Bounded Independence Plus Noise by Chin Ho Lee A Thesis Submitted in Partial Fulfilment of the Requirements for the Degree of Doctor of Philosophy in Computer Science Northeastern University August 2019 c 2019 Chin Ho Lee All rights reserved. To my parents Abstract Bounded Independence Plus Noise by Chin Ho Lee Doctor of Philosophy Khoury College of Computer Science Northeastern University Derandomization is a fundamental research area in theoretical computer science. In the past decade, researchers have been able to derandomize a number of computational problems, leading to breakthrough discoveries such as proving SL = L, explicit constructions of Ramsey graphs and optimal-rate error-correcting codes. Bounded independent and small-bias distributions are two pseudorandom primitives that are used extensively in derandomization. This thesis studies the power of these primitives under the perturbation of noise. We give positive and negative results on these perturbed distributions. In particular, we show that they are significantly more powerful than the unperturbed ones, and have the potential to solve long standing open problems such as proving RL = L and AC0[⊕] lower bounds. As applications, we give new lower bounds on the complexity of decoding error-correcting codes, nearly-optimal pseudorandom generators for old and new classes of tests, and limita- tions on the sum of small-bias distributions. i Acknowledgements Foremost, I would like to thank Manu, Daniel, Jon and Omer for taking the time to serve on my thesis committee. This thesis would have been impossible without the wise guidance of Manu. I am truly grateful for his patience during my six years of PhD. His unique perspectives on many matters, whether they are related to research or not, have made a great impact on my life. His striving for simplicity and his clarity of writing will always be examples for me to pursue in the future. I thank Ravi Boppana, Elad Haramaty, Johan H˚astadand Manu, with whom I have collaborated on several results related to this thesis, for sharing their ideas with me. I have learned a great deal of doing research through these collaborations. I am extremely grateful to Amnon Ta-Shma for hosting me at Tel-Aviv University, and to Dean Doron and Gil Cohen for many stimulating discussions during my visit. I thank Salil Vadhan for his excellent course on pseudorandomness at Harvard. My understanding of pseudorandomness would not have been the same without it. I thank Andrej Bogdanov for teaching me Fourier Analysis during my masters, a tool that is used extensively in this thesis. I also thank Andrej for being available for discussions whenever I went home for a visit, and for giving me the opportunities to give talks at the theory seminars in CUHK. My PhD life would have been miserable without my friends at Harvard and Khoury, in particular those at MD138 and WVH266. I thank them for keeping the office spaces full of positive energy, and organizing all kinds of activities to put my mind at ease when getting depressed from research. Finally, I thank my beloved Sabrina for her endless support, patience and love on the other side of the world. ii Contents Abstract.........................................i Acknowledgements................................... ii Table of Contents.................................... iii 1 Introduction1 1.1 Contribution of this thesis............................3 1.2 Organization of this thesis............................4 2 Bounded Independence Plus Noise Fools Products7 2.1 Our results....................................9 2.1.1 Application: The complexity of decoding................ 10 2.1.2 Application: Pseudorandomness..................... 14 2.1.3 Techniques................................. 16 2.2 Bounded independence plus noise fools products................ 17 2.2.1 Preliminaries............................... 20 2.2.2 Proof of Theorem 2.22.......................... 22 2.3 Proofs for Section 2.1.1.............................. 24 2.4 Pseudorandomness: I.............................. 26 2.5 Pseudorandomness, II.............................. 29 2.5.1 Proof of Theorem 2.38.......................... 29 2.5.2 A recursive generator........................... 32 2.6 Pseudorandomness, III.............................. 35 2.7 A lower bound on b and η ............................ 36 3 Pseudorandom Generators for Read-Once Polynomials 37 3.1 Our results.................................... 37 3.1.1 Techniques................................. 40 3.2 Bounded independence plus noise fools products................ 44 3.3 Pseudorandom generators............................ 48 3.4 On almost k-wise independent variables with small total-variance...... 52 3.4.1 Preliminaries............................... 