A PSEUDORANDOM GENERATOR CONSTRUCTION BASED ON RANDOMNESS EXTRACTORS AND COMBINATORIAL DESIGNS Ladan A. Mahabadi B.Sc., University of British Columbia, Vancouver, Canada, 2003 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in the School of Computing Science @ Ladan A. Mahabadi 2006 SIMON FRASER UNIVERSITY Spring 2006 All rights reserved. This work may not be reproduced in whole or in part, by photocopy or other means, without the pernlission of the author. APPROVAL Name: Ladan A. Mahabadi Degree: Master of' Science Title of thesis: A Pseudorandom Generator Construction based on Randomness Extractors and Combinatorial Designs Examining Committee: Dr. Arthur Kirkpatrick Chair Dr. Arvind Gupta, Supervisor, Professor of Conlputing Science Simon F'raser University Dr. Valentine Kabanets, Senior Supervisor, Professor of Computing Science Simon Fraser University Dr. GBbor Tardos, Examiner, Professor of Computing Science, Simon Fraser University Date Approved: hr-/3A% .. I1 SIMON FRASER !? &-J2 !? UNIVERSITYIibra ry DECLARATION OF PARTIAL COPYRIGHT LICENCE The author, whose copyright is declared on the title page of this work, has granted to Simon Fraser University the right to lend this thesis, project or extended essay to users of the Simon Fraser University Library, and to make partial or single copies only for such users or in response to a request from the library of any other university, or other educational institution, on its own behalf or for one of its users. The author has further granted permission to Simon Fraser University to keep or make a digital copy for use in its circulating collection, and, without changing the content, to translate the thesislproject or extended essays, if technically possible, to any medium or format for the purpose of preservation of the digital work. The author has further agreed that permission for multiple copying of this work for scholarly purposes may be granted by either the author or the Dean of Graduate Studies. It is understood that copying or publication of this work for financial gain shall not be allowed without the author's written permission. Permission for public performance, or limited permission for private scholarly use, of any multimedia materials forming part of this work, may have been granted by the author. This information may be found on the separately catalogued multimedia material and in the signed Partial Copyright License. The original Partial Copyright License attesting to these terms, and signed by this author, may be found in the original bound copy of this work, retained in the Simon Fraser University Archive. Simon Fraser University Library Burnaby, BC, Canada Abstract Nisan and Wigderson in their seminal work introduced a new (conditional) pseudorandom generator construction which since then has been extensively used in con~plexitytheory and has led to extensive further research. Impagliazzo and Wigderson (1997), and Sudan, Trevisan, and Vadhan (2001) have shown how this construction can be utilized to prove conditional derandomization results under weaker hardness assumptions. We study the construction of pseudorandom generators, and use an observation of Sudan et al. to re- cast the Impagliazzo-Wigderson construction in terms of weak sources of randomness; such a source is a distribution on binary strings that is "random" in the sense of having high "entropy". We will then use an efficient algorithm of Gabizon et al. to extract almost all of the randomness present, obtaining a pseudorandom generator that stretches O(n) bits to R(n2n) bits. Keywords: pseudorandom generator, derandomization, pseudorandomness, oblivious bit-fixing weak sources, extractors, preservation of indistinguishability To my parents, my sister, and mom Badri. "Only those who will risk going too far can possibly find out how far me can go." - T. S. Eliot Acknowledgments Guidance, support and assistance of numerous individuals have made this thesis possible, and there are many to whom I am forever grateful. Mainly, this thesis is based on joint work with my senior supervisor, Dr. Valentine Kabanets. Naturally, I 'd like to thank Dr. Kabanets and Dr. Arvind Gupta for introducing me to this field, and for their great input and guidance. I 'd like to extend my gratitude to Dr. GBbor Tardos for serving as my examiner on the examining committee. I owe special thanks to my friends Jano Manuch and Peter Zubcsek for their unconditional support and help throughout this thesis. I truly enjoyed my many research meetings with Jano, and am grateful for his patience and meticulous attention to mathematical details. My sincere thanks go to Antonina Kolokolova for her great and timely advice. Most importantly, I owe everything to my parents and sister. They have been with me through it all, and without their love, support and everlasting faith in me I would not have succeeded. Contents Approval Abstract iii Dedication Quotation Acknowledgments vi Contents vii List of Figures ix 1 Introduction 1.1 Blum-Micali-Yao's pseudorandom generator . 