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Cryptology and Error Correction

An Algebraic Introduction and Real-World Applications

  • Textbook
  • © 2019

Overview

Authors:
  1. Lindsay N. Childs

    1. Department of Mathematics and Statistics, University at Albany, State University of New York, Albany, USA

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  • Exercises in each chapter are real-world application based
  • Provides solid mathematical preparation for more specialized applied courses on cryptography/error correction
  • Presents some of the remarkable strategies for dealing with information in the computer age and the basic algebraic ideas behind those strategies
  • Solutions manual is available to instructors who adopt the text for their course
  • Request lecturer material: sn.pub/lecturer-material

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Table of contents (19 chapters)

  1. Front Matter

    Pages i-xiv
    Download chapter PDF

  2. Secure, Reliable Information

    • Lindsay N. Childs
    Pages 1-11

  3. Modular Arithmetic

    • Lindsay N. Childs
    Pages 13-26

  4. Linear Equations Modulo m

    • Lindsay N. Childs
    Pages 27-49

  5. Unique Factorization in \(\mathbb {Z}\)

    • Lindsay N. Childs
    Pages 51-64

  6. Rings and Fields

    • Lindsay N. Childs
    Pages 65-82

  7. Polynomials

    • Lindsay N. Childs
    Pages 83-91

  8. Matrices and Hamming Codes

    • Lindsay N. Childs
    Pages 93-115

  9. Orders and Euler’s Theorem

    • Lindsay N. Childs
    Pages 117-133

  10. RSA Cryptography and Prime Numbers

    • Lindsay N. Childs
    Pages 135-151

  11. Groups, Cosets and Lagrange’s Theorem

    • Lindsay N. Childs
    Pages 153-169

  12. Solving Systems of Congruences

    • Lindsay N. Childs
    Pages 171-193

  13. Homomorphisms and Euler’s Phi Function

    • Lindsay N. Childs
    Pages 195-213

  14. Cyclic Groups and Cryptography

    • Lindsay N. Childs
    Pages 215-239

  15. Applications of Cosets

    • Lindsay N. Childs
    Pages 241-257

  16. An Introduction to Reed–Solomon Codes

    • Lindsay N. Childs
    Pages 259-272

  17. Blum-Goldwasser Cryptography

    • Lindsay N. Childs
    Pages 273-292

  18. Factoring by the Quadratic Sieve

    • Lindsay N. Childs
    Pages 293-312

  19. Polynomials and Finite Fields

    • Lindsay N. Childs
    Pages 313-330

  20. Reed-Solomon Codes II

    • Lindsay N. Childs
    Pages 331-342
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Keywords

About this book

This text presents a careful introduction to methods of cryptology and error correction in wide use throughout the world and the concepts of abstract algebra and number theory that are essential for  understanding these methods.  The objective is to provide a thorough understanding of RSA, Diffie–Hellman, and Blum–Goldwasser cryptosystems and Hamming and Reed–Solomon error correction: how they are constructed, how they are made to work efficiently, and also how they can be attacked.   To reach that level of understanding requires and motivates many ideas found in a first course in abstract algebra—rings, fields, finite abelian groups, basic theory of numbers, computational number theory, homomorphisms, ideals, and cosets.  Those who complete this book will have gained a solid mathematical foundation for more specialized applied courses on cryptology or error correction, and should also be well prepared, both in concepts and in motivation, to pursue more advanced study in algebra and number theory.

This text is suitable for classroom or online use or for independent study. Aimed at students in mathematics, computer science, and engineering, the prerequisite includes one or two years of a standard calculus sequence. Ideally the reader will also take a concurrent course in linear algebra or elementary matrix theory. A solutions manual for the 400 exercises in the book is available to instructors who adopt the text for their course.


Reviews

“This is a really nice way of introducing students to abstract algebra. It is evident that the author has spent much time polishing the presentation including large amount of details (and numerous examples) which make the book ideal for self-study. Even though the book starts out very elementary (basically requiring no prior knowledge beyond integer arithmetic), it gets to some sophisticated results towards the end. So in summary, I can warmly recommend this book as a first introduction to algebra.” (G. Teschl, Monatshefte für Mathematik, Vol. 196 (3), November, 2021)

Authors and Affiliations

  • Department of Mathematics and Statistics, University at Albany, State University of New York, Albany, USA

    Lindsay N. Childs

About the author

Lindsay N. Childs is Professor Emeritus at the University of Albany where he earned recognition as a much-loved mentor of students, and as an expert in Galois field theory. Capping his tenure at Albany, he was named a Collins Fellow for his extraordinary devotion to the University at Albany and the people in it over a sustained period of time. Post University of Albany, Professor Childs has taught a sequence of online courses whose content evolved into this book. Lindsay Childs is author of A Concrete Introduction to Higher Algebra, published in Springer's Undergraduate Texts in Mathematics series, as well as a monograph, Taming Wild Extensions: Hopf Algebras and Local Galois Module Theory (American Mathematical Society), and more than 60 research publications in abstract algebra.


Bibliographic Information

  • Book Title: Cryptology and Error Correction

  • Book Subtitle: An Algebraic Introduction and Real-World Applications

  • Authors: Lindsay N. Childs

  • Series Title: Springer Undergraduate Texts in Mathematics and Technology

  • DOI: https://doi.org/10.1007/978-3-030-15453-0

  • Publisher: Springer Cham

  • eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)

  • Copyright Information: Springer Nature Switzerland AG 2019

  • Hardcover ISBN: 978-3-030-15451-6Published: 02 May 2019

  • eBook ISBN: 978-3-030-15453-0Published: 18 April 2019

  • Series ISSN: 1867-5506

  • Series E-ISSN: 1867-5514

  • Edition Number: 1

  • Number of Pages: XIV, 351

  • Number of Illustrations: 6 b/w illustrations, 1 illustrations in colour

  • Topics: Algebra, Cryptology, Number Theory

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