Primality Testing And Abelian Varieties Over Finite Fields

Author : Leonard M. Adleman,Ming-Deh A. Huang
Publisher : Springer
Page : 149 pages
File Size : 45,6 Mb
Release : 2006-11-15
Category : Mathematics
ISBN : 9783540470212

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Primality Testing and Abelian Varieties Over Finite Fields by Leonard M. Adleman,Ming-Deh A. Huang Pdf

From Gauss to G|del, mathematicians have sought an efficient algorithm to distinguish prime numbers from composite numbers. This book presents a random polynomial time algorithm for the problem. The methods used are from arithmetic algebraic geometry, algebraic number theory and analyticnumber theory. In particular, the theory of two dimensional Abelian varieties over finite fields is developed. The book will be of interest to both researchers and graduate students in number theory and theoretical computer science.

Author : Leonard M. Adleman,Ming-Deh A. Huang
Publisher : Unknown
Page : 152 pages
File Size : 47,7 Mb
Release : 2014-01-15
Category : Electronic
ISBN : 3662170590

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Primality Testing and Abelian Varieties Over Finite Fields by Leonard M. Adleman,Ming-Deh A. Huang Pdf

Author : Kristin Estella Lauter
Publisher : American Mathematical Soc.
Page : 146 pages
File Size : 46,5 Mb
Release : 2008
Category : Algebraic number theory
ISBN : 9780821843208

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Computational Arithmetic Geometry by Kristin Estella Lauter Pdf

With the recent increase in available computing power, new computations are possible in many areas of arithmetic geometry. To name just a few examples, Cremona's tables of elliptic curves now go up to conductor 120,000 instead of just conductor 1,000, tables of Hilbert class fields are known for discriminant up to at least 5,000, and special values of Hilbert and Siegel modular forms can be calculated to extremely high precision. In many cases, these experimental capabilities haveled to new observations and ideas for progress in the field. They have also led to natural algorithmic questions on the feasibility and efficiency of many computations, especially for the purpose of applications in cryptography. The AMS Special Session on Computational Arithmetic Geometry, held onApril 29-30, 2006, in San Francisco, CA, gathered together many of the people currently working on the computational and algorithmic aspects of arithmetic geometry. This volume contains research articles related to talks given at the session. The majority of articles are devoted to various aspects of arithmetic geometry, mainly with a computational approach.

Author : Igor Shparlinski
Publisher : Springer Science & Business Media
Page : 532 pages
File Size : 40,9 Mb
Release : 2013-03-09
Category : Mathematics
ISBN : 9789401592390

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Finite Fields: Theory and Computation by Igor Shparlinski Pdf

This book is mainly devoted to some computational and algorithmic problems in finite fields such as, for example, polynomial factorization, finding irreducible and primitive polynomials, the distribution of these primitive polynomials and of primitive points on elliptic curves, constructing bases of various types and new applications of finite fields to other areas of mathematics. For completeness we in clude two special chapters on some recent advances and applications of the theory of congruences (optimal coefficients, congruential pseudo-random number gener ators, modular arithmetic, etc.) and computational number theory (primality testing, factoring integers, computation in algebraic number theory, etc.). The problems considered here have many applications in Computer Science, Cod ing Theory, Cryptography, Numerical Methods, and so on. There are a few books devoted to more general questions, but the results contained in this book have not till now been collected under one cover. In the present work the author has attempted to point out new links among different areas of the theory of finite fields. It contains many very important results which previously could be found only in widely scattered and hardly available conference proceedings and journals. In particular, we extensively review results which originally appeared only in Russian, and are not well known to mathematicians outside the former USSR.

Author : Henri Cohen,Gerhard Frey,Roberto Avanzi,Christophe Doche,Tanja Lange,Kim Nguyen,Frederik Vercauteren
Publisher : CRC Press
Page : 843 pages
File Size : 45,9 Mb
Release : 2005-07-19
Category : Mathematics
ISBN : 9781420034981

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Handbook of Elliptic and Hyperelliptic Curve Cryptography by Henri Cohen,Gerhard Frey,Roberto Avanzi,Christophe Doche,Tanja Lange,Kim Nguyen,Frederik Vercauteren Pdf

