Boolean Functions

Boolean functions are essential to systems for secure and reliable communication. This comprehensive survey of Boolean functions for cryptography and coding covers the whole domain and all important results, building on the author's influential articles with additional topics and recent results. A useful resource for researchers and graduate students, the book balances detailed discussions of properties and parameters with examples of various types of cryptographic attacks that motivate the consideration of these parameters. It provides all the necessary background on mathematics, cryptography, and coding, and an overview on recent applications, such as side channel attacks on smart cards, cloud computing through fully homomorphic encryption, and local pseudo-random generators. The result is a complete and accessible text on the state of the art in single and multiple output Boolean functions that illustrates the interaction between mathematics, computer science, and telecommunications

The essential guide showing how the unbounded delay model of computation of the Boolean functions may be used in the analysis of the Boolean networks Boolean Functions: Topics in Asynchronicity&n

This book focuses on the different representations and cryptographic properties of Booleans functions, presents constructions of Boolean functions with some good cryptographic properties. More specifi

This is a graduate-level introduction to the theory of Boolean functions, an exciting area lying on the border of probability theory, discrete mathematics, analysis, and theoretical computer science. Certain functions are highly sensitive to noise; this can be seen via Fourier analysis on the hypercube. The key model analyzed in depth is critical percolation on the hexagonal lattice. For this model, the critical exponents, previously determined using the now-famous Schramm–Loewner evolution, appear here in the study of sensitivity behavior. Even for this relatively simple model, beyond the Fourier-analytic set-up, there are three crucially important but distinct approaches: hypercontractivity of operators, connections to randomized algorithms, and viewing the spectrum as a random Cantor set. This book assumes a basic background in probability theory and integration theory. Each chapter ends with exercises, some straightforward, some challenging.

This is a graduate-level introduction to the theory of Boolean functions, an exciting area lying on the border of probability theory, discrete mathematics, analysis, and theoretical computer science. Certain functions are highly sensitive to noise; this can be seen via Fourier analysis on the hypercube. The key model analyzed in depth is critical percolation on the hexagonal lattice. For this model, the critical exponents, previously determined using the now-famous Schramm–Loewner evolution, appear here in the study of sensitivity behavior. Even for this relatively simple model, beyond the Fourier-analytic set-up, there are three crucially important but distinct approaches: hypercontractivity of operators, connections to randomized algorithms, and viewing the spectrum as a random Cantor set. This book assumes a basic background in probability theory and integration theory. Each chapter ends with exercises, some straightforward, some challenging.

Boolean functions are perhaps the most basic objects of study in theoretical computer science. They also arise in other areas of mathematics, including combinatorics, statistical physics, and mathematical social choice. The field of analysis of Boolean functions seeks to understand them via their Fourier transform and other analytic methods. This text gives a thorough overview of the field, beginning with the most basic definitions and proceeding to advanced topics such as hypercontractivity and isoperimetry. Each chapter includes a 'highlight application' such as Arrow's theorem from economics, the Goldreich–Levin algorithm from cryptography/learning theory, Håstad's NP-hardness of approximation results, and 'sharp threshold' theorems for random graph properties. The book includes roughly 450 exercises and can be used as the basis of a one-semester graduate course. It should appeal to advanced undergraduates, graduate students and researchers in computer science theory and related mat

Written by prominent experts in the field, this monograph provides the first comprehensive, unified presentation of the structural, algorithmic and applied aspects of the theory of Boolean functions. The book focuses on algebraic representations of Boolean functions, especially disjunctive and conjunctive normal form representations. This framework looks at the fundamental elements of the theory (Boolean equations and satisfiability problems, prime implicants and associated short representations, dualization), an in-depth study of special classes of Boolean functions (quadratic, Horn, shellable, regular, threshold, read-once functions and their characterization by functional equations) and two fruitful generalizations of the concept of Boolean functions (partially defined functions and pseudo-Boolean functions). Several topics are presented here in book form for the first time. Because of the depth and breadth and its emphasis on algorithms and applications, this monograph will have sp

Boolean functions are the building blocks of symmetric cryptographic systems. Symmetrical cryptographic algorithms are fundamental tools in the design of all types of digital security systems (i.e. co

The two internationally renowned authors elucidate the structure of "fast" parallel computation. Its complexity is emphasised through a variety of techniques ranging from finite combinatorics, probabi

Cryptographic Boolean Functions and Applications 2nd edition is designed to be a comprehensive reference for the use of Boolean functions in modern cryptography. While the vast majority of research on

This book discusses cryptographic and combinatorial properties of Boolean functions. Boolean functions are very easy to understand (but deep in analysis) and the subject covers significant amount of m

Presents a large number of recent research results previously unavailable in book form. Initially deals with the wee-known computation models, and goes on to special types of circuits, parallel comput

This book is concerned with cardinal number valued functions defined for any Boolean algebra. Examples of such functions are independence, which assigns to each Boolean algebra the supremum of the car