1.2 Nisan-Wigderson's Pseudorandom Generator . 1.3 Distinction between the NW and BMY PRGs . 1.4 Relationship between hardness and randomness . 1.4.1 Hardness implies Randomness . 1.4.2 Randomness implies Hardness . 1.5 Amplification of Hardness . 1.5.1 Yao's XOR Lemma . 1.5.2 Amplification from worst-case hardness to Mild hardness . 1.5.3 Impagliazzo-Wigderson's Randomness efficient hardness amplification 1.6 Brief introduction to extractors . 1.7 Sudan-Trevisan-Vadhan Pseudorandom Generator Construction . vii 1.8 Significance of our work .............................. 14 2 Preliminaries 18 2.1 Preliminary definitions ............................... 18 2.2 Hardness vs . Indistinguishability ......................... 20 2.3 Hardness vs . Randomness ............................. 20 2.3.1 Randomness implies hardness ....................... 20 2.3.2 Hardness implies randomness ....................... 22 2.4 Nisan-Wigderson Pseudorandom Generator Construction ........... 24 2.5 Main Definitions .................................. 25 2.5.1 Weak sources and extractors ....................... 25 2.5.2 Expanders .................................. 27 3 Our Construction 30 4 Preservation of Indistinguishability 34 4.1 Preservation of indistinguishability in the cryptography setting ........ 35 4.2 Preservation of indistinguishability in the derandomization setting .............................. 39 5 Correctness of our construction 46 6 Conclusion 53 A Auxiliary Proofs 5 5 A.l Hardness vs . Indistinguishabilit. ......................... 55 Bibliography 5 8 ... Vlll List of Figures Derandomization of a randomized algorithm using a pseudorandom generator G . (0' l)d + (0. l)T(n)............................... Overview of Nisan-Wigderson's pseudorandom generator [NW94] ....... Yao's amplification of hardness using the xor function [Yao82] ......... Overview of the Impagliazzo-Wigderson Construction[IW97] .......... Derandomization of a randomized algorithm using an extractor ........ Sudan-Trevisan-Vadhan's pseudorandom generator construction using an ex- tractor [STVOl] ................................... Existing hardness amplification techniques .................... Nisan-Wigderson's pseudorandom generator [NW94] .............. Impagliazzo-Wigderson 's generator used for derandomizing Yao's XOR lemma [IW97] ........................................ An overview of our PRG construction G : (0. l)O(n)+ (0. l)(1-o(1))6n2n... Pseudorandon1 Generator G(z): (0. l)O(n)+ (0. l)(1-o(1))6n2n........ Preservation of Indistinguishability in different computational settings .... Chapter 1 Introduction The problem of improving efficiency of algorithms through more econonlical use of resources such as time, space and randomness has fascinated computer scientists for decades. For in- stance, randomness appears to be vital for the efficiency of some algorithms (e.g., polynonlial identity testing) and in other cases can be reduced or eliminated (e.g., primality testing). However, there are applications such as cryptography or zero-knowledge proofs where ran- domness is inherently necessary. Indeed, one of the most fundamental questions in com- plexity theory is determining the necessity of randomness in efficient algorithms. Namely, is the aesthetic speed up produced by utilizing randomness in algorithm design just due to lack of efficient, advanced deterministic algorithms or does randomness really help? Resolving this fundamental question is intriguing both philosophically and theoretically. Philosophically, does adding information independent from the input help speed up some computations? Theoretically, elimination of randomness translates into efficient determin- istic simulation of randomized conlplexity classes such as BPP (2-sided probabilistic poly- nomial time algorithms). Deciphering the role of randomness requires elucidation of the relationship between conlplexity classes BPP, P (deterministic polynomial time) and EXP (deterministic exponential time). This relationship has been studied from different angles using tools such as pseudo-random generators (referred to as PRGs). A randomized algorithm R, besides its regular input string x, needs a secondary input r as the random choices used throughout its computation. Throughout its computation, if a random coin flip is needed, the random bit will be looked up from r and depending on this bit one of two actions takes place. In practice, access to truly random bits is very limited, and