Cryptography is concerned with the construction of schemes that withstand any abuse. A cryptographic scheme is constructed so as to maintain a desired functionality, even under malicious attempts aimed at making it deviate from its prescribed behavior. The design of cryptographic systems must be based on firm foundations, whereas ad hoc approaches and heuristics are a very dangerous way to go. These foundations were developed mostly in the 1980s, in works that are all co-authored by Shafi Goldwasser and/or Silvio Micali. These works have transformed cryptography from an engineering discipline, lacking sound theoretical foundations, into a scientific field possessing a well-founded theory, which influences practice as well as contributes to other areas of theoretical computer science. This book celebrates these works, which were the basis for bestowing the 2012 A.M. Turing Award upon Shafi Goldwasser and Silvio Micali. A significant portion of this book reproduces some of these works, and another portion consists of scientific perspectives by some of their former students. The highlight of the book is provided by a few chapters that allow the readers to meet Shafi and Silvio in person. These include interviews with them, their biographies and their Turing Award lectures.

The book focuses on three related areas in the theory of computation. The areas are modern cryptography, the study of probabilistic proof systems, and the theory of computational pseudorandomness. The common theme is the interplay between randomness and computation. The book offers an introduction and extensive survey to each of these areas, presenting both the basic notions and the most important (sometimes advanced) results. The presentation is focused on the essentials and does not elaborate on details. In some cases it offers a novel and illuminating perspective. The reader may obtain from the book 1. A clear view of what each of these areas is all above. 2. Knowledge of the basic important notions and results in each area. 3. New insights into each of these areas. It is believed that the book may thus be useful both to a beginner (who has only some background in the theory of computing), and an expert in any of these areas.

The focus of this book is the P-versus-NP Question and the theory of NP-completeness. It also provides adequate preliminaries regarding computational problems and computational models. The P-versus-NP Question asks whether or not finding solutions is harder than checking the correctness of solutions. An alternative formulation asks whether or not discovering proofs is harder than verifying their correctness. It is widely believed that the answer to these equivalent formulations is positive, and this is captured by saying that P is different from NP. Although the P-versus-NP Question remains unresolved, the theory of NP-completeness offers evidence for the intractability of specific problems in NP by showing that they are universal for the entire class. Amazingly enough, NP-complete problems exist, and furthermore hundreds of natural computational problems arising in many different areas of mathematics and science are NP-complete.

This book presents a collection of 36 pieces of scientific work in the areas of complexity theory and foundations of cryptography: 20 research contributions, 13 survey articles, and 3 programmatic and reflective viewpoint statements. These so far formally unpublished pieces were written by Oded Goldreich, some in collaboration with other scientists.
The articles included in this book essentially reflect the topical scope of the scientific career of Oded Goldreich now spanning three decades. In particular the topics dealt with include average-case complexity, complexity of approximation, derandomization, expander graphs, hashing functions, locally testable codes, machines that take advice, NP-completeness, one-way functions, probabilistically checkable proofs, proofs of knowledge, property testing, pseudorandomness, randomness extractors, sampling, trapdoor permutations, zero-knowledge, and non-iterative zero-knowledge.
All in all, this potpourri of studies in complexity and cryptography constitutes a most valuable contribution to the field of theoretical computer science centered around the personal achievements and views of one of its outstanding representatives.

Cryptography is one of the most active areas in current mathematics research and applications. This book focuses on cryptography along with two related areas: the study of probabilistic proof systems, and the theory of computational pseudorandomness. Following a common theme that explores the interplay between randomness and computation, the important notions in each field are covered, as well as novel ideas and insights.

This volume commemorates Shimon Even, one of founding fathers of Computer Science in Israel, who passed away on May 1, 2004. This Festschrift contains research contributions, surveys and educational essays in theoretical computer science, written by former students and close collaborators of Shimon. The essays address natural computational problems and are accessible to most researchers in theoretical computer science.

Cryptography is concerned with the conceptualization, definition and construction of computing systems that address security concerns. The design of cryptographic systems must be based on firm foundations. Foundations of Cryptography presents a rigorous and systematic treatment of foundational issues, defining cryptographic tasks and solving cryptographic problems. The emphasis is on the clarification of fundamental concepts and on demonstrating the feasibility of solving several central cryptographic problems, as opposed to describing ad-hoc approaches. This second volume contains a thorough treatment of three basic applications: Encryption, Signatures, and General Cryptographic Protocols. It builds on the previous volume, which provided a treatment of one-way functions, pseudorandomness, and zero-knowledge proofs. It is suitable for use in a graduate course on cryptography and as a reference book for experts. The author assumes basic familiarity with the design and analysis of algorithms; some knowledge of complexity theory and probability is also useful.

Cryptography is concerned with the conceptualization, definition and construction of computing systems that address security concerns. The design of cryptographic systems must be based on firm foundations. This book presents a rigorous and systematic treatment of the foundational issues: defining cryptographic tasks and solving new cryptographic problems using existing tools. It focuses on the basic mathematical tools: computational difficulty (one-way functions), pseudorandomness and zero-knowledge proofs. The emphasis is on the clarification of fundamental concepts and on demonstrating the feasibility of solving cryptographic problems, rather than on describing ad-hoc approaches. The book is suitable for use in a graduate course on cryptography and as a reference book for experts. The author assumes basic familiarity with the design and analysis of algorithms; some knowledge of complexity theory and probability is also useful.

