Number theory Engineering & Materials Science and error correcting codes, and is accessible to a broad audience. As mentioned above, relationships between the two subjects of the title are emphasized. I would be happy to teach a course out of this book. An instructor may want students to become comfortable with these topics initially through computations, using the Euclidean Algorithm. One of the chapters whose size increased considerably is the one on congruences. Depending on a course's focus, this could be done fairly easily. A Computational Introduction to Number Theory and Algebra . I imagine that most classes would skip the background material and head straight for the computational chapters, with the background there "as needed" for the students. It also includes an introduction to discrete probability theory—this material is needed to properly treat the topics of probabilistic algorithms and cryptographic applications. PDF | On Sep 1, 2007, Igor Shparlinski published A computational introduction to number theory and algebra. This is an introductory course on the computational aspects of number theory and algebra. I could not use it in a graduate number theory class; it assumes no background at all and does not do some standard topics. The Euclidean Algorithm is presented after sections on solving linear congruences modulo n, and the Chinese Remainder Theorem; applications of the Euclidean Algorithm to these topics are presented later. The titles of the chapters are essentially the same in the two editions, but the contents have undergone change-some topics were added, while some were deleted or modified. Online Computing Reviews Service, It's a pleasure to find a book that is so masterful and so well written that it has all the hallmarks of a classic. These first chapters set the scene, so to speak, for the number theory with which the text is concerned. A book introducing basic concepts from computational number theory and algebra, including all the necessary mathematical background. One regrettable omission is computational tools¿neither Shoup's own C++ NTL library for computational number theory and algebra nor a computer algebra system. Topics covered in the book include: basic properties of integers; congruences; computing with large integers; Euclid's algorithm; the distribution of primes; Abelian groups; rings; finite and discrete probability distributions; probabilistic algorithms; probabilistic primality testing; finding generators and discrete logarithms; quadratic reciprocity and computing modular square roots; modules and vector spaces; matrices; subexponential-time discrete logarithms and factoring; more rings; polynomial arithmetic and applications; linearly generated sequences and applications; finite fields; algorithms for finite fields; and deterministic primality testing. Many examples are given and difficult ideas are introduced gradually. Publication date 2008 Usage Attribution-Noncommercial-No Derivative Works 3.0 Topics Number theory, Algebra Collection opensource; community Language English. I'm trying to solve this exercise from A Computational Introduction to Number Theory and Algebra by Victor Shoup (image below). Ideal as a textbook for introductory courses in number theory and algebra, especially Compared to the previous edition, this one incorporates 150 new exercises. Interdependences among chapters are clearly indicated. I don't think it would be appropriate in any class to start at Chapter 1 and and work through all (or even most) of the content. Each chapter is written in a logical manner, referencing previous material as needed. The author wanted to include all of the mathematics required beyond a standard calculus sequence. I read a standard PDF file. The mathematical coverage includes the basics of number theory, abstract algebra and discrete probability theory. We use cookies to ensure that we give you the best experience on our website. Rather than giving brief introductions to a wide variety of topics, this book provides an in-depth introduction to the field of Number Theory. A Computational Introduction to Number Theory and Algebra provides an introduction to number theory and algebra, with an emphasis on algorithms and applications. Thus the book can serve several purposes. I was pleased to see copious exercises; a student who has completed the book and mastered the exercises will be in a very strong position to embark on some advanced studies. A Computational Introduction to Number Theory and Algebra (BETA version 2) Victor Shoup. The mathematical material covered includes the basics of number theory (including unique factorization, congruences, the distribution of primes, and quadratic reciprocity) and of abstract algebra (including groups, rings, fields, and vector spaces). There are a few sections that are marked with a “(∗),” indicating that the material covered in that section is a bit technical, and is not needed else- where. This will, of course, become outdated with new research in computer science. For many students this will detract from clarity because they do not yet have the mathematical sophistication to work at this level. I first taught an undergraduate class at Harvard in maybe 2002 and went over the first 20 pages of Swinnerton-Dyer's brief course on algebraic number theory book -- expanding it into course-length notes. A Computational Introduction to Number Theory and Algebra - by Victor Shoup April 2005 Skip to main content Accessibility help We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Even though "some topics could simply not be covered," it manages to include a wide range of topics. Reviewed by Michelle Manes, Associate Professor, University of Hawaii on 8/21/16, The text is so comprehensive that it feels overwhelming. read more. The