UvA-DARE (Digital Academic Repository) Cryptography in a quantum world Wehner, S.D.C. Publication date 2008 Document Version Final published version Link to publication Citation for published version (APA): Wehner, S. D. C. (2008). Cryptography in a quantum world. General rights It is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s) and/or copyright holder(s), other than for strictly personal, individual use, unless the work is under an open content license (like Creative Commons). Disclaimer/Complaints regulations If you believe that digital publication of certain material infringes any of your rights or (privacy) interests, please let the Library know, stating your reasons. In case of a legitimate complaint, the Library will make the material inaccessible and/or remove it from the website. Please Ask the Library: https://uba.uva.nl/en/contact, or a letter to: Library of the University of Amsterdam, Secretariat, Singel 425, 1012 WP Amsterdam, The Netherlands. You will be contacted as soon as possible. UvA-DARE is a service provided by the library of the University of Amsterdam (https://dare.uva.nl) Download date:26 Sep 2021 Cryptography in a Quantum World Stephanie Wehner Cryptography in a Quantum World ILLC Dissertation Series DS-2008-01 For further information about ILLC-publications, please contact Institute for Logic, Language and Computation Universiteit van Amsterdam Plantage Muidergracht 24 1018 TV Amsterdam phone: +31-20-525 6051 fax: +31-20-525 5206 e-mail: [email protected] homepage: http://www.illc.uva.nl/ Cryptography in a Quantum World Academisch Proefschrift ter verkrijging van de graad van doctor aan de Universiteit van Amsterdam op gezag van de Rector Magnificus prof.dr. D.C. van den Boom ten overstaan van een door het college voor promoties ingestelde commissie, in het openbaar te verdedigen in de Agnietenkapel op woensdag 27 februari 2008, te 14.00 uur door Stephanie Dorothea Christine Wehner geboren te W¨urzburg, Duitsland. Promotiecommissie: Promotor: prof.dr. H.M. Buhrman Overige leden: prof.dr.ir. F.A. Bais prof.dr. R.J.F. Cramer prof.dr. R.H. Dijkgraaf prof.dr. A.J. Winter dr. R.M. de Wolf Faculteit der Natuurwetenschappen, Wiskunde en Informatica The investigations were supported by EU projects RESQ IST-2001-37559, QAP IST 015848 and the NWO vici project 2004-2009. Copyright c 2008 by Stephanie Wehner Cover design by Frans Bartels. Printed and bound by PrintPartners Ipskamp. ISBN: 90-6196-544-6 Parts of this thesis are based on material contained in the following papers: • Cryptography from noisy storage S. Wehner, C. Schaffner, and B. Terhal Submitted (Chapter 11) • Higher entropic uncertainty relations for anti-commuting observ- ables S. Wehner and A. Winter Submitted (Chapter 4) • Security of Quantum Bit String Commitment depends on the in- formation measure H. Buhrman, M. Christandl, P. Hayden, H.K. Lo and S. Wehner In Physical Review Letters, 97, 250501 (2006) (long version submitted to Physical Review A) (Chapter 10) • State Discrimination with Post-Measurement Information M. Ballester, S. Wehner and A. Winter To appear in IEEE Transactions on Information Theory (Chapter 3) • Entropic uncertainty relations and locking: tight bounds for mu- tually unbiased bases M. Ballester and S. Wehner In Physical Review A, 75, 022319 (2007) (Chapters 4 and 5) • Tsirelson bounds for generalized CHSH inequalities S. Wehner In Physical Review A, 73, 022110 (2006) (Chapter 7) • Entanglement in Interactive Proof Systems with Binary Answers S. Wehner In Proceedings of STACS 2006, LNCS 3884, pages 162-171 (2006) (Chapter 9) Other papers to which the author contributed during her time as a PhD student: v • The quantum moment problem A. Doherty, Y. Liang, B. Toner and S. Wehner Submitted • Security in the Bounded Quantum Storage Model S. Wehner and J. Wullschleger Submitted • A simple family of non-additive codes J.A. Smolin, G. Smith and S. Wehner In Physical Review Letters, 99, 130505 (2007) • Analyzing Worms and Network Traffic using Compression S. Wehner Journal of Computer Security, Vol 15, Number 3, 303-320 (2007) • Implications of Superstrong Nonlocality for Cryptography H. Buhrman, M. Christandl, F. Unger, S. Wehner and A. Winter In Proceedings of the Royal Society A, vol. 462 (2071), pages 1919-1932 (2006) • Quantum Anonymous Transmissions M. Christandl and S. Wehner In Proceedings of ASIACRYPT 2005, LNCS 3788, pages 217-235 (2005) vi C’est v´eritablement utile puisque c’est joli. Le Petit Prince, Antoine de Saint-Exup´ery vii Contents Acknowledgments xv I Introduction 1 1 Quantum cryptography 3 1.1 Introduction . 