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UCLA Electronic Theses and Dissertations UCLA UCLA Electronic Theses and Dissertations Title How to Rewind with Minimal Interaction Permalink https://escholarship.org/uc/item/53s2w021 Author Khurana, Dakshita Publication Date 2018 Peer reviewed|Thesis/dissertation eScholarship.org Powered by the California Digital Library University of California UNIVERSITY OF CALIFORNIA Los Angeles How to Rewind with Minimal Interaction A dissertation submitted in partial satisfaction of the requirements for the degree Doctor of Philosophy in Computer Science by Dakshita Khurana 2018 c Copyright by Dakshita Khurana 2018 ABSTRACT OF THE DISSERTATION How to Rewind with Minimal Interaction by Dakshita Khurana Doctor of Philosophy in Computer Science University of California, Los Angeles, 2018 Professor Rafail Ostrovsky, Co-Chair Professor Amit Sahai, Co-Chair The notion of simulation is central to cryptography: often, to demonstrate that an adversary did not recover any information about private inputs of other participants, we exhibit the existence of a simulator that generates the adversary's view without access to inputs of honest participants. The primary method used to build simulators is rewinding, where a simulator resets the adversary to a previous point in the protocol and tries to complete the protocol tree multiple times until it achieves a favorable outcome. First introduced in the context of zero-knowledge proof systems and secure computa- tion, today the rewinding technique is synonymous with protocol security and polynomial simulation. Prior to this work, all known rewinding techniques in the plain model required multiple rounds of back-and-forth interaction between participants. In this thesis, we demonstrate the first rewinding techniques that require only a single message from each participant. Using these techniques, we overcome several barriers from literature to construct for the first time, based on standard sub-exponential cryptographic assumptions, the following core protocols, and several subsequent applications: • Two message commitments satisfying non-malleability (with respect to commitment). • Two-message delayed-input weak zero-knowledge arguments for NP. These imply ar- guments for NP satisfying witness hiding and strong witness indistinguishability. ii The dissertation of Dakshita Khurana is approved. Suhas N. Diggavi Alexander Sherstov Amit Sahai, Committee Co-Chair Rafail Ostrovsky, Committee Co-Chair University of California, Los Angeles 2018 iii To my parents, for their unconditional love. iv TABLE OF CONTENTS 1 Introduction :::::::::::::::::::::::::::::::::::::: 1 1.1 Overview of Non-Malleability..........................3 1.1.1 Summary of Results...........................6 1.1.2 Related Work...............................7 1.1.3 Technical Overview............................8 1.2 Overview of Zero-Knowledge Arguments.................... 23 1.2.1 Summary of Results........................... 26 1.2.2 Discussion................................. 29 1.2.3 Related Work............................... 30 1.2.4 Technical Overview............................ 31 2 Constructing Non-Malleable Commitments :::::::::::::::::: 38 2.1 Preliminaries................................... 38 2.1.1 ZK With Super-polynomial Simulation.................. 38 2.1.2 Special Two-Party Computation..................... 39 2.2 Definitions..................................... 40 2.2.1 Non-Malleability w.r.t. Commitment.................. 41 2.2.2 Extractable Commitments........................ 42 2.2.3 ZK with Super-polynomial Strong Simulation............. 44 2.2.4 Secure Computation........................... 44 2.3 Extractable Commitments............................ 45 2.3.1 Construction............................... 45 2.3.2 SPSS Zero Knowledge from Extractable Commitments........ 55 v 2.4 Non-malleable Commitments........................... 59 2.4.1 Non-Malleable Commitments for Two Tags............... 59 2.4.2 Non-Malleable Commitments for Four Tags.............. 61 2.4.3 Bounded-Concurrent Non-malleability for Four Tags.......... 66 2.4.4 Round-Preserving Tag Amplification.................. 68 3 Constructing Weak Distributional Zero-Knowledge Arguments :::::: 77 3.1 Preliminaries................................... 77 3.2 Definitions..................................... 79 3.2.1 Proof Systems............................... 79 3.3 Two Round Argument Systems......................... 83 3.3.1 Construction............................... 83 3.3.2 Adaptive Soundness........................... 83 3.3.3 Witness Indistinguishability....................... 86 3.3.4 Distributional Weak Zero Knowledge.................. 97 3.3.5 Witness Hiding.............................. 104 3.3.6 Strong Witness Indistinguishability................... 