text, as one would expect, contains several chapters that discuss the basic theory of abelian groups and rings, including a fine proof of the fundamental theorem of the structure of finite abelian groups as sums of cyclic groups. However, the mathematical maturity required to read and learn from this text is quite... It is unavoidable that this will happen in any book that treats both subjects seriously, and the author is careful with notation and keeps potential confusion to a minimum. Become a reviewer for Computing Reviews. Of course, this dichotomy between theory and applications is not perfectly maintained: the chapters that focus mainly on applications include the development of some of the mathematics that is specific to a particular application, and very occasionally, some of the chapters that focus mainly on mathematics include a discussion of related algorithmic ideas as well. There is a very brief “Preliminaries” chapter, which fixes a bit of notation and recalls a few standard facts. Shoup steers clear of this recipe approach and, instead, places the entire theory into a formal algebraic setting. I suppose it would be useful for self-study by a very advanced student who already knew a good deal of mathematics and wanted to explore the computational side. All material necessary in future sections is included in the appropriate section. What makes this book unique is the way in which several different mathematical strands?number theory and algebra?are interwoven and made into a masterful whole. The material has also been reorganized to improve clarity of exposition and presentation. Introduction The first part of this book is an introduction to group theory.It begins with a study of permutation groups in chapter 3.Historically this was one of the starting points of group theory.In fact it was in the context of permutations of the roots of a polynomial that they first appeared (see7.4). This PDF document contains hyperlinks, and one may navigate through it by click- ... was to provide an introduction to number theory and algebra, with an emphasis on algorithms and applications, that would be accessible to a broad audience. The first few chapters contain as much mathematics as many other entire cryptography texts and yet, at this stage, we are not even one fifth into the book. (Note: the entire text is available under a Creative Commons license on the author's Web site (http://www.shoup.net/). A Computational Introduction to Number Theory and Algebra March 24, 2006 A book introducing basic concepts from computational number theory and algebra, including all the necessary mathematical background. H. Cohen, A course in computational algebraic number theory, Springer-Verlag. It is not "friendly" or "chatty" as you will find with many number theory books targeted to undergraduates. DOI: 10.1017/CBO9780511814549.004 Corpus ID: 2574706. This should be skimmed over by the reader. They play an essential role in modern computer science, as evidenced by applications such as coding theory and cryptography. The prose is very lucid and easy to follow. The book is geared toward students of cryptography and coding theory, as it covers the most relevant material in those disciplines. It is highly recommended that the reader work these exercises, or at the very least, read and understand their statements. The presentation switches between theory and applications. The author could have included some thought-provoking exercises, programming challenges, open problems, and additional references, and emphasized the implementation of algorithms. This edition now includes over 150 new exercises, The second edition corrects some errors in the first edition and includes new examples. This introductory book emphasizes algorithms and applications, such as cryptography Algebra and number theory are important subdisciplines of mathematics. One of the chapters that was reduced is, surprisingly, the chapter on probabilistic primality testing. Zhang J, Yang Y, Chen Y, Chen J and Zhang Q, Benhamouda F, Herranz J, Joye M and Libert B, Shi W, Bao Z, Wang J, Lu N, Zhu F and Shen J, Dwivedi S Computing Modular Exponentiation for Fixed-Exponent Proceedings of the 8th Annual ACM India Conference, (89-94), Bannister M, Devanny W, Eppstein D and Goodrich M The Galois Complexity of Graph Drawing Revised Selected Papers of the 22nd International Symposium on Graph Drawing - Volume 8871, (149-161), Groß T Efficient Certification and Zero-Knowledge Proofs of Knowledge on Infrastructure Topology Graphs Proceedings of the 6th edition of the ACM Workshop on Cloud Computing Security, (69-80), Krenn S, Pietrzak K and Wadia A A counterexample to the chain rule for conditional HILL entropy Proceedings of the 10th theory of cryptography conference on Theory of Cryptography, (23-39), Fleming S and Thomas D Hardware acceleration of matrix multiplication over small prime finite fields Proceedings of the 9th international conference on Reconfigurable Computing: architectures, tools, and applications, (103-114), Barthe G, Grégoire B, Heraud S, Olmedo F and Zanella Béguelin S Verified indifferentiable hashing into elliptic curves Proceedings of the First international conference on Principles of Security and Trust, (209-228), Bright C and Storjohann A Vector rational number reconstruction Proceedings of the 36th international symposium on Symbolic and algebraic computation, (51-58), Agrawal S, Freeman D and Vaikuntanathan V Functional encryption for inner product predicates from learning with errors Proceedings of the 17th international conference on The Theory and Application of Cryptology and Information Security, (21-40), Lipton R, Regan K and Rudra A Symmetric functions capture general functions Proceedings of the 36th international conference on Mathematical