Book•
A Course in Computational Algebraic Number Theory
01 Jan 1993-
TL;DR: The first seven chapters guide readers to the heart of current research in computational algebraic number theory, including recent algorithms for computing class groups and units, as well as elliptic curve computations, while the last three chapters survey factoring and primality testing methods.
Abstract: A description of 148 algorithms fundamental to number-theoretic computations, in particular for computations related to algebraic number theory, elliptic curves, primality testing and factoring. The first seven chapters guide readers to the heart of current research in computational algebraic number theory, including recent algorithms for computing class groups and units, as well as elliptic curve computations, while the last three chapters survey factoring and primality testing methods, including a detailed description of the number field sieve algorithm. The whole is rounded off with a description of available computer packages and some useful tables, backed by numerous exercises. Written by an authority in the field, and one with great practical and teaching experience, this is certain to become the standard and indispensable reference on the subject.
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Book•
01 Jan 1996TL;DR: A valuable reference for the novice as well as for the expert who needs a wider scope of coverage within the area of cryptography, this book provides easy and rapid access of information and includes more than 200 algorithms and protocols.
Abstract: From the Publisher:
A valuable reference for the novice as well as for the expert who needs a wider scope of coverage within the area of cryptography, this book provides easy and rapid access of information and includes more than 200 algorithms and protocols; more than 200 tables and figures; more than 1,000 numbered definitions, facts, examples, notes, and remarks; and over 1,250 significant references, including brief comments on each paper.
13,597 citations
Cites background or methods from "A Course in Computational Algebraic..."
...Cohen [263] provides a more comprehensive treatment of the rightto-left and left-to-right (Algorithm 14....
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...See also Chapters 8 and 10 of Cohen [263], and the books by Bressoud [198] and Koblitz [697]....
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...Two more recent books exclusively devoted to this subject are Bach and Shallit [70] and Cohen [263]....
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...Cohen [263] gives a comprehensive treatment of modern primality tests....
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...Cohen [263] provides a similar discussion, but without motivation....
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Book•
01 Jan 2004
TL;DR: This guide explains the basic mathematics, describes state-of-the-art implementation methods, and presents standardized protocols for public-key encryption, digital signatures, and key establishment, as well as side-channel attacks and countermeasures.
Abstract: After two decades of research and development, elliptic curve cryptography now has widespread exposure and acceptance. Industry, banking, and government standards are in place to facilitate extensive deployment of this efficient public-key mechanism. Anchored by a comprehensive treatment of the practical aspects of elliptic curve cryptography (ECC), this guide explains the basic mathematics, describes state-of-the-art implementation methods, and presents standardized protocols for public-key encryption, digital signatures, and key establishment. In addition, the book addresses some issues that arise in software and hardware implementation, as well as side-channel attacks and countermeasures. Readers receive the theoretical fundamentals as an underpinning for a wealth of practical and accessible knowledge about efficient application. Features & Benefits: * Breadth of coverage and unified, integrated approach to elliptic curve cryptosystems * Describes important industry and government protocols, such as the FIPS 186-2 standard from the U.S. National Institute for Standards and Technology * Provides full exposition on techniques for efficiently implementing finite-field and elliptic curve arithmetic* Distills complex mathematics and algorithms for easy understanding* Includes useful literature references, a list of algorithms, and appendices on sample parameters, ECC standards, and software toolsThis comprehensive, highly focused reference is a useful and indispensable resource for practitioners, professionals, or researchers in computer science, computer engineering, network design, and network data security.
2,893 citations
Cites background from "A Course in Computational Algebraic..."
...Cohen [99] discusses right-to-left and left-to-right algorithms with base 2k ....
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Book•
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24 Oct 2001
TL;DR: Digital Watermarking covers the crucial research findings in the field and explains the principles underlying digital watermarking technologies, describes the requirements that have given rise to them, and discusses the diverse ends to which these technologies are being applied.
Abstract: Digital watermarking is a key ingredient to copyright protection. It provides a solution to illegal copying of digital material and has many other useful applications such as broadcast monitoring and the recording of electronic transactions. Now, for the first time, there is a book that focuses exclusively on this exciting technology. Digital Watermarking covers the crucial research findings in the field: it explains the principles underlying digital watermarking technologies, describes the requirements that have given rise to them, and discusses the diverse ends to which these technologies are being applied. As a result, additional groundwork is laid for future developments in this field, helping the reader understand and anticipate new approaches and applications.
2,849 citations
TL;DR: An efficient closest point search algorithm, based on the Schnorr-Euchner (1995) variation of the Pohst (1981) method, is implemented and is shown to be substantially faster than other known methods.
Abstract: In this semitutorial paper, a comprehensive survey of closest point search methods for lattices without a regular structure is presented. The existing search strategies are described in a unified framework, and differences between them are elucidated. An efficient closest point search algorithm, based on the Schnorr-Euchner (1995) variation of the Pohst (1981) method, is implemented. Given an arbitrary point x /spl isin/ /spl Ropf//sup m/ and a generator matrix for a lattice /spl Lambda/, the algorithm computes the point of /spl Lambda/ that is closest to x. The algorithm is shown to be substantially faster than other known methods, by means of a theoretical comparison with the Kannan (1983, 1987) algorithm and an experimental comparison with the Pohst (1981) algorithm and its variants, such as the Viterbo-Boutros (see ibid. vol.45, p.1639-42, 1999) decoder. Modifications of the algorithm are developed to solve a number of related search problems for lattices, such as finding a shortest vector, determining the kissing number, computing the Voronoi (1908)-relevant vectors, and finding a Korkine-Zolotareff (1873) reduced basis.
