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The Pennsylvania State University the Graduate School The Pennsylvania State University The Graduate School QUANTUM COMPUTING: A CRYPTOGRAPHIC PERSPECTIVE A Dissertation in Computer Science and Engineering by Fang Song c 2013 Fang Song Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy August 2013 The dissertation of Fang Song was reviewed and approved∗ by the following: Sean Hallgren Associate Professor of Computer Science and Engineering Department Dissertation Advisor, Chair of Committee Adam Smith Associate Professor of Computer Science and Engineering Department Piotr Berman Associate Professor of Computer Science and Engineering Department Jason Morton Assistant Professor of Mathematics and Statistics Lee Coraor Associate Professor of Computer Science and Engineering Department Graduate Officer ∗Signatures are on file in the Graduate School. Abstract Quantum computing provides a new computational model that is only restricted by the laws of quantum physics. It has caused fundamental changes in many scientific fields. This thesis studies the strengths and limits of quantum computing from a cryptographic perspective. Our main focus is on investigating how secure computation, a central subject in classical cryptography, changes in a quantum world. In another endeavor, we also examine the potential of efficient quantum algorithms for solving some computational problems, which are regarded hard classically and are used in cryptographic constructions. Secure computation allows a group of players to jointly perform a computa- tional task on their private inputs without revealing more information about their inputs beyond what the output values imply. There are secure protocols that can realize any poly-time computation task under various conditions [Yao86, GMW87, CLOS02, Kil88]. However, these classical protocols may become insecure in pres- ence of quantum attacks. First of all, quantum algorithms, e.g., Shor's quantum factoring algorithm [?], can solve some computational problems efficiently, which are otherwise assumed hard classically to construct cryptographic protocols. Even if we assume some problems are also hard for quantum computers, classically se- cure protocols based on them can still be broken by a quantum attacker without solving the hard problem necessarily. On the other hand, we can also exploit quan- tum computing capability in a positive way. For example, Bennett et al. [BBCS91] constructed a quantum protocol for oblivious transfer (OT) that is secure against unbounded attackers, assuming existence of an ideal commitment protocol. Re- markably, such a goal is unattainable using purely classical protocols. In view of these challenges and opportunities, we make two major contribu- tions in the realm of secure computation: first we prove that there are classical protocols for computing any poly-time function securely against poly-time quan- tum attackers and show that a large family of classical protocols can be made iii secure against quantum attackers. This means that the classical feasibility pic- ture largely remains unchanged in the presence of quantum attacks. Second, we construct a quantum OT protocol assuming a task called 2-bit cut-and-choose is al- ready realized, which gives another demonstration of separation between quantum and classical protocols in the flavor of [BBCS91]. Along the way we also develop new tools that are useful when analyzing quantum cryptographic protocols. As an application of our two-fold investigation into secure computation in a quan- tum world, we are able to give an almost thorough categorization of an important class of two-party secure computation tasks. Very roughly speaking, if we consider quantum poly-time attackers, there is a zero-one law, analogous to the classical setting [MPR10]: every task is either feasible or it can be used to realize any other task. Whereas in the presence of unbounded attackers, every task belongs into one of three mutually exclusive subclasses, in contrast with the more complicated classical picture [MPR09, KMQ11]. As we have seen, certain computational assumptions can be broken by effi- cient quantum algorithms. It is thus crucial to understand which problems are easy or hard for quantum computers. In this direction, we show evidence that an important number-theoretical problem can by solved by an efficient quantum algorithm. Specifically, we show that finding the group of units in a number field of arbitrary degree (Unit-Finding) can be reduced to an hidden subgroup problem (HSP). The best classical algorithm requires super-polynomial time to compute the units. Therefore it has also been used to as a candidate of hardness assumption for cryptographic constructions. However, our result indicates that finding units is potentially easy on a quantum computer, as the HSP instance to which we reduce is a natural generalization of the HSP instances for which there exist efficient quan- tum algorithms (e.g., HSP over finite Abelien groups and