Bent Functions: Results and Applications to Cryptography offers a unique survey of the objects of discrete mathematics known as Boolean bent functions. These maximal, nonlinear Boolean functions and t

This book describes representations of Boolean functions that have small size for many important functions and which allow efficient work with the represented functions. Efficient algorithms for opera

This graduate textbook explains the mathematical basis for Boolean reasoning, describes syllogistic and functional approaches to the solution of Boolean and propositional problems, and presents applic

This collection of papers presents a series of in-depth examinations of a variety of advanced topics related to Boolean functions and expressions. The chapters are written by some of the most prominent experts in their respective fields and cover topics ranging from algebra and propositional logic to learning theory, cryptography, computational complexity, electrical engineering, and reliability theory. Beyond the diversity of the questions raised and investigated in different chapters, a remarkable feature of the collection is the common thread created by the fundamental language, concepts, models, and tools provided by Boolean theory. Many readers will be surprised to discover the countless links between seemingly remote topics discussed in various chapters of the book. This text will help them draw on such connections to further their understanding of their own scientific discipline and to explore new avenues for research.

This introduction to Boolean algebra begins with an intuitive approach to set theory and an axiomatic account of the fundamentals of Boolean algebra, proceeding to concise accounts of applications to

This elementary treatment by a distinguished mathematician begins with the algebra of classes and proceeds to discussions of several different axiomatizations and Boolean algebra in the setting of the

This undergraduate-level describes the mathematics most needed in computer science. Bender and Williamson (both mathematics, U. of California, San Diego) cover Boolean functions and computer arithmeti

Mathematics for Electrical Engineering and Computing embraces many applications of modern mathematics, such as Boolean Algebra and Sets and Functions, and also teaches both discrete and continuous sys

By considering the size of the logical network needed to perform a given computational task, the intrinsic difficulty of that task can be examined. Boolean function complexity, the combinatorial study of such networks, is a subject that started back in the 1950s and has today become one of the most challenging and vigorous areas of theoretical computer science. The papers in this book stem from the London Mathematical Society Symposium on Boolean Function Complexity held at Durham University in July 1990. The range of topics covered will be of interest to the newcomer to the field as well as the expert, and overall the papers are representative of the research presented at the Symposium. Anyone with an interest in Boolean Function complexity will find that this book is a necessary purchase.

Communication Complexity describes a new intuitive model for studying circuit networks that captures the essence of circuit depth. Although the complexity of boolean functions has been studied for alm

This presentation on the basics of Boolean algebra has ranked among the fundamental books on this important subject in mathematics and computing science since its initial publication in 1963. Concise

This book represents a significant step towards bridging the areas of Boolean satisfiability and constraint satisfaction by providing theoretical explanations for why SAT-solvers are efficient on cert

This self-contained monograph explores a new theory centered around boolean representations of simplicial complexes leading to a class of complexes and featuring matroids as central to the theory. The

Boolean circuit complexity is the combinatorics of computer science and involves many intriguing problems that are easy to state and explain, even for the layman. This book is a comprehensive descrip

This volume concentrates on the structure of Boolean algebras and rings as developed through simpler algebraic systems. The algebra of logic and set theory appears as applications or illustrations thr

This book is an informal, although systematic presentation of lectures given by the authors on Boolean algebras, intended for advanced undergraduates and beginning graduate students. In a bold and ref

The Boolean network (BN) is a mathematical model of genetic networks and other biological networks. Although extensive studies have been done on BNs from a viewpoint of complex systems, not so many st

In this new text, Steven Givant—the author of several acclaimed books, including works co-authored with Paul Halmos and Alfred Tarski—develops three theories of duality for Boolean algebras with opera

This book brings to bear a body of logic synthesis techniques, in order to contribute to the analysis and control of Boolean Networks (BN) for modeling genetic diseases such as cancer. The authors pro

This book provides an overview of automatic test pattern generation (ATPG) and introduces novel techniques to complement classical ATPG, based on Boolean Satisfiability (SAT). A fast and highly fault

This third edition, now available in paperback, is a follow up to the author's classic Boolean-Valued Models and Independence Proofs in Set Theory. It provides an exposition of some of the most import

In Test Pattern Generation using Boolean Proof Engines, we give an introduction to ATPG. The basic concept and classical ATPG algorithms are reviewed. Then, the formulation as a SAT problem is conside

Written for students and researchers in the fields of behavioral sciences, history and logic, this textbook on comparative methods with Boolean algebra explains how these mathematical theories are use

An investigation of the logical flexibility principles needed for a formal semantic account of coordination, plurality, and scope in natural language. Since the early work of Montague, Boolean seman

Mathematician Harald Bohr, motivated by questions about which functions could be represented by a Dirichlet series, devised the theory of almost periodic functions during the 1920s. His groundbreaking

For courses in college algebra. Ties concepts together using a functions approach The Concepts Through Functions Series introduces functions at the start of each text, and maintains a continuo