A Computational Introduction to Number Theory and Algebra
| Date: 06 May 2011, 00:46
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Number theory and algebra play an increasingly significant role in computing and communications, as evidenced by the striking applications of these subjects to such fields as cryptography and coding theory. This introductory book emphasises algorithms and applications, such as cryptography and error correcting codes, and is accessible to a broad audience. The mathematical prerequisites are minimal: nothing beyond material in a typical undergraduate course in calculus is presumed, other than some experience in doing proofs - everything else is developed from scratch. Thus the book can serve several purposes. It can be used as a reference and for self-study by readers who want to learn the mathematical foundations of modern cryptography. It is also ideal as a textbook for introductory courses in number theory and algebra, especially those geared towards computer science students.
Review
"This is an outstanding and well-written book whose aim is to introduce the reader to a broad range of material -- ranging from basic to relatively advanced -- without requiring any prior knowledge on the part of the reader other than calculus and mathematical maturity. That the book succeeds at this goal is quite an accomplishment! ...this book is a must-read for anyone interested in