Primality Testing for Beginners
Primality Testing for Beginners

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

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.

Primality Testing in Polynomial Time
Primality Testing in Polynomial Time

Author: Martin Dietzfelbinger

Publisher: Springer

Published: 2004-08-17

Total Pages: 153

ISBN-13: 3540259333

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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 and Abelian Varieties Over Finite Fields
Primality Testing and Abelian Varieties Over Finite Fields

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.

A Course in Computational Algebraic Number Theory
A Course in Computational Algebraic Number Theory

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.

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

Author: Manindra Agrawal

Publisher: Springer

Published: 2014-03-12

Total Pages: 368

ISBN-13: 9783662214039

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This volume consists of the proceedings of the 22nd International Conference on the Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2002), organized under the auspices of the Indian Association for Research in Computing Science (IARCS). The conference was held at the Indian Institute of Technology, Kanpur during December 12–14, 2002. The conference attracted 108 submissions (of which two were withdrawn). Of these, a total of 26 papers were selected for presentation in the conference. As in the last year, the PC meeting was held electronically (stretching over nearly three weeks in August 2002) and was a great success. In addition to the contributed papers, we had ?ve invited speakers this year: Hendrik Lenstra, Jr., Harry Mairson, Dale Miller, Chih-Hao Luke Ong, and Margus Veanes. We thank them for accepting our invitation and for providing abstracts (or even full papers) for the proceedings. Two workshops were organized in conjunction with the conference – both in Kanpur. A workshop on Parameterized Complexity was held during December 10–11, organized by Mike Fellows and Venkatesh Raman. The second workshop actually consisted of three miniworkshops: on Coding Theory by Madhu Sudan; on Finite Field Algorithms by Hendrik Lenstra, Jr.; and on Sieve Theory by R. Balasubramanian. We wish to thank all the reviewers and PC members who contributed greatly to making the conference a success. We also wish to thank the team at Springer- Verlag for their help in preparing the proceedings.

An Illustrated Theory of Numbers
An Illustrated Theory of Numbers

Author: Martin H. Weissman

Publisher: American Mathematical Soc.

Published: 2020-09-15

Total Pages: 341

ISBN-13: 1470463717

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News about this title: — Author Marty Weissman has been awarded a Guggenheim Fellowship for 2020. (Learn more here.) — Selected as a 2018 CHOICE Outstanding Academic Title — 2018 PROSE Awards Honorable Mention An Illustrated Theory of Numbers gives a comprehensive introduction to number theory, with complete proofs, worked examples, and exercises. Its exposition reflects the most recent scholarship in mathematics and its history. Almost 500 sharp illustrations accompany elegant proofs, from prime decomposition through quadratic reciprocity. Geometric and dynamical arguments provide new insights, and allow for a rigorous approach with less algebraic manipulation. The final chapters contain an extended treatment of binary quadratic forms, using Conway's topograph to solve quadratic Diophantine equations (e.g., Pell's equation) and to study reduction and the finiteness of class numbers. Data visualizations introduce the reader to open questions and cutting-edge results in analytic number theory such as the Riemann hypothesis, boundedness of prime gaps, and the class number 1 problem. Accompanying each chapter, historical notes curate primary sources and secondary scholarship to trace the development of number theory within and outside the Western tradition. Requiring only high school algebra and geometry, this text is recommended for a first course in elementary number theory. It is also suitable for mathematicians seeking a fresh perspective on an ancient subject.

Factorization and Primality Testing
Factorization and Primality Testing

Author: David M. Bressoud

Publisher: Springer Science & Business Media

Published: 2012-12-06

Total Pages: 252

ISBN-13: 1461245443

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"About binomial theorems I'm teeming with a lot of news, With many cheerful facts about the square on the hypotenuse. " - William S. Gilbert (The Pirates of Penzance, Act I) The question of divisibility is arguably the oldest problem in mathematics. Ancient peoples observed the cycles of nature: the day, the lunar month, and the year, and assumed that each divided evenly into the next. Civilizations as separate as the Egyptians of ten thousand years ago and the Central American Mayans adopted a month of thirty days and a year of twelve months. Even when the inaccuracy of a 360-day year became apparent, they preferred to retain it and add five intercalary days. The number 360 retains its psychological appeal today because it is divisible by many small integers. The technical term for such a number reflects this appeal. It is called a "smooth" number. At the other extreme are those integers with no smaller divisors other than 1, integers which might be called the indivisibles. The mystic qualities of numbers such as 7 and 13 derive in no small part from the fact that they are indivisibles. The ancient Greeks realized that every integer could be written uniquely as a product of indivisibles larger than 1, what we appropriately call prime numbers. To know the decomposition of an integer into a product of primes is to have a complete description of all of its divisors.

A Beginner's Guide to Algorithms: For Programming
A Beginner's Guide to Algorithms: For Programming

Author: Karl Beeston

Publisher: Richards Education

Published:

Total Pages: 102

ISBN-13:

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Unlock the secrets of algorithmic thinking and revolutionize your programming skills with A Beginner's Guide to Algorithms: For Programming. This comprehensive and accessible guide is designed for aspiring programmers and computer science enthusiasts who are eager to delve into the world of algorithms. Embark on a journey through the essential concepts of algorithm development, starting from the basics and progressing to advanced topics. Each chapter offers clear explanations, practical examples, and step-by-step instructions to help you master fundamental data structures, sorting and searching techniques, dynamic programming, graph theory, and much more. Discover how to: Understand and apply different types of algorithms Choose the right data structure for your specific problem Implement and optimize sorting and searching algorithms Harness the power of recursion and dynamic programming Solve complex problems using graph and greedy algorithms Explore advanced topics like computational geometry and quantum algorithms With detailed case studies and practical applications, you'll see how algorithms play a crucial role in fields such as machine learning, cryptography, bioinformatics, and game development. Whether you're a student, a self-taught programmer, or a seasoned developer looking to refresh your knowledge, this book provides the tools and insights you need to excel in the ever-evolving landscape of programming. Join the ranks of proficient programmers who can tackle any challenge with confidence. Dive into A Beginner's Guide to Algorithms: For Programming and take the first step towards becoming an algorithmic thinker today.