Primality Testing in Polynomial Time

Primality Testing in Polynomial Time PDF

Author: Martin Dietzfelbinger

Publisher: Springer Science & Business Media

Published: 2004-06-29

Total Pages: 153

ISBN-13: 3540403442

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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.

Primality Testing in Polynomial Time: from Randomized Algorithms to Primes Is in P

Primality Testing in Polynomial Time: from Randomized Algorithms to Primes Is in P PDF

Author: M. Dietzfelbinger

Publisher:

Published: 2004

Total Pages: 147

ISBN-13: 9781280307959

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This book is devoted to algorithms for the venerable primality problem: Given a natural number n, decide whether it is prime or composite. The problem is basic in number theory, efficient algorithms that solve it, i.e., algorithms that run in a number of computational steps which is polynomial in the number of digits needed to write n, are important for theoretical computer science and for applications in algorithmics and cryptology. This book gives a self-contained account of theoretically and practically important efficient algorithms for the primality problem, covering 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 textbook is written for students of computer science, in particular for 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.

Primality Testing for Beginners

Primality Testing for Beginners PDF

Author: Lasse Rempe-Gillen

Publisher: American Mathematical Soc.

Published: 2013-12-11

Total Pages: 258

ISBN-13: 0821898833

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How can you tell whether a number is prime? What if the number has hundreds or thousands of digits? This question may seem abstract or irrelevant, but in fact, primality tests are performed every time we make a secure online transaction. In 2002, Agrawal, Kayal, and Saxena answered a long-standing open question in this context by presenting a deterministic test (the AKS algorithm) with polynomial running time that checks whether a number is prime or not. What is more, their methods are essentially elementary, providing us with a unique opportunity to give a complete explanation of a current mathematical breakthrough to a wide audience. Rempe-Gillen and Waldecker introduce the aspects of number theory, algorithm theory, and cryptography that are relevant for the AKS algorithm and explain in detail why and how this test works. This book is specifically designed to make the reader familiar with the background that is necessary to appreciate the AKS algorithm and begins at a level that is suitable for secondary school students, teachers, and interested amateurs. Throughout the book, the reader becomes involved in the topic by means of numerous exercises.

Primality Testing and Integer Factorization in Public-Key Cryptography

Primality Testing and Integer Factorization in Public-Key Cryptography PDF

Author: Song Y. Yan

Publisher: Springer Science & Business Media

Published: 2013-06-29

Total Pages: 249

ISBN-13: 1475738161

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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.

FST TCS 2002: Foundations of Software Technology and Theoretical Computer Science

FST TCS 2002: Foundations of Software Technology and Theoretical Computer Science PDF

Author: Manindra Agrawal

Publisher: Springer Science & Business Media

Published: 2002-11-29

Total Pages: 372

ISBN-13: 3540002251

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This book constitutes the refereed proceedings of the 22nd Conference on Foundations of Software Technology and Theoretical Computer Science, FST TCS 2002, held in Kanpur, India in December 2002. The 26 revised full papers presented together with 5 invited contributions were carefully reviewed and selected from 108 submissions. A broad variety of topics from the theory of computing are addressed, from algorithmics and discrete mathematics as well as from logics and programming theory.

Primality Testing and Abelian Varieties Over Finite Fields

Primality Testing and Abelian Varieties Over Finite Fields PDF

Author: Leonard M. Adleman

Publisher: Springer

Published: 2006-11-15

Total Pages: 149

ISBN-13: 3540470212

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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.

An Exposition of the Deterministic Polynomial-time Primality Testing Algorithm of Agrawal-Kayal-Saxena

An Exposition of the Deterministic Polynomial-time Primality Testing Algorithm of Agrawal-Kayal-Saxena PDF

Author: Robert Lawrence Anderson

Publisher:

Published: 2005

Total Pages: 80

ISBN-13:

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I present a thorough examination of the unconditional deterministic polynomial-time algorithm for determining whether an input number is prime or composite proposed by Agrawal, Kayal and Saxena in their paper [1]. All proofs cited have been reworked with full details for the sake of completeness and readability.

A Course in Computational Algebraic Number Theory

A Course in Computational Algebraic Number Theory PDF

Author: Henri Cohen

Publisher: Springer Science & Business Media

Published: 2013-04-17

Total Pages: 556

ISBN-13: 3662029456

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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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