105 4 Survey of Subsequent Applications ::::::::::::::::::::::: 106 4.1 Two-Message Non-Malleable Commitments................... 106 4.2 Two-Message Distributional WZK/Strong WI/WH Arguments........ 108 5 Cryptographic Foundations for Practical Security :::::::::::::: 110 References ::::::::::::::::::::::::::::::::::::::::: 115 vi LIST OF FIGURES 1.1 A simple non-malleable commitment scheme for two tags............. 14 1.2 Illustrative natural (but incomplete) extension of basic scheme to four tags... 16 2.1 Two-Round Extractable Commitments....................... 47 2.2 Two Round SPSS ZK Arguments.......................... 56 2.3 Non-malleable Commitments for Four Tags..................... 62 2.4 Round-Preserving Tag Amplification........................ 69 3.1 Two Round Argument System for NP....................... 84 3.2 A Single Repetition of a Two Round Argument System for NP.......... 87 3.3 Approximately Learning the Verifier’s Challenge for WI.............. 89 3.4 Approximately Learning the Verifier’s Challenge for WZK............ 99 vii ACKNOWLEDGMENTS I would like to express my sincere gratitude to my advisors Rafail Ostrovsky and Amit Sahai for admitting me as their student and being constant sources of encouragement, advice and inspiration. I am extremely fortunate to have had them as my advisors, and grateful for the wonderful experience that culminated in this thesis. I look forward to continuing to learn from them through my career. I am grateful to have learned from Rafi’s broad perspective on research and the impor- tance of problems. His invaluable feedback on my ideas prevented my research from going astray and helped shape my perspective on what directions to pursue. Every time I run out of ideas, I can turn to Rafi for new directions. I am also really grateful to Rafi for his mentorship and advice throughout my PhD. Amit has taught me how to think: he has the amazing ability to understand my half- baked ideas when even I dont fully understand them myself. He patiently listened to my jumbled thoughts, and instead of just finding the right solution for me, taught me how to reason better. He has helped me evolve from someone who loved solving puzzles to someone who loves doing research. I am grateful for his support, and hope that some day I can be half as effective as him at advising students. I am also thankful for his advice and perspective that has helped me make many decisions, both in research and in life. Many thanks, also, to Alexander Sherstov and Suhas Diggavi for their feedback and support as members of my thesis committee. I am grateful to Vipul Goyal for hosting me for multiple visits and internships during the course of my graduate studies. Vipul has had a strong influence on my research, and I am lucky to have benefitted countless number of times from his perspective, guidance and friendship. I also spent two wonderful summers at MSR New England, for which I am thankful to Yael Tauman Kalai. Yael inspires me to keep looking when all seems hopeless. Often, when I am stuck on a problem, I find myself wondering, what would Yael do? viii I am thankful to Maji for his mentorship and guidance, especially during the early years of graduate school. He spent many consecutive hours tearing down my manuscripts and teaching me to write. I cannot thank Saikrishna enough for being a generous and dependable friend. I greatly cherish our collaboration. I am especially thankful to Aayush Jain and Abhishek Jain for being great collaborators and friends. I would also like to thank my officemates Abishek, Akshay, Alain, Alessandra, Ashutosh, Divya, Eric, Nathan, Peter, Prabhanjan, Ran, Rex, Sanjam and Wutichai for all the wonderful times we spent together. I have benefitted greatly from interactions with Shafi Goldwasser and Yuval Ishai, for which I am thankful. I am also grateful to Shweta Agrawal, Omkant Pandey, Anat Paskin- Cherniavsky, Rafael Pass, Manoj Prabhakaran, Alon Rosen, Ron Rothblum, Ivan Visconti, Vinod Vaikuntanathan, Brent Waters, Daniel Wichs, and many other researchers for en- lightening technical discussions that have taught me so much. Beyond collaborations, I am grateful to the crypto community and the Simons cryptography program for giving me great friends and peers in Apoorvaa, Antigoni, Pratyay, Rishab, Shashank, Venkata and many others that I am sure I missed writing about. The work in this thesis is based on joint works with Amit Sahai, with Abhishek Jain, Yael Tauman Kalai and Ron Rothblum, and with Rafail Ostrovsky. I am thankful to all of them for their collaboration. Finally, I am extremely grateful to my parents, grandparents and my sister Prashi for their love and support, and for always teaching me to be a better person. I am also blessed to have wonderfully supportive friends in Anand, Ishita, Pragati and Stuti. Without Abhishek,
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