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Learning Noisy Characters, Multiplication Codes, and Cryptographic Hardcore Predicates by Adi Akavia

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Learning Noisy Characters, Multiplication Codes, and Cryptographic Hardcore Predicates by Adi Akavia Submitted to the Department of Electrical Engineering and Computer Science in partial fulfillment of the requirements for the degree of Doctor of Philosophy at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY February 2008 @ Massachusetts Institute of Technology 2008. All rights reserved. A uthor ....... ................................ Department of Electrical Engineering and Computer Science January 30, 2008 Certified by............. ........ Shafi Goldwasser RSA Professor of Electrical Engineering and Computer Science Thesis Supervisor Accepted by .. ............. ...................... ............. Chairman, Department Committee on Graduate Students MASSACHUSETTS INSW OF TEOHNOLOGY APR 0 7 2008 ARCHIVEs LIBRARIES Learning Noisy Characters, Multiplication Codes, and Cryptographic Hardcore Predicates by Adi Akavia Submitted to the Department of Electrical Engineering and Computer Science on January 30, 2008, in partial fulfillment of the requirements for the degree of Doctor of Philosophy Abstract We present results in cryptography, coding theory and sublinear algorithms. In cryptography, we introduce a unifying framework for proving that a Boolean predicate is hardcore for a one-way function and apply it to a broad family of functions and predicates, showing new hardcore predicates for well known one-way function candidates such as RSA and discrete-log as well as reproving old results in an entirely different way. Our proof framework extends the list-decoding method of Goldreich and Levin [38] for showing hardcore predicates, by introducing a new class of error correcting codes and new list-decoding algorithm we develop for these codes. In coding theory, we introduce a novel class of error correcting codes that we name: Multiplication codes (MPC). We develop decoding algorithms for MPC codes, showing they achieve desirable combinatorial and algorithmic properties, including: (1) binary MPC of constant distance and exponential encoding length for which we provide efficient local list decoding and local self correctingalgorithms; (2) binary MPC of constant distance and polynomial encoding length for which we provide efficient decoding algorithm in random noise model; (3) binary MPC of constant rate and distance. MPC codes are unique in particular in achieving properties as above while having a large group as their underlying algebraic structure. In sublinear algorithms, we present the SFT algorithm for finding the sparse Fourier approximation of complex multi-dimensional signals in time logarithmic in the signal length. We also present additional algorithms for related settings, differing in the model by which the input signal is given, in the considered approximation measure, and in the class of addressed signals. The sublinear algorithms we present are central components in achieving our results in cryptography and coding theory. Reaching beyond theoretical computer science, we suggest employing our algorithms as tools for performance enhancement in data intensive applications, in particular, we suggest replacing the O(Nlog N)-time FFT algorithm with our e(log N)-time SFT algorithm for settings where a sparse approximation suffices. Thesis Supervisor: Shafi Goldwasser Title: RSA Professor of Electrical Engineering and Computer Science Acknowledgments My deepest thanks are to Shafi Goldwasser who has been a wonderful advisor, mentor and friend in all these years since I first set foot in her office. I feel blissed and privileged to have had her guidance through the paths of research. Her insights and perspectives on research along with her determination in conducting one have taught me invaluable lessons. Her kindness, support and encouragement while being my advisor have been indispensable. No words can express how indebted I am to her. Much of this thesis is due to a joint work with Shafi and with Muli Safra; I'm indebted to Shafi and to Muli for their vital contribution to this work. Parts of this thesis are due to a joint work with Ramarathnam Venkatesan; it's a pleasure to thank Venkie for these parts in particular and for his collaboration as a whole. I am indebted to Oded Goldreich and to Vinod Vaikuntanathan for endless dis- cussions influencing my entire academic education and this dissertation in particular. I benefited a great deal from discussions with Nati Linial, Alon Rosen, Adi Shamir, Madhu Sudan, Salil Vadhan, Avi Wigderson, as well as with the fellow students of the theory lab at MIT - many thanks to you all. I am grateful to my thesis committee members Ron Rivest and Madhu Sudan. Finally, many thanks to Sara, Marty and Erik, Michal, Yonatan, Alon and Yuval for being my home away from home; to Eema and Aba, Uri, Tamar, Ella and the rest of the family for their endless support and encouragement; and last but not least, many many thanks to Doron for his constant support and love even during the long months I disappeared into this dissertation. Contents 1 Introduction 13 1.1 Cryptographic Hardcore Predicates ...... ........... 