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Special Signature Schemes and Key Agreement Protocols

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Special Signature Schemes and Key Agreement Protocols Special Signature Schemes and Key Agreement Protocols Caroline J. Kudla Thesis submitted to the University of London for the degree of Doctor of Philosophy Information Security Group Department of Mathematics Royal Holloway, University of London 2006 Declaration These doctoral studies were conducted under the supervision of Kenneth G. Paterson. The work presented in this thesis is the result of original research carried out by myself, in collaboration with others, whilst enrolled in the Department of Mathematics as a candidate for the degree of Doctor of Philosophy. This work has not been submitted for any other degree or award in any other university or educational establishment. Caroline J. Kudla January, 2006 2 Acknowledgements My sincere thanks go to my supervisor, Kenny Paterson, whose dedication has been out- standing, and who has given generously of his time to guide and encourage me in my work over the last 3 years. His invaluable comments, constructive criticism and unwavering support have been crucial in helping me to achieve my goals and develop as a researcher. I would also like to thank my advisor, Chris Mitchell, for his encouragement in my various endeavors. I would like to express my deepest gratitude to HP Labs in Bristol, whose generous financial support has enabled me to pursue my studies. HP also permitted me to work from their labs, allowing me insight into the world of industrial research and providing me with invaluable experience. In particular, I would like to thank Keith Harrison for all his encouragement and advice during my time at HP, and Liqun Chen for her guidance and support. It has been great working with and learning from you both. I would also like to thank my friends and family, near and far, for all their encourage- ment over the last 3 years. I have finished studying - at last! Finally, I want to thank my beloved husband, Guillaume. He has been fantastic in every way, and I couldn’t have done this without his constant love and support. 3 Abstract This thesis is divided into two distinct parts. The first part of the thesis explores various deniable signature schemes and their applications. Such schemes do not bind a unique public key to a message, but rather specify a set of entities that could have created the signature, so each entity involved in the signature can deny having generated it. The main deniable signature schemes we examine are ring signature schemes. Ring signatures can be used to construct designated verifier signature schemes, which are closely related to designated verifier proof systems. We provide previously lacking formal definitions and security models for designated verifier proofs and signatures and examine their relationship to undeniable signature schemes. Ring signature schemes also have applications in the context of fair exchange of signa- tures. We introduce the notion of concurrent signatures, which can be constructed using ring signatures, and which provide a “near solution” to the problem of fair exchange. Concurrent signatures are more efficient than traditional solutions for fair exchange at the cost of some of the security guaranteed by traditional solutions. The second part of the thesis is concerned with the security of two-party key agreement protocols. It has traditionally been difficult to prove that a key agreement protocol satisfies a formal definition of security. A modular approach to constructing provably secure key agreement protocols was proposed, but the approach generally results in less efficient protocols. We examine the relationships between various well-known models of security and in- troduce a modular approach to the construction of proofs of security for key agreement protocols in such security models. Our approach simplifies the proof process, enabling us to provide proofs of security for several efficient key agreement protocols in the literature that were previously unproven. 