This beginning graduate textbook describes both recent achievements and classical results of computational complexity theory. Requiring essentially no background apart from mathematical maturity, the book can be used as a reference for self-study for anyone interested in complexity, including physicists, mathematicians, and other scientists, as well as a textbook for a variety of courses and seminars. More than 300 exercises are included with a selected hint set.

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Computational Complexity: A Modern Approach

In the data stream scenario, input arrives very rapidly and there is limited memory to store the input. Algorithms have to work with one or few passes over the data, space less than linear in the input size or time significantly less than the input size. In the past few years, a new theory has emerged for reasoning about algorithms that work within these constraints on space, time, and number of passes. Some of the methods rely on metric embeddings, pseudo-random computations, sparse approximation theory and communication complexity. The applications for this scenario include IP network traffic analysis, mining text message streams and processing massive data sets in general. Researchers in Theoretical Computer Science, Databases, IP Networking and Computer Systems are working on the data stream challenges. This article is an overview and survey of data stream algorithmics and is an updated version of [1].

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Data streams: algorithms and applications

1 Introduction 2 Map 3 The Data Stream Phenomenon 4 Data Streaming: Formal Aspects 5 Foundations: Basic Mathematical Ideas 6 Foundations: Basic Algorithmic Techniques 7 Foundations: Summary 8 Streaming Systems 9 New Directions 10 Historic Notes 11 Concluding Remarks Acknowledgements References

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Data Streams: Algorithms and Applications

Algebraically Closed Fields- Real Closed Fields- Semi-Algebraic Sets- Algebra- Decomposition of Semi-Algebraic Sets- Elements of Topology- Quantitative Semi-algebraic Geometry- Complexity of Basic Algorithms- Cauchy Index and Applications- Real Roots- Cylindrical Decomposition Algorithm- Polynomial System Solving- Existential Theory of the Reals- Quantifier Elimination- Computing Roadmaps and Connected Components of Algebraic Sets- Computing Roadmaps and Connected Components of Semi-algebraic Sets

Algorithms in real algebraic geometry

We provide formal definitions and efficient secure techniques for turning noisy information into keys usable for any cryptographic application, and, in particular, reliably and securely authenticating biometric data. Our techniques apply not just to biometric information, but to any keying material that, unlike traditional cryptographic keys, is (1) not reproducible precisely and (2) not distributed uniformly. We propose two primitives: a fuzzy extractor reliably extracts nearly uniform randomness $R$ from its input; the extraction is error-tolerant in the sense that $R$ will be the same even if the input changes, as long as it remains reasonably close to the original. Thus, $R$ can be used as a key in a cryptographic application. A secure sketch produces public information about its input $w$ that does not reveal $w$ and yet allows exact recovery of $w$ given another value that is close to $w$. Thus, it can be used to reliably reproduce error-prone biometric inputs without incurring the security risk inherent in storing them. We define the primitives to be both formally secure and versatile, generalizing much prior work. In addition, we provide nearly optimal constructions of both primitives for various measures of “closeness” of input data, such as Hamming distance, edit distance, and set difference.

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Fuzzy Extractors: How to Generate Strong Keys from Biometrics and Other Noisy Data

Computer algebra systems are now ubiquitous in all areas of science and engineering. This highly successful textbook, widely regarded as the 'bible of computer algebra', gives a thorough introduction to the algorithmic basis of the mathematical engine in computer algebra systems. Designed to accompany one- or two-semester courses for advanced undergraduate or graduate students in computer science or mathematics, its comprehensiveness and reliability has also made it an essential reference for professionals in the area. Special features include: detailed study of algorithms including time analysis; implementation reports on several topics; complete proofs of the mathematical underpinnings; and a wide variety of applications (among others, in chemistry, coding theory, cryptography, computational logic, and the design of calendars and musical scales). A great deal of historical information and illustration enlivens the text. In this third edition, errors have been corrected and much of the Fast Euclidean Algorithm chapter has been renovated.

http://www.gbv.de/dms/goettingen/246716959.pdf

Modern Computer Algebra

A selective classified bibliography of symbolic computation in some areas of chemistry is provided together with some examples of computer algebra algorithms and techniques to facilitate future joint work of chemists and computer scientists. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2004

Symbolic calculation in chemistry: Selected examples

Joachim von zur Gathen Jurgen Gerhard Fachbereich 17 Mathematik-Informatik Universitat-GH Paderborn D-33095 Paderborn, Germany {gathen, jngerhar}@uni-paderborn. de http://www.uni-paderborn .de/cs/gathen .html

Fast algorithms for Taylor shifts and certain difference equations

We describe algorithms for polynomial multiplication and polynomial factorization over the binary field IF2.and their implementation. They allow polynomials of degree up to 100,000 to be factored in about one dqy of CPU time.

Jürgen Gerhard

Papers

Modern Computer Algebra

Symbolic calculation in chemistry: Selected examples

Fast algorithms for Taylor shifts and certain difference equations

Arithmetic and factorization of polynomial over F2 (extended abstract)

Modern Computer Algebra: Gauß