14 1.2 A Study of Multiplication Codes ... ..... .......... 21 1.3 Learning Characters with Noise ... ..... .......... 28 1.4 Algorithms for Data Intensive Applications .. .......... 36 1.5 Conclusions & Thesis Organization .. .... .......... 38 2 Preliminaries 39 2.1 Notations and Terminology .. ..... ... 39 2.2 Fourier Transform over Finite Abelian Groups 40 2.3 Error Correcting Codes . ..... ..... .. 42 2.4 One-Way Functions and Hardcore Predicates . 45 2.5 Tail Inequalities in Probability . ... ..... 46 3 Learning Characters with Noise in Query Access Model 49 3.1 Introduction ............................ .... 49 3.2 The SFT Algorithm ........................ .... 51 3.2.1 The SFT Algorithm over N ............... ... 52 3.2.2 The SFT Algorithm over ZN, x ... x Zk. ....... ... 55 3.2.3 The SFT Algorithm over Finite Abelian Groups . ... ... 59 3.3 Analysis of the SFT Algorithm ................. ... 59 3.3.1 Analysis of the SFT Algorithm over ZZN ........ ... 59 3.3.2 Analysis of the SFT Algorithm over ZN, X ... X ZNk . .. 67 3.3.3 Analysis of SFT Algorithm over Finite Abelian Groups ... 72 3.4 Properties of Filters ha,b and h ',a,'b ............... ... 73 3.4.1 Properties of ha,b: Proof of Lemma 3.19 . ....... ... 73 3.4.2 Technical Proposition on Consecutive Characters Sum ... 75 3.4.3 Properties of h",a,b: Proof of Lemma 3.27 . ...... ... 77 4 Learning Characters with Noise in Random Samples Access Model 4.1 Introduction. ......................... 4.2 Overview of Definitions and Results ................... 4.2.1 rLCN in Bounded Distance Noise Models . .......... 4.2.2 rLCN in Noise Models Simulating Product Group Structure 4.3 Proof of Tractability Results . ...................... 4.4 Proof of Random Self Reducibility Result . 4.5 Algorithm for rLCN: Proof of Theorem 4.10 5 Learning Characters with Noise in Intermediate Access Models 5.1 Introduction . ... ................. ... 96 5.2 Overview of Definitions and Results ...... .. ... 97 5.2.1 Interval Access Model ......... .. ... 98 5.2.2 GP-Access Model and DH-Access Model . ... 99 5.2.3 Subset Access Model ............ .... 100 5.2.4 Further Results ............... ... 102 5.3 Omitted Proofs .. .................. ... 103 6 Codes for Finite Abelian Groups: Multiplication Codes 105 6.1 Introduction ........................ ........ 105 6.2 Overview of Results and Techniques ..... .. .......... 113 6.2.1 List Decoding via Learning . .... ............... 113 6.2.2 Self Correcting via Testing . ... ... .... .. 119 6.2.3 Distance Bounding using Fourier Analysis . .... .... .. 122 6.2.4 Codes for Groups with Small Generating Set . ..... .... 123 6.2.5 Linear Encoding Length via Restrictions .. .... .... .. 127 6.2.6 Soft Local Error Reduction for Concatenated ABNNR Codes . 129 6.3 List Decoding via Learning . ....................... 132 6.3.1 Combinatorial List Decoding Bound .... ... ..... 133 6.3.2 List Decoding Algorithm . ..................... 133 6.3.3 Concentration and Agreement Lemma . ... .... ... .. 135 6.3.4 List Decoding Codes CP for Well Concentrated P ... .... 136 6.3.5 List Decoding Chal f . ... .... 138 6.4 Distance Bounding using Fourier Spectrum .. ... .. ........ 138 6.5 Self Correcting via Testing . ....................... 141 6.6 Linear Encoding Length via Restrictions . .... .. ... .. 147 6.7 Codes for Groups of Small Generating Sets . ... .. ... ...... 149 6.8 Soft Error Reduction for Concatenated ABNNR Codes . .... ... 150 7 Cryptographic Hardcore Predicates via List Decoding 157 7.1 Introduction . ........................ .... 158 7.2 Hardcore Predicates via List Decoding .......... .. 163 7.3 Number Theoretic Hardcore Predicates . ......... .. 167 7.3.1 Segment Predicates . ................ .... 167 7.3.2 Proving Segment Predicates Hardcore . .. ... .... 168 7.4 Simultaneous Bits Security . ................ .. 173 7.5 On Diffie-Hellman Hardcore Predicates . ......... .. 176 List of Figures 6-1 ABNNR code concatenated with binary code .............. 130 List of Tables 1.1 Example of predicates and codes .. ................ .. .. 17 1.2 Algorithms for Learning Characters with Noise ............. 32 7.1 Example of predicates and codes .. ................ .. .. 161 Chapter 1 Introduction This thesis presents results in three areas: cryptography, coding theory and sublinear algorithms. In cryptography, we introduce a unifying framework for proving that a Boolean predicate is hardcore for a one-way function and apply
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