foundations of computer science, (436-447), Brázdil T, Brožek V, Etessami K, Kučera A and Wojtczak D One-counter Markov decision processes Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete algorithms, (863-874), Agrawal S, Boneh D and Boyen X Efficient lattice (H)IBE in the standard model Proceedings of the 29th Annual international conference on Theory and Applications of Cryptographic Techniques, (553-572), Catrina O and De Hoogh S Improved primitives for secure multiparty integer computation Proceedings of the 7th international conference on Security and cryptography for networks, (182-199), Barthe G, Daubignard M, Kapron B, Lakhnech Y and Laporte V On the equality of probabilistic terms Proceedings of the 16th international conference on Logic for programming, artificial intelligence, and reasoning, (46-63), Feng Q, Liu Y, Lu S and Wang J Improved Deterministic Algorithms for Weighted Matching and Packing Problems Proceedings of the 6th Annual Conference on Theory and Applications of Models of Computation, (211-220), Pieters W and Tang Q Data Is Key Proceedings of the 23rd Annual IFIP WG 11.3 Working Conference on Data and Applications Security XXIII, (240-251). I would think that a book targeted at this level of mathematical sophistication would assume students are comfortable with (for example) the most basic notions of group theory or the idea of equivalence classes. Although this approach might seem at first to be unnecessarily obtuse, it is in fact the most natural way to introduce these algorithms, as it places them squarely in a generalized algebraic theory. Another chapter discusses the distribution of primes, including a proof of Bertrand's postulate and a discussion of the prime number theorem. Description. Asecond starting point was He is the author of a widely used textbook, "A Computational Introduction to Number Theory and Algebra". Most of the algorithms studied are quite "classical" (as much as that makes sense for computer science), with modern ideas and developments usually relegated to "Notes" at the end of the computational chapters. The ACM Digital Library is published by the Association for Computing Machinery. However, the mathematical maturity required to read and learn from this text is quite high. The content is very accurate ad up to date. Fingerprint Dive into the research topics of 'A computational introduction to number theory and Algebra'. Consequently, books that introduce the computational aspects of number theory and algebra will help novices appreciate such applications. A computational introduction to number theory and algebra @inproceedings{Shoup2005ACI, title={A computational introduction to number theory and algebra}, author={V. Shoup}, year={2005} } What would be helpful would be some suggested paths through the text for various purposes. It includes a total of 650 exercises and 280 worked examples. There are natural places of overlap (like Euclid's algorithm), and they are separated rather than treated more holistically. As promised by the title, the book gives a very nice overview of a side range of topics in It is clear and concise yet thorough. [Main Text] V. Shoup, A computational introduction to number theory and algebra, Cambridge University Press. A computational introduction to number theory and algebra @inproceedings{Shoup2005ACI, title={A computational introduction to number theory and algebra}, author={V. Shoup}, year={2005} } Essential exercises are underlined (a very nice feature!) Victor Shoup is a Professor in the Department of Computer Science at the Courant Institute of Mathematical Sciences, New York University. A Computational Introduction to Number Theory and Algebra (Version 2) by Victor Shoup. In solving exercises, the reader is free to use any previously stated results in the text, including those in previous exercises. There are many examples in the text, which form an integral part of the book, and should not be skipped. All of the mathematics required beyond basic calculus is developed “from scratch.” Moreover, the book generally alternates between “theory” and “applications”: one or two chapters on a particular set of purely mathematical concepts are followed by one or two chapters on algorithms and applications; the mathematics provides the theoretical underpinnings for the applications, while the applications both motivate and illustrate the mathematics. Victor Shoup. The last three or four decades have seen an interesting array of applications of algebra and number theory to computer science and related areas, from securing the interchange of information (public key cryptography) to error-correcting codes (widely used in the storage, retrieval and … Journalism, Media Studies & Communications, 8 Finite and discrete probability distributions, 11 Finding generators and discrete logarithms in Z∗p, 12 Quadratic reciprocity and computing modular square roots, 15 Subexponential-time discrete logarithms and factoring, 17 Polynomial arithmetic and applications, 19 Linearly generated sequences and applications. The Table of Contents indicates a few sections that are not required for future material. Algebra and number theory are important subdisciplines of mathematics. Cambridge Core - Algorithmics, Complexity, Computer Algebra, Computational Geometry - A Computational Introduction to Number Theory and Algebra - by Victor Shoup I see no signs of bias. The book's terminology and mathematical frameworks appear to be consistent. This is not relevant for a mathematics text, but I saw nothing that would be offensive to a reader of any ethnic background. I'll raise you the work to arrive at the solutions, as well. Reviewed by William McGovern, Professor, University of Washingon on 8/21/16, As promised by the title, the book gives a very nice