Cryptology and Error Correction: An Algebraic Introduction and Real-World Applications

Description

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. Front Matter ....Pages i-xiv Secure, Reliable Information (Lindsay N. Childs)....Pages 1-11 Modular Arithmetic (Lindsay N. Childs)....Pages 13-26 Linear Equations Modulo m (Lindsay N. Childs)....Pages 27-49 Unique Factorization in \(\mathbb {Z}\) (Lindsay N. Childs)....Pages 51-64 Rings and Fields (Lindsay N. Childs)....Pages 65-82 Polynomials (Lindsay N. Childs)....Pages 83-91 Matrices and Hamming Codes (Lindsay N. Childs)....Pages 93-115 Orders and Euler’s Theorem (Lindsay N. Childs)....Pages 117-133 RSA Cryptography and Prime Numbers (Lindsay N. Childs)....Pages 135-151 Groups, Cosets and Lagrange’s Theorem (Lindsay N. Childs)....Pages 153-169 Solving Systems of Congruences (Lindsay N. Childs)....Pages 171-193 Homomorphisms and Euler’s Phi Function (Lindsay N. Childs)....Pages 195-213 Cyclic Groups and Cryptography (Lindsay N. Childs)....Pages 215-239 Applications of Cosets (Lindsay N. Childs)....Pages 241-257 An Introduction to Reed–Solomon Codes (Lindsay N. Childs)....Pages 259-272 Blum-Goldwasser Cryptography (Lindsay N. Childs)....Pages 273-292 Factoring by the Quadratic Sieve (Lindsay N. Childs)....Pages 293-312 Polynomials and Finite Fields (Lindsay N. Childs)....Pages 313-330 Reed-Solomon Codes II (Lindsay N. Childs)....Pages 331-342 Back Matter ....Pages 343-351