This volume contains a collection of studies in the areas of complexity theory and property testing. The 21 pieces of scientific work included were conducted at different times, mostly during the last decade. Although most of these works have been cited in the literature, none of them was formally published before. Within complexity theory the topics include constant-depth Boolean circuits, explicit construction of expander graphs, interactive proof systems, monotone formulae for majority, probabilistically checkable proofs (PCPs), pseudorandomness, worst-case to average-case reductions, and zero-knowledge proofs. Within property testing the topics include distribution testing, linearity testing, lower bounds on the query complexity (of property testing), testing graph properties, and tolerant testing. A common theme in this collection is the interplay between randomness and computation.

Property testing is concerned with the design of super-fast algorithms for the structural analysis of large quantities of data. The aim is to unveil global features of the data, such as determining whether the data has a particular property or estimating global parameters. Remarkably, it is possible for decisions to be made by accessing only a small portion of the data. Property testing focuses on properties and parameters that go beyond simple statistics. This book provides an extensive and authoritative introduction to property testing. It provides a wide range of algorithmic techniques for the design and analysis of tests for algebraic properties, properties of Boolean functions, graph properties, and properties of distributions.

The focus of this book is the P-versus-NP Question and the theory of NP-completeness. It also provides adequate preliminaries regarding computational problems and computational models. The P-versus-NP Question asks whether or not finding solutions is harder than checking the correctness of solutions. An alternative formulation asks whether or not discovering proofs is harder than verifying their correctness. It is widely believed that the answer to these equivalent formulations is positive, and this is captured by saying that P is different from NP. Although the P-versus-NP Question remains unresolved, the theory of NP-completeness offers evidence for the intractability of specific problems in NP by showing that they are universal for the entire class. Amazingly enough, NP-complete problems exist, and furthermore hundreds of natural computational problems arising in many different areas of mathematics and science are NP-complete.

Complexity theory is a central field of the theoretical foundations of computer science. It is concerned with the general study of the intrinsic complexity of computational tasks; that is, it addresses the question of what can be achieved within limited time (and/or with other limited natural computational resources). This book offers a conceptual perspective on complexity theory. It is intended to serve as an introduction for advanced undergraduate and graduate students, either as a textbook or for self-study. The book will also be useful to experts, since it provides expositions of the various sub-areas of complexity theory such as hardness amplification, pseudorandomness and probabilistic proof systems. In each case, the author starts by posing the intuitive questions that are addressed by the sub-area and then discusses the choices made in the actual formulation of these questions, the approaches that lead to the answers, and the ideas that are embedded in these answers.

Cryptography is concerned with the construction of schemes that withstand any abuse. A cryptographic scheme is constructed so as to maintain a desired functionality, even under malicious attempts aimed at making it deviate from its prescribed behavior. The design of cryptographic systems must be based on firm foundations, whereas ad hoc approaches and heuristics are a very dangerous way to go. These foundations were developed mostly in the 1980s, in works that are all co-authored by Shafi Goldwasser and/or Silvio Micali. These works have transformed cryptography from an engineering discipline, lacking sound theoretical foundations, into a scientific field possessing a well-founded theory, which influences practice as well as contributes to other areas of theoretical computer science. This book celebrates these works, which were the basis for bestowing the 2012 A.M. Turing Award upon Shafi Goldwasser and Silvio Micali. A significant portion of this book reproduces some of these works, and another portion consists of scientific perspectives by some of their former students. The highlight of the book is provided by a few chapters that allow the readers to meet Shafi and Silvio in person. These include interviews with them, their biographies and their Turing Award lectures.

An interactive proof system is called doubly-efficient if the prescribed prover strategy can be implemented in polynomial-time and the verifier's strategy can be implemented in almost-linear time. Such proof systems make the benefits of interactive proof system available to real-life agents who are restricted to polynomial-time computation. This book surveys some of the known results regarding doubly-efficient interactive proof systems. It starts by presenting two simple constructions for t-no-CLIQUE, where the first construction offers the benefit of being generalized to any "locally characterizable" set, and the second construction offers the benefit of preserving the combinatorial flavor of the problem. It then turns to two more general constructions of doubly-efficient interactive proof system: the proof system for sets having (uniform) bounded-depth circuits and the proof system for sets that are recognized in polynomial-time and small space. The presentation of the GKR construction is complete and is somewhat different from the original presentation. A brief overview is provided for the RRR construction.

Various types of probabilistic proof systems have played a central role in the development of computer science in the last couple of decades. These proof systems deviate from the traditional concept of a proof by introducing randomization and interaction into the verification process. Probabilistic proof systems carry an error probability (which is explicitly bounded and can be decreased by repetitions), but they offer various advantages over deterministic proof systems. This primer concentrates on three types of probabilistic proof systems: interactive proofs, zero-knowledge proofs, and probabilistically checkable proofs (PCP). Surveying the basic results regarding these proof systems, the primer stresses the essential role of randomness in each of them.