over constant-dimension real spaces). Thus we have made the first substantial step towards a complete quantum algorithm solving Unit-Finding. iv Acknowledgments First of all, my deep appreciation goes to my advisor, Sean Hallgren, a great advisor and a great person. He led me into the fascinating field of quantum computing and has guided me throughout the past five years towards being an independent researcher. He was always available for questions I had, and was always willing to help me improve my oral presentation and my writing. I have learned from him what critical thinking means, and, not at all obvious as it may first appear, what research means. His patience, encouragement and support, especially during the hard times I had, have been and will continue motivating me. Beside a guide in research, he is also a good friend in life, from whom I got to know many interesting facets of the American culture. I would like to thank Adam Smith for his enormous help in various aspects of my research. His intriguing course on cryptography helped me lay my research interest on the inter-discipline of cryptography in a quantum world. In collabora- tion with him, I benefited a lot from his broad knowledge and invaluable insights in cryptography. His intuitive explanation to my questions was often enlightening. He helped me polish my bad writings and practice my conference and job talks. He was always full of energy, and his enthusiasm and devotion into research always gave me a boost. I am grateful to the other two committee members: Piotr Berman and Jason Morton. Piotr, with his unique way of reasoning, was often able to raise the right questions through which I benefited with new perspectives and new insights on problems. In addition, biking with him around the mountains in Happy Valley area, in a random-walk fashion, has been a lot fun. Jason carefully proofread the thesis and provided very useful feedback from a mathematician's perspective. Discussions with him often gave me the opportunity to think in a more abstract view. I was fortunate to know and have enjoyable discussions with other peer scholars (friends too): Kai-Min Chung, Xiaodi Wu and Hong-Sheng Zhou. They selflessly brought me into interesting research questions they encounter and shared their v experiences and reflections in research. I want to thank other collaborators of mine. Jon Katz always responded quickly on my questions, and he also showed me what a concise scientific writing should look like. Sergh Fehr can always explain technical puzzles in a crystal-clear way and kindly shared with me job opportunities. I learned from Vassilis Zikas that hard working and enjoying life do not necessarily conflict. I took math courses under and collaborated with Kirsten Eisentr¨ager.Her deep understanding in algebra and number theory often amazed me. Alexei Kitaev, during his short visit, shared with us his unique and illuminating view on quantum algorithms. My PhD career benefited from other professors in our theoretical computer science group as well. Sofya Raskhodnikova demonstrated me in her computational complexity course what good teaching is and opened my eye to the interesting area. This interest was further expanded in an advanced course offered by Martin F¨urer. I want to thank other theory students as well. Life in State College would never be the same without discussions, chatting and laughters with you guys. Many thanks to my friends. The basketball nights, party nights, and trips on the roads and on the campgrounds always recharged me. Your support of all kinds will never be forgotten. Finally, my parents, Wanlin Song and Qiaoyun Wang, nothing can express my gratefulness to you. My sister, Ting Song, your optimism can cure anything. Sining Wang, my little kid and my sunshine, thanks for everything you did for me. vi Table of Contents Acknowledgments v Chapter 1 Introduction 1 Chapter 2 Preliminaries 9 2.1 Notations and Basic Definitions . 9 2.2 Distinguishing Quantum States and Quantum Machines . 11 2.3 A (partial) List of Cryptographic Functionalities . 14 Chapter 3 Modeling Security in Presence of Quantum Attacks 18 3.1 A General Quantum Stand-Alone Security Model . 18 3.1.1 The Model . 20 3.1.2 Modular Composition Theorem . 25 3.1.3 Proof of Modular Composition Theorem . 26 3.2 Variants of Quantum Stand-Alone Models: A Unified Framework . 28 3.2.1 Z1 takes quantum advice . 30 3.2.2 Z1 does not take quantum advice . 37 3.2.3 A special constraint: Markov Condition . 38 3.3 Quantum UC Model: An Overview . 41 Chapter 4 Secure Computation against Quantum Attacks: Computational Setting 44 4.1 Founding Quantum UC Secure Computation on FZK . 45 vii 4.1.1 Simple Hybrid Argument. 46 4.1.2 Lifting CLOS to Quantum UC Security. 48 4.2 Realizing FZK with Stand-alone Security . 50 4.3 Putting It Together . 57 Chapter 5 Secure Computation against Quantum Attacks: Statistical Set- ting 59 5.1 Founding Q-SUC Secure Comp. on FOT and FCOM . 60 5.2 Founding Q-SUC Secure Comp.
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