4 Contents 1 Introduction 9 1.1 Contributions . 9 1.2 Overall Structure . 11 1.3 Publications . 12 I Special Signature Schemes 13 2 Preliminary Topics 14 2.1 Mathematical Background . 14 2.1.1 Complexity Theory . 14 2.1.2 Abstract Algebra . 15 2.1.3 The Discrete Logarithm and Diffie-Hellman Problems . 16 2.1.4 Additional Notation . 18 2.2 Public Key Cryptography . 18 2.2.1 Basic Terminology and Cryptographic Goals . 18 2.2.2 Key Infrastructures . 19 2.2.3 Digital Signature Schemes . 20 2.3 Provable Security . 21 2.3.1 Security Definition for Digital Signatures . 24 2.3.2 Cryptographic Hash Functions . 25 2.3.3 The Random Oracle Model . 26 2.3.4 Rewinding Oracles and the Forking Lemma . 27 2.3.5 The Non-Generic Forking Lemma . 29 3 Ring Signature Schemes 32 3.1 Introduction . 32 3.2 Preliminaries . 32 3.3 Ring Signature Definitions . 33 3.3.1 Related Work . 33 3.4 Security Model . 34 5 CONTENTS 3.4.1 Correctness . 34 3.4.2 Anonymity . 34 3.4.3 Unforgeability . 35 3.4.4 Notes on the Security Definitions for Ring Signatures . 35 3.5 A Concrete Scheme . 36 3.6 Security of the Concrete Scheme . 37 4 Non-interactive Designated Verifier Proofs and Their Applications 40 4.1 Introduction . 40 4.1.1 Related Work and Notions . 43 4.2 NIDV Proofs . 44 4.3 Security for NIDV Proofs . 45 4.3.1 Correctness . 45 4.3.2 Non-transferability . 45 4.3.3 Soundness . 45 4.3.4 Notes on the Security Definitions for NIDV Proof Systems . 46 4.4 NIDV Undeniable Signatures . 48 4.5 Security for NIDV Undeniable Signatures . 49 4.5.1 The Security of the Core Signature Scheme . 49 4.5.2 Notes on the Security Definitions for Undeniable Signatures . 51 4.6 A Concrete NIDV Undeniable Signature Scheme . 53 4.6.1 The Core Signature . 53 4.6.2 The Confirmation and Denial Proofs . 54 4.6.3 A Concrete NIDV EDL Proof System . 54 4.6.4 A Concrete NIDV IDL Proof System . 55 4.7 Security of the Concrete Scheme . 56 4.7.1 Security of the NIDV EDL proof system . 56 4.7.2 Security of the NIDV IDL proof system . 64 4.7.3 Application to the core signature scheme . 71 4.8 DV Signatures . 77 4.9 Security for DV Signatures . 78 4.9.1 Correctness . 78 4.9.2 Non-transferability . 78 4.9.3 Unforgeability . 79 4.9.4 Notes on the Security Definitions for DV Signatures . 80 4.9.5 Comparison to Other Work . 80 4.10 Concrete DV Signature Schemes . 81 4.10.1 DV Signatures from Ring Signatures . 81 4.10.2 DV Signatures from NIDV Undeniable Signatures . 83 4.10.3 A Concrete DV Signature Scheme from an NIDV EDL Proof . 83 6 CONTENTS 4.10.4 Security of our Concrete DV Signature Scheme . 84 4.11 Conclusions and Open Problems . 89 5 Concurrent Signatures 91 5.1 Introduction . 91 5.1.1 Technical Approach . 95 5.1.2 Published Work . 96 5.2 Formal Definitions . 97 5.2.1 Concurrent Signature Algorithms . 97 5.2.2 Concurrent Signature Protocol . 98 5.3 Formal Security Model . 99 5.3.1 Correctness . 99 5.3.2 Non-transferability . 99 5.3.3 Unforgeability . 100 5.3.4 Fairness . 101 5.4 A Concrete Concurrent Signature Scheme . 102 5.5 Security of the Concrete Concurrent Signature Scheme . 103 5.6 Extensions and Open Problems . 111 5.6.1 The Scheme Can Use a Variety of Keys . 111 5.6.2 The Multi-party Case . 111 5.6.3 Extensions to Concurrent Signatures . 112 5.7 Conclusion . 113 II Key Agreement Protocols 114 6 Introduction to Key Agreement 115 6.1 Basic Concepts . 115 6.1.1 Adversarial Assumptions . 116 6.2 The Diffie-Hellman Protocol . 116 6.2.1 Man-in-the-Middle Attacks . 117 6.3 Authenticated Key Agreement . 118 6.3.1 Security Attributes . 120 6.3.2 Authenticated Diffie-Hellman Protocols . 121 6.3.3 Security Attributes of Protocols 2 and 3 . 122 7 Models of Security for Key Agreement Protocols 125 7.1 Introduction . 125 7.2 The BJM Model . 126 7.2.1 Protocol Participants . 127 7.2.2 Oracle Queries . 128 7 CONTENTS 7.2.3 Matching Conversations . 129 7.2.4 Freshness . 130 7.2.5 The BJM Game and Test Query . 130 7.2.6 AK security . 131 7.2.7.
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