Cryptology and Error Correction

1. Front Matter

   Pages i-xiv

   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

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.

• Caeser ciphers
• Chinese Remainder Theorem
• El Gamal cryptography
• Lagrange's Theorem
• Luhn's formula
• Vernam cipher

“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)

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

  Lindsay N. Childs

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.

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