The discrete logarithm problem based on elliptic and hyperelliptic curves has gained a lot of popularity as a cryptographic primitive. The main reason is that no subexponential algorithm for computing discrete logarithms on small genus curves is currently available, except in very special cases. Therefore curve-based cryptosystems require much smaller key sizes than RSA to attain the same security level. This makes them particularly attractive for implementations on memory-restricted devices like smart cards and in high-security applications. The Handbook of Elliptic and Hyperelliptic Curve Cryptography introduces the theory and algorithms involved in curve-based cryptography. After a very detailed exposition of the mathematical background, it provides ready-to-implement algorithms for the group operations and computation of pairings. It explores methods for point counting and constructing curves with the complex multiplication method and provides the algorithms in an explicit manner. It also surveys generic methods to compute discrete logarithms and details index calculus methods for hyperelliptic curves. For some special curves the discrete logarithm problem can be transferred to an easier one; the consequences are explained and suggestions for good choices are given. The authors present applications to protocols for discrete-logarithm-based systems (including bilinear structures) and explain the use of elliptic and hyperelliptic curves in factorization and primality proving. Two chapters explore their design and efficient implementations in smart cards. Practical and theoretical aspects of side-channel attacks and countermeasures and a chapter devoted to (pseudo-)random number generation round off the exposition. The broad coverage of all- important areas makes this book a complete handbook of elliptic and hyperelliptic curve cryptography and an invaluable reference to anyone interested in this exciting field.

Author : Michael Walker
Publisher : Springer Science & Business Media
Page : 323 pages
File Size : 49,9 Mb
Release : 1999-12-08
Category : Computers
ISBN : 9783540668879

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Cryptography and Coding by Michael Walker Pdf

This book constitutes the refereed proceedings of the 7th IMA Conference on Cryptography and Coding held in Cirencester, UK, in December 1999. The 35 revised full papers presented were carefully reviewed and selected for inclusion in the proceedings. Among the topics covered are error-correcting coding, arithmetic coding for data compression and encryption, image coding, biometric authentication, broadcast channel access, graph and trellis decoding, turbo codes, convolution codes, Reed Solomon codes, elliptic curve cryptography, primality testing, finite-field arithmetic, and cryptographic protocols.

Author : Ian F. Blake,G. Seroussi,N. Smart
Publisher : Cambridge University Press
Page : 228 pages
File Size : 43,5 Mb
Release : 1999-07-08
Category : Computers
ISBN : 0521653746

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Elliptic Curves in Cryptography by Ian F. Blake,G. Seroussi,N. Smart Pdf

This book summarizes knowledge built up within Hewlett-Packard over a number of years, and explains the mathematics behind practical implementations of elliptic curve systems. Due to the advanced nature of the mathematics there is a high barrier to entry for individuals and companies to this technology. Hence this book will be invaluable not only to mathematicians wanting to see how pure mathematics can be applied but also to engineers and computer scientists wishing (or needing) to actually implement such systems.

Author : Martin Dietzfelbinger
Publisher : Springer Science & Business Media
Page : 153 pages
File Size : 48,9 Mb
Release : 2004-06-29
Category : Mathematics
ISBN : 9783540403449

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Primality Testing in Polynomial Time by Martin Dietzfelbinger Pdf

A self-contained treatment of theoretically and practically important efficient algorithms for the primality problem. The text covers the randomized algorithms by Solovay-Strassen and Miller-Rabin from the late 1970s as well as the recent deterministic algorithm of Agrawal, Kayal and Saxena. The volume is written for students of computer science, in particular those with a special interest in cryptology, and students of mathematics, and it may be used as a supplement for courses or for self-study.

Author : Henri Cohen
Publisher : Springer Science & Business Media
Page : 422 pages
File Size : 41,8 Mb
Release : 1996-08-07
Category : Computers
ISBN : 3540615814

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Algorithmic Number Theory by Henri Cohen Pdf

This book constitutes the refereed post-conference proceedings of the Second International Algorithmic Number Theory Symposium, ANTS-II, held in Talence, France in May 1996. The 35 revised full papers included in the book were selected from a variety of submissions. They cover a broad spectrum of topics and report state-of-the-art research results in computational number theory and complexity theory. Among the issues addressed are number fields computation, Abelian varieties, factoring algorithms, finite fields, elliptic curves, algorithm complexity, lattice theory, and coding.

Author : Song Y. Yan
Publisher : John Wiley & Sons
Page : 432 pages
File Size : 44,9 Mb
Release : 2013-01-29
Category : Computers
ISBN : 9781118188583

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Computational Number Theory and Modern Cryptography by Song Y. Yan Pdf

The only book to provide a unified view of the interplay between computational number theory and cryptography Computational number theory and modern cryptography are two of the most important and fundamental research fields in information security. In this book, Song Y. Yang combines knowledge of these two critical fields, providing a unified view of the relationships between computational number theory and cryptography. The author takes an innovative approach, presenting mathematical ideas first, thereupon treating cryptography as an immediate application of the mathematical concepts. The book also presents topics from number theory, which are relevant for applications in public-key cryptography, as well as modern topics, such as coding and lattice based cryptography for post-quantum cryptography. The author further covers the current research and applications for common cryptographic algorithms, describing the mathematical problems behind these applications in a manner accessible to computer scientists and engineers. Makes mathematical problems accessible to computer scientists and engineers by showing their immediate application Presents topics from number theory relevant for public-key cryptography applications Covers modern topics such as coding and lattice based cryptography for post-quantum cryptography Starts with the basics, then goes into applications and areas of active research Geared at a global audience; classroom tested in North America, Europe, and Asia Incudes exercises in every chapter Instructor resources available on the book’s Companion Website Computational Number Theory and Modern Cryptography is ideal for graduate and advanced undergraduate students in computer science, communications engineering, cryptography and mathematics. Computer scientists, practicing cryptographers, and other professionals involved in various security schemes will also find this book to be a helpful reference.

