Exponential Separation for One-Way Quantum Communication Complexity, with Applications to Cryptography

We give an exponential separation between one-way quantum and classical communication protocols for twopartial Boolean functions, both of which are variants of the Boolean Hidden Matching Problem of Bar-Yossef et al. Earlier such an exponential separation was known only for a relational version of the Hidden Matching Problem. Our proofs use the Fourier coefficients inequality of Kahn, Kalai, and Linial. We give a number of applications of this separation. In particular, in the bounded-storage model of cryptography we exhibita scheme that is secure against adversaries with a certain amount of classical storage, but insecure against adversaries with a similar (or even much smaller) amount of quantum storage; in the setting of privacy amplification, we show that there are strong extractors that yield a classically secure key, but are insecure against a quantum adversary.

[1]  Renato Renner,et al.  Security of quantum key distribution , 2005, Ausgezeichnete Informatikdissertationen.

[2]  Ran Raz,et al.  Fourier analysis for probabilistic communication complexity , 1995, computational complexity.

[3]  Eyal Kushilevitz,et al.  A zero-one law for Boolean privacy , 1989, STOC '89.

[4]  Ueli Maurer,et al.  On the power of quantum memory , 2005, IEEE Transactions on Information Theory.

[5]  J. Kuelbs.  Probability on Banach spaces , 1978 .

[6]  Avi Wigderson,et al.  Quantum vs. classical communication and computation , 1998, STOC '98.

[7]  Gilles Brassard,et al.  Privacy Amplification by Public Discussion , 1988, SIAM J. Comput..

[8]  David Chaum,et al.  Multiparty unconditionally secure protocols , 1988, STOC '88.

[9]  Michel Talagrand,et al.  How much are increasing sets positively correlated? , 1996, Comb..

[10]  Yonatan Aumann,et al.  Everlasting security in the bounded storage model , 2002, IEEE Trans. Inf. Theory.

[11]  Silvio Micali,et al.  How to play ANY mental game , 1987, STOC.

[12]  Vojtech Rödl,et al.  Geometrical realization of set systems and probabilistic communication complexity , 1985, 26th Annual Symposium on Foundations of Computer Science (sfcs 1985).

[13]  Peter W. Shor,et al.  Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer , 1995, SIAM Rev..

[14]  Chi-Jen Lu.  Encryption against Storage-Bounded Adversaries from On-Line Strong Extractors , 2003, Journal of Cryptology.

[15]  Andrew Chi-Chih Yao,et al.  Protocols for secure computations , 1982, FOCS 1982.

[16]  François Le Gall,et al.  Exponential separation of quantum and classical online space complexity , 2006, SPAA '06.

[17]  Zvi Galil,et al.  Lower bounds on communication complexity , 1984, STOC '84.

[18]  Ronen Shaltiel,et al.  Recent Developments in Explicit Constructions of Extractors , 2002, Bull. EATCS.

[19]  Andrew Chi-Chih Yao,et al.  Some complexity questions related to distributive computing(Preliminary Report) , 1979, STOC.

[20]  Ziv Bar-Yossef,et al.  Exponential separation of quantum and classical one-way communication complexity , 2004, STOC '04.

[21]  Robert König,et al.  Universally Composable Privacy Amplification Against Quantum Adversaries , 2004, TCC.

[22]  Ueli Maurer.  Conditionally-perfect secrecy and a provably-secure randomized cipher , 2004, Journal of Cryptology.

[23]  Noga Alon,et al.  The space complexity of approximating the frequency moments , 1996, STOC '96.

[24]  Gabriel Bracha,et al.  An O(log n) expected rounds randomized byzantine generals protocol , 1987, JACM.

[25]  Andrew C. Yao,et al.  Lower bounds by probabilistic arguments , 1983, 24th Annual Symposium on Foundations of Computer Science (sfcs 1983).

[26]  Hartmut Klauck.  Lower Bounds for Quantum Communication Complexity , 2007, SIAM J. Comput..

[27]  Nathan Linial,et al.  The influence of variables on Boolean functions , 1988, [Proceedings 1988] 29th Annual Symposium on Foundations of Computer Science.

[28]  Thierry Paul,et al.  Quantum computation and quantum information , 2007, Mathematical Structures in Computer Science.

[29]  Scott Aaronson,et al.  The learnability of quantum states , 2006, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[30]  Ilan Newman,et al.  Private vs. Common Random Bits in Communication Complexity , 1991, Inf. Process. Lett..

[31]  Andrew Chi-Chih Yao,et al.  Probabilistic computations: Toward a unified measure of complexity , 1977, 18th Annual Symposium on Foundations of Computer Science (sfcs 1977).

[32]  Ran Raz,et al.  Exponential Separation for One-Way Quantum Communication Complexity, with Applications to Cryptography , 2008, SIAM J. Comput..

[33]  S. Muthukrishnan,et al.  Data streams: algorithms and applications , 2005, SODA '03.

[34]  Ronald de Wolf,et al.  Bounded-error quantum state identification and exponential separations in communication complexity , 2005, STOC '06.

[35]  A. Holevo.  Bounds for the quantity of information transmitted by a quantum communication channel , 1973 .

[36]  W. Beckner.  Inequalities in Fourier analysis , 1975 .

[37]  A. Yao,et al.  Fair exchange with a semi-trusted third party (extended abstract) , 1997, CCS '97.

[38]  Leonid A. Levin,et al.  Pseudo-random generation from one-way functions , 1989, STOC '89.

[39]  Oded Goldreich,et al.  Unbiased Bits from Sources of Weak Randomness and Probabilistic Communication Complexity , 1988, SIAM J. Comput..

[40]  Salil P. Vadhan,et al.  Constructing Locally Computable Extractors and Cryptosystems in the Bounded-Storage Model , 2003, Journal of Cryptology.

[41]  Ueli Maurer,et al.  Optimal Randomizer Efficiency in the Bounded-Storage Model , 2003, Journal of Cryptology.

[42]  Avi Wigderson,et al.  Completeness theorems for non-cryptographic fault-tolerant distributed computation , 1988, STOC '88.

[43]  Daniel Lehmann,et al.  On the advantages of free choice: a symmetric and fully distributed solution to the dining philosophers problem , 1981, POPL '81.

[44]  A. Bonami.  Étude des coefficients de Fourier des fonctions de $L^p(G)$ , 1970 .

[45]  Silvio Micali,et al.  Optimal algorithms for Byzantine agreement , 1988, STOC '88.

[46]  Ronald de Wolf,et al.  Quantum communication and complexity , 2002, Theor. Comput. Sci..

[47]  Barbara M. Terhal,et al.  The Bounded-Storage Model in the Presence of a Quantum Adversary , 2008, IEEE Transactions on Information Theory.

[48]  M. Habib.  Probabilistic methods for algorithmic discrete mathematics , 1998 .

[49]  C. McDiarmid.  Concentration , 1862, The Dental register.

[50]  R. Cleve,et al.  Quantum fingerprinting. , 2001, Physical review letters.

[51]  Ran Raz,et al.  Exponential separation of quantum and classical communication complexity , 1999, STOC '99.