overview of a side range of topics in number theory and algebra (primarily the former, but with quite a bit of attention to the latter as well), with special emphasis to the areas in which... Given the importance of primes to modern cryptography, these are vital topics, so it is refreshing to see them treated so carefully. The text includes an effective index. The book appears to be up-to-date, and includes some interesting applications of theoretical material to topics relevant in cryptography (e.g., the RSA cryptosystem, and primality testing). The book is exceedingly well written, though it is at a very high level. Topics whose exclusion from the book is notable include the theory of lattices, along with algorithms for and applications of lattice basis reduction. Number theory and algebra play an increasingly signiflcant role in computing ... course on computational number theory at NYU in the fall semester of 2003, read more. This introductory book is a revised second edition of a book that first appeared in 2005. There is an appendix that contains a few useful facts; where such a fact is used in the text, there is a reference such as “see §An,” which refers to the item labeled “An” in the appendix. There do not appear to be major grammatical errors in the text. However, except where otherwise noted, any result in a section marked with a “(∗),” or in §5.5, need not and should not be used outside the section in which it appears. The treatment of all these topics is more or less standard, except that the text only deals with commutative structures (i.e., abelian groups and commutative rings with unity) — this is all that is really needed for the purposes of this text, and the theory of these structures is much simpler and more transparent than that of more general, non-commutative structures. The number of citations to the literature has increased marginally, by just six-from 105 to 111; the author should have provided a more substantial number of references to add value to the book. Due to the topics in this text, this question does not appear to be applicable. This is a truly magnificent text, deserving of a place on the shelves of any mathematician or computer scientist working in these areas. The mathematical presentation is rigorous, clear, and well-explained. The material moves swiftly¿while never compromising rigor¿and the multiple strands assume considerable ability on the part of the reader. It can be used as a reference and for self-study by readers who want to learn the mathematical foundations of modern cryptography. I can't imagine an appropriate audience for this text: one with the ability to read and work entirely at this abstract level but without any (or most) of the mathematical preparation provided in at least half the chapters. A Computational Introduction to Number Theory and Algebra A Computational Introduction to Number Theory and Algebra, Victor Shoup: Auteur: Victor Shoup: Édition: illustrée, réimprimée: Éditeur: Cambridge University Press, 2005: ISBN: 0521851548, 9780521851541: Longueur: 517 pages : Exporter la citation: BiBTeX EndNote RefMan Nonetheless, as an ingenious use of much number theory and algebra, the AKS algorithm is a lovely example with which to finish the text. Find all the textbook answers and step-by-step explanations below Chapters. 英文数学书合集第一部分 Algebra 文件名: A Computational Introduction To Number Theory And Algebra - Victor Shoups.pdf 附件大小: 13.36 MB 有奖举报问题资料 下载通道游客无法下载， 注册 登 … But any faculty member who keeps up with the relevant research will be able to mention new developments to students, and it will not interrupt the flow of the ideas at all. The book jumps from chapters on purely algebraic topics to those focused on applications. However, much of the subsequent material is discussed in terms of the general theory of groups and rings. No jargon is used and terminology is carefully explained. The author does not recommend any specific chapter sequence for a semester's study, but an astute teacher could pull some parts from this text for an initial course of study, and then complete the text in an advanced course. Attribution-NonCommercial-NoDerivs Topics logically related, such as the chapters on probabilistic algorithms and probabilistic primality testing, have been brought together. As promised by the title, the book gives a very nice overview of a side range of topics in number theory and algebra (primarily the former, but with quite a bit of attention to the latter as well), with special emphasis to the areas in which computational techniques have proved useful. J. von zur Gathen and J. Gerhard, Modern computer algebra… A. Das, Computational Number Theory, CRC Press. Some sections are terse, and an instructor may want to supplement the theoretical exercises with some more computational ones. Author of "Algebraic Number Theory: A Computational Approach" here, in case anybody has any questions. BibTeX @MISC{Shoup04acomputational, author = {Victor Shoup}, title = {A COMPUTATIONAL INTRODUCTION TO NUMBER THEORY AND ALGEBRA}, year = {2004}} There were a few hyperlinks (from the table of contents to section headings, for example), but not much else in the way of interface. This, and other topics, are tools for interesting computational applications. Here's the history of that book. The text is so comprehensive that it feels overwhelming. The book (now in its second edition) is published by Cambridge University Press. The computational chapters use pseudocode, so they will not be quickly outdated when new languages become fashionable. From my research in writing this review, I have not come across any major errors. Online Computing Reviews Service. I found no mathematical errors. This is rather surprising, considering that lattice basis reduction plays a significant