Author : Neal Koblitz
Publisher : Springer Science & Business Media
Page : 214 pages
File Size : 43,5 Mb
Release : 2012-12-06
Category : Computers
ISBN : 9783662036426

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Algebraic Aspects of Cryptography by Neal Koblitz Pdf

From the reviews: "This is a textbook in cryptography with emphasis on algebraic methods. It is supported by many exercises (with answers) making it appropriate for a course in mathematics or computer science. [...] Overall, this is an excellent expository text, and will be very useful to both the student and researcher." Mathematical Reviews

Author : Song Y. Yan
Publisher : Springer Science & Business Media
Page : 249 pages
File Size : 45,7 Mb
Release : 2013-06-29
Category : Computers
ISBN : 9781475738162

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Primality Testing and Integer Factorization in Public-Key Cryptography by Song Y. Yan Pdf

Primality Testing and Integer Factorization in Public-Key Cryptography introduces various algorithms for primality testing and integer factorization, with their applications in public-key cryptography and information security. More specifically, this book explores basic concepts and results in number theory in Chapter 1. Chapter 2 discusses various algorithms for primality testing and prime number generation, with an emphasis on the Miller-Rabin probabilistic test, the Goldwasser-Kilian and Atkin-Morain elliptic curve tests, and the Agrawal-Kayal-Saxena deterministic test for primality. Chapter 3 introduces various algorithms, particularly the Elliptic Curve Method (ECM), the Quadratic Sieve (QS) and the Number Field Sieve (NFS) for integer factorization. This chapter also discusses some other computational problems that are related to factoring, such as the square root problem, the discrete logarithm problem and the quadratic residuosity problem.

Author : Michiel Hazewinkel
Publisher : Springer Science & Business Media
Page : 595 pages
File Size : 54,8 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9789401512886

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Encyclopaedia of Mathematics by Michiel Hazewinkel Pdf

This is the first Supplementary volume to Kluwer's highly acclaimed Encyclopaedia of Mathematics. This additional volume contains nearly 600 new entries written by experts and covers developments and topics not included in the already published 10-volume set. These entries have been arranged alphabetically throughout. A detailed index is included in the book. This Supplementary volume enhances the existing 10-volume set. Together, these eleven volumes represent the most authoritative, comprehensive up-to-date Encyclopaedia of Mathematics available.

Author : N.E. Hurt
Publisher : Springer Science & Business Media
Page : 368 pages
File Size : 46,5 Mb
Release : 2013-11-11
Category : Mathematics
ISBN : 9789401702515

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Many Rational Points by N.E. Hurt Pdf

This volume provides a source book of examples with relationships to advanced topics regarding Sato-Tate conjectures, Eichler-Selberg trace formula, Katz-Sarnak conjectures and Hecke operators." "The book will be of use to mathematicians, physicists and engineers interested in the mathematical methods of algebraic geometry as they apply to coding theory and cryptography."--Jacket

Author : S.D. Chatterji
Publisher : Birkhäuser
Page : 1669 pages
File Size : 41,7 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9783034890786

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Proceedings of the International Congress of Mathematicians by S.D. Chatterji Pdf

Since the first ICM was held in Zürich in 1897, it has become the pinnacle of mathematical gatherings. It aims at giving an overview of the current state of different branches of mathematics and its applications as well as an insight into the treatment of special problems of exceptional importance. The proceedings of the ICMs have provided a rich chronology of mathematical development in all its branches and a unique documentation of contemporary research. They form an indispensable part of every mathematical library. The Proceedings of the International Congress of Mathematicians 1994, held in Zürich from August 3rd to 11th, 1994, are published in two volumes. Volume I contains an account of the organization of the Congress, the list of ordinary members, the reports on the work of the Fields Medalists and the Nevanlinna Prize Winner, the plenary one-hour addresses, and the invited addresses presented at Section Meetings 1 - 6. Volume II contains the invited address for Section Meetings 7 - 19. A complete author index is included in both volumes. '...the content of these impressive two volumes sheds a certain light on the present state of mathematical sciences and anybody doing research in mathematics should look carefully at these Proceedings. For young people beginning research, this is even more important, so these are a must for any serious mathematics library. The graphical presentation is, as always with Birkhäuser, excellent....' (Revue Roumaine de Mathematiques pures et Appliquées)