1,616 citations
TL;DR: A novel public key cryptosystem in which the public key of a subscriber can be chosen to be a publicly known value, such as his identity, which is related to the difficulty of solving the quadratic residuosity problem.
Abstract: We present a novel public key cryptosystem in which the public key of a subscriber can be chosen to be a publicly known value, such as his identity. We discuss the security of the proposed scheme, and show that this is related to the difficulty of solving the quadratic residuosity problem.
1,228 citations
References
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Book•
01 Jan 1902
TL;DR: The volume now gives a somewhat exhaustive account of the various ramifications of the subject, which are set out in an attractive manner and should become indispensable, not only as a textbook for advanced students, but as a work of reference to those whose aim is to extend the knowledge of analysis.
Abstract: This classic work has been a unique resource for thousands of mathematicians, scientists and engineers since its first appearance in 1902 Never out of print, its continuing value lies in its thorough and exhaustive treatment of special functions of mathematical physics and the analysis of differential equations from which they emerge The book also is of historical value as it was the first book in English to introduce the then modern methods of complex analysis This fifth edition preserves the style and content of the original, but it has been supplemented with more recent results and references where appropriate All the formulas have been checked and many corrections made A complete bibliographical search has been conducted to present the references in modern form for ease of use A new foreword by Professor SJ Patterson sketches the circumstances of the book's genesis and explains the reasons for its longevity A welcome addition to any mathematician's bookshelf, this will allow a whole new generation to experience the beauty contained in this text
8,965 citations
TL;DR: A method for multiplying two integers modulo N while avoiding division by N, a representation of residue classes so as to speed modular multiplication without affecting the modular addition and subtraction algorithms.
Abstract: Let N > 1. We present a method for multiplying two integers (called N-residues) modulo N while avoiding division by N. N-residues are represented in a nonstandard way, so this method is useful only if several computations are done modulo one N. The addition and subtraction algorithms are unchanged. 1. Description. Some algorithms (1), (2), (4), (5) require extensive modular arith- metic. We propose a representation of residue classes so as to speed modular multiplication without affecting the modular addition and subtraction algorithms. Other recent algorithms for modular arithmetic appear in (3), (6). Fix N > 1. Define an A'-residue to be a residue class modulo N. Select a radix R coprime to N (possibly the machine word size or a power thereof) such that R > N and such that computations modulo R are inexpensive to process. Let R~l and N' be integers satisfying 0 N then return t - N else return t ■ To validate REDC, observe mN = TN'N = -Tmod R, so t is an integer. Also, tR = Tmod N so t = TR'X mod N. Thirdly, 0 < T + mN < RN + RN, so 0 < t < 2N. If R and N are large, then T + mN may exceed the largest double-precision value. One can circumvent this by adjusting m so -R < m < 0. Given two numbers x and y between 0 and N - 1 inclusive, let z = REDC(xy). Then z = (xy)R~x mod N, so (xR-l)(yR~x) = zRx mod N. Also, 0 < z < N, so z is the product of x and y in this representation. Other algorithms for operating on N-residues in this representation can be derived from the algorithms normally used. The addition algorithm is unchanged, since xR~x + yR~x = zR~x mod N if and only if x + y = z mod N. Also unchanged are
2,647 citations
Book•
01 Jan 1994
TL;DR: In this article, the authors continue the study of elliptic curves by presenting six important, but somewhat more specialized topics: Elliptic and modular functions for the full modular group.
Abstract: In The Arithmetic of Elliptic Curves, the author presented the basic theory culminating in two fundamental global results, the Mordell-Weil theorem on the finite generation of the group of rational points and Siegel's theorem on the finiteness of the set of integral points. This book continues the study of elliptic curves by presenting six important, but somewhat more specialized topics: I. Elliptic and modular functions for the full modular group. II. Elliptic curves with complex multiplication. III. Elliptic surfaces and specialization theorems. IV. Neron models, Kodaira-N ron classification of special fibres, Tate's algorithm, and Ogg's conductor-discriminant formula. V. Tate's theory of q-curves over p-adic fields. VI. Neron's theory of canonical local height functions.
1,853 citations
TL;DR: In this paper, the authors show that searching through an ellipsoid is in many cases much more efficient than enumerating all vectors of Z'.. in a suitable box.
Abstract: The standard methods for calculating vectors of short length in a lattice use a reduction procedure followed by enumerating all vectors of Z'.. in a suitable box. However, it suffices to consider those x E Z'" which lie in a suitable ellipsoid having a much smaller volume than the box. We show in this paper that searching through that ellipsoid is in many cases much more efficient. If combined with an appropriate reduction procedure our method allows to do computations in lattices of much higher dimensions. Several randomly constructed numerical examples illustrate the superiority of our new method over the known ones.
1,538 citations
TL;DR: Empirical tests show that the strongest of these algorithms solves almost all subset sum problems with up to 66 random weights of arbitrary bit length within at most a few hours on a UNISYS 6000/70 or within a couple of minutes on a SPARC1 + computer.
Abstract: We report on improved practical algorithms for lattice basis reduction. We present a variant of the L3-algorithm with “deep insertions” and a practical algorithm for blockwise Korkine-Zolotarev reduction, a concept extending L3-reduction, that has been introduced by Schnorr (1987). Empirical tests show that the strongest of these algorithms solves almost all subset sum problems with up to 58 random weights of arbitrary bit length within at most a few hours on a UNISYS 6000/70 or within a couple of minutes on a SPARC 2 computer.
1,390 citations