role in computational number theory, algebra, and cryptography. The book starts with several standard integer-based number theory concepts: divisibility; congruences and modular arithmetic, including quadratic residues (but reciprocity is treated later); large integer arithmetic; Euclid's algorithm and its association with the Chinese remainder theorem; and a brief discussion of the RSA cryptosystem, including a particularly elegant proof of its correctness. There are a number of exercises in the text that serve to reinforce, as well as to develop important applications and generalizations of, the material presented in the text. My main comment about the structure is that the mathematics chapters and the computational chapters seem to be separated. As I read, I often felt "now we are doing mathematics... now we are concerned with computational questions." Copyright © 2021 ACM, Inc. A Computational Introduction to Number Theory and Algebra, All Holdings within the ACM Digital Library. mathematics. All pages display very well on my screen, with no legibility or distortion issues that I could see. The book also includes an introduction to probability. A Computational Introduction to Number Theory and Algebra March 24, 2006 A book introducing basic concepts from computational number theory and algebra, including all the necessary mathematical background. The book starts with basic properties of integers (e.g., divisibility, unique factorization), and touches on topics in elementary number theory (e.g., arithmetic modulo n, the distribution of primes, discrete logarithms, primality testing, quadratic reciprocity) and abstract algebra (e.g., groups, rings, ideals, modules, fields and vector spaces, some linear algebra, polynomial rings and their quotients). Apart from rings and fields, there is much linear algebra (modules, vector spaces, and matrices), as well as a considerable amount of probability distributions and probabilistic algorithms, culminating in the Miller-Rabin test for primality and a few applications. However, even among the remaining sections, an instructor would need to carefully choose sections that include all necessary prerequisite material. DOI: 10.1017/CBO9780511814549.004 Corpus ID: 2574706. For this reason, an instructor may want to choose certain sections in a chapter to cover as prerequisites for an application, instead of covering the material linearly. The book does an excellent job of consistency of notation. The material has been reorganized to improve clarity. Whenever there is the potential for confusion (for example, in using "a mod b" as a binary operation as is common in computer science versus using "a is congruent to x mod b" as is more standard in mathematics) the author is careful to point out the dual meanings and to warn the reader that there is some overloading of terminology. The presentation A Computational Introduction to Number Theory and Algebra - April 2005 This book is a marvel. in the body of the text, and which further develop the theory and present new applications. If you follow me on Twitter, you've probably known that I've been into "A computational introduction to number theory and algebra" aka NTB for the last two or three months.IMHO, NTB is the best introductory-level book on number theory and algebra, especially for those who want to study these two mathematic subjects from a computer science and cryptography perspective. Together they form a unique fingerprint. The book is nicely broken up into manageable sections that would fit well into a lecture course. There are a few sections indicated that are not required for future material. The author is obviously a bit of an obsessive compulsive, he has found the shortest paths from the clearest definitions to the most important results, each given with the cleanest, most insight-inducing proofs ... the results (and definitions) he gives are the ones any student (practitioner!) It can be terse at times, skipping steps and making conceptual leaps that will be challenging for all but the very best students. For example, the chapter on "Congruences" covers a tremendous amount of number theory, not all of which falls naturally (in my mind) under that heading. We should also point out that mathematical induction is a prerequisite for this text, and some of the material is presented using pseudocode, which is different than many texts on these topics. This text is an introduction to number theory and abstract algebra; based on its presentation, it appears appropriate for students coming from computer science. There is a lot of interesting history and "cultural" notes in the computing chapters, and almost none in the more mathematical chapters. References are given to websites as well as books. The format of the book makes it especially easy to update as advances in the subjects occur, particularly computational advances. read more. Consequently, books that introduce the computational aspects of number theory and algebra will help novices appreciate such applications. The text is so comprehensive that it feels overwhelming. A Computational Introduction to Number Theory and Algebra (Version 2) Victor Shoup. There is a very good index and glossary and a good review of notation and basic facts in the first chapter. The book ends with some chapters on finite fields and their various algorithms and a chapter on the Agrawal-Kayal-Saxena (AKS) deterministic primality test; the author carefully observes that the probabilistic Miller-Rabin test is much faster than AKS and, hence, should be preferred for all practical purposes. The book covers both standard background that will always be relevant for these topics: the number theory and algebra background, the probability theory.

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