/ncomms1392 – volume: 6 start-page: 176 issue: 5 year: 1959 ident: 2303_CR37 publication-title: IRE Trans. Circuit Theory doi: 10.1109/TCT.1959.1086595 – ident: 2303_CR8 – volume: 2017 start-page: 161 year: 2017 ident: 2303_CR4 publication-title: ASIACRYPT – ident: 2303_CR10 – volume: 43 start-page: 169 year: 1986 ident: 2303_CR32 publication-title: Theor. Comput. Sci. doi: 10.1016/0304-3975(86)90174-X – ident: 2303_CR40 doi: 10.1088/1367-2630/16/9/093026 – volume-title: Analysis of boolean functions year: 2014 ident: 2303_CR19 doi: 10.1017/CBO9781139814782 – ident: 2303_CR12 doi: 10.46586/tosc.v2016.i1.71-94 – ident: 2303_CR2 – volume: 52 start-page: 3457 year: 1995 ident: 2303_CR36 publication-title: Phys. Rev. A doi: 10.1103/PhysRevA.52.3457 – volume: 45 start-page: 653 issue: 4 year: 1998 ident: 2303_CR20 publication-title: J. ACM doi: 10.1145/285055.285060 – start-page: 323 volume-title: STACS 2002 year: 2002 ident: 2303_CR28 doi: 10.1007/3-540-45841-7_26 – volume: 57 start-page: 3971 issue: 6 year: 2011 ident: 2303_CR30 publication-title: IEEE Trans. Inf. Theory doi: 10.1109/TIT.2011.2134250 – ident: 2303_CR11 doi: 10.1007/978-3-662-53008-5_8 – volume: 2017 start-page: 211 year: 2017 ident: 2303_CR3 publication-title: ASIACRYPT – ident: 2303_CR15 – volume: 26 start-page: 1474 issue: 5 year: 1997 ident: 2303_CR5 publication-title: SIAM J. Comput. doi: 10.1137/S0097539796298637 – volume-title: Boolean Functions and Their Applications in Cryptography year: 2016 ident: 2303_CR18 doi: 10.1007/978-3-662-48865-2 – ident: 2303_CR38 – volume: 78 start-page: 042336 year: 2008 ident: 2303_CR33 publication-title: Phys. Rev. A doi: 10.1103/PhysRevA.78.042336 – volume: 305 start-page: 53 year: 2002 ident: 2303_CR24 publication-title: Contemp. Math. doi: 10.1090/conm/305/05215 – ident: 2303_CR34 – volume: 17 start-page: 160 year: 2018 ident: 2303_CR22 publication-title: Quantum Inf. Proc. doi: 10.1007/s11128-018-1932-8 – volume: 41 start-page: 303 issue: 2 year: 1999 ident: 2303_CR1 publication-title: SIAM Rev. doi: 10.1137/S0036144598347011 – volume: 7 start-page: 1 year: 2016 ident: 2303_CR16 publication-title: Theory Comput. Libr. Gr. Surv. – volume: 62 start-page: 22501 year: 2019 ident: 2303_CR7 publication-title: Sci. China Inf. Sci. doi: 10.1007/s11432-017-9436-7 – volume: 471 start-page: 1 issue: 2181 year: 2015 ident: 2303_CR31 publication-title: Proc. R. Soc. Lond. A (Math. Phys. Eng. Sci.) – ident: 2303_CR43 doi: 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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