53 3.4.2 Proof of Lemma 3.12........................... 54 3.5 Improved bound for bounded independence plus noise fools products.... 68 3.5.1 Noise reduces variance of bounded complex-valued functions..... 69 iii 3.5.2 XOR Lemma for bounded independence................ 71 3.5.3 Proof of Theorem 3.9........................... 73 3.5.4 Proof of Theorem 3.11.......................... 76 3.6 Small-bias plus noise fools degree-2 polynomials................ 77 3.7 Proof of Claim 3.8................................ 78 3.8 Moment bounds for sum of almost d-wise independent variables....... 80 4 Fourier Bounds and Pseudorandom Generators for Product Tests 85 4.1 Our results.................................... 85 4.1.1 Techniques................................. 89 4.2 Fourier spectrum of product tests........................ 92 4.2.1 Schur-concavity of g ........................... 97 4.2.2 Lower bound............................... 99 4.3 Pseudorandom generators............................ 99 4.3.1 Generator for product tests....................... 101 4.3.2 Almost-optimal generator for XOR of Boolean functions....... 102 4.4 Level-d inequalities................................ 105 5 Some Limitations of the Sum of Small-Bias Distributions 109 5.1 Our results.................................... 110 5.2 Our techniques.................................. 113 5.3 Our counterexamples............................... 116 5.3.1 General circuits.............................. 116 5.3.2 NC2 circuits............................... 117 5.3.3 One-way log-space computation..................... 117 5.3.4 Depth 3 circuits, DNF formulas and AC0 circuits........... 118 5.3.5 Mod 3 linear functions.......................... 119 5.3.6 Sum of k copies of small-bias distributions............... 123 5.4 Mod 3 rank of k-wise independence....................... 124 5.4.1 Lower bound for almost k-wise independence.............. 124 5.4.2 Pairwise independence.......................... 125 5.5 Complexity of decoding.............................. 127 Bibilography 129 A Fooling read-once DNF formulas 141 iv Chapter 1 Introduction The theory of pseudorandomness studies explicit constructions of objects that appear random to restricted classes of tests. It has numerous connections to computer science and math- ematics, including algorithms, computational complexity, cryptography and combinatorics. In particular, the study of pseudorandomness is indispensable in understanding the power of randomness in computation, a fundamental research area in theoretical computer science. While it is known that certain tasks in areas such as cryptography and distributed computing are impossible without randomness, researchers have been able to derandomize a number of computational problems, showing that randomness often does not give us significant savings in computational resources over determinism. One central open question in derandomization is the BPP vs. P question. It asks whether probabilitistic polynomial-time algorithms can be made deterministic without a drastic draw- back in its running time. This question is largely open, as resolving this question would imply circuit lower bounds that seem beyond reach given our current techniques [IKW02, KI04]. Because of this, research in derandomization can be divided into two directions. • Conditional results: One line of research, pioneered by Blum and Micali [BM84], and Yao [Yao82], shows that derandomization can be realized under the assumption that hard functions exist. This approach has found a lot of success in cryptography, where cryptographic primitives are constructed based on the intractability of several specific computational problems. Indeed, this approach was first proposed for cryptographic applications, and the idea of trading hardness for randomness was first suggested by Shamir [Sha81], who constructed pseudorandom sequences assuming the hardness of the RSA encryption function. The seminal work of Nisan and Wigderson [NW94] shows that derandomizing BPP is possible under weaker complexity assumptions. This is followed by the subsequent work of Impagliazzo and Wigderson [IW99], which showed that if any problem in the class E requires circuits of super-polynomial size to approximate, then BPP = P. Since then, researchers have tried to optimize the hardness and randomness trade-off. • Unconditional results: Another line of research turns to derandomizing restricted classes of computation models for which we can prove unconditional lower bounds. 1 Two major classes of tests that have received a lot of attention since the late 80s are constant-depth circuits [AW89] and space-bounded computation [AKS87]. A major open problem in derandomizing space-bounded computation is the RL vs. L question, which is a space-analogue of BPP vs.