3 1.2 Setting the state . 5 1.2.1 Terminology . 5 1.2.2 Assumptions . 6 1.2.3 Quantum properties . 7 1.3 Primitives . 9 1.3.1 Bit commitment . 9 1.3.2 Secure function evaluation . 11 1.3.3 Secret sharing . 17 1.3.4 Anonymous transmissions . 18 1.3.5 Other protocols . 19 1.4 Challenges . 19 1.5 Conclusion . 20 II Information in quantum states 23 2 Introduction 25 2.1 Quantum mechanics . 25 2.1.1 Quantum states . 25 2.1.2 Multipartite systems . 27 2.1.3 Quantum operations . 29 2.2 Distinguishability . 32 ix 2.3 Information measures . 36 2.3.1 Classical . 36 2.3.2 Quantum . 37 2.4 Mutually unbiased bases . 39 2.4.1 Latin squares . 39 2.4.2 Generalized Pauli matrices . 41 2.5 Conclusion . 42 3 State discrimination with post-measurement information 43 3.1 Introduction . 43 3.1.1 Outline . 45 3.1.2 Related work . 46 3.2 Preliminaries . 47 3.2.1 Notation and tools . 47 3.2.2 Definitions . 47 3.2.3 A trivial bound: guessing the basis . 48 3.3 No post-measurement information . 49 3.3.1 Two simple examples . 49 3.3.2 An upper bound for all Boolean functions . 50 3.3.3 AND function . 50 3.3.4 XOR function . 51 3.4 Using post-measurement information . 54 3.4.1 A lower bound for balanced functions . 54 3.4.2 Optimal bounds for the AND and XOR function . 57 3.5 Using post-measurement information and quantum memory . 63 3.5.1 An algebraic framework for perfect prediction . 63 3.5.2 Using two bases . 66 3.5.3 Using three bases . 70 3.6 Conclusion . 72 4 Uncertainty relations 75 4.1 Introduction . 75 4.2 Limitations of mutually unbiased bases . 78 4.2.1 MUBs in square dimensions . 79 4.2.2 MUBs based on Latin squares . 80 4.2.3 Using a full set of MUBs . 80 4.3 Good uncertainty relations . 83 4.3.1 Preliminaries . 84 4.3.2 A meta-uncertainty relation . 89 4.3.3 Entropic uncertainty relations . 89 4.4 Conclusion . 91 x 5 Locking classical information 93 5.1 Introduction . 93 5.1.1 A locking protocol . 94 5.1.2 Locking and uncertainty relations . 95 5.2 Locking using mutually unbiased bases . 96 5.2.1 An example . 96 5.2.2 MUBs from generalized Pauli matrices . 99 5.2.3 MUBs from Latin squares . 101 5.3 Conclusion . 101 III Entanglement 103 6 Introduction 105 6.1 Introduction . 105 6.1.1 Bell’s inequality . 106 6.1.2 Tsirelson’s bound . 108 6.2 Setting the stage . 109 6.2.1 Entangled states . 109 6.2.2 Other Bell inequalities . 110 6.2.3 Non-local games . 110 6.3 Observations . 113 6.3.1 Simple structural observations . 113 6.3.2 Vectorizing measurements . 115 6.4 The use of post-measurement information . 116 6.5 Conclusion . 119 7 Finding optimal quantum strategies 121 7.1 Introduction . 121 7.2 A simple example: Tsirelson’s bound . 123 7.3 The generalized CHSH inequality . 125 7.4 General approach and its applications . 128 7.4.1 General approach . 128 7.4.2 Applications . 129 7.5 Conclusion . 130 8 Bounding entanglement in NL-games 131 8.1 Introduction . 131 8.2 Preliminaries . 132 8.2.1 Random access codes . 132 8.2.2 Non-local games and state discrimination . 134 8.3 A lower bound . 134 8.4 Upper bounds . 136 xi 8.5 Conclusion . 138 9 Interactive Proof Systems 139 9.1 Introduction . 139 9.1.1 Classical interactive proof systems . 139 9.1.2 Quantum multi-prover interactive proof systems . 140 9.2 Proof systems and non-local games . 142 9.2.1 Non-local games . 142 9.2.2 Multiple classical provers . 143 9.2.3 A single quantum prover . 145 9.3 Simulating two classical provers with one quantum prover . 145 9.4 Conclusion . 148 IV Consequences for Crytography 149 10 Limitations 151 10.1 Introduction . 151 10.2 Preliminaries . 152 10.2.1 Definitions . 152 10.2.2 Model . 153 10.2.3 Tools . 154 10.3 Impossibility of quantum string commitments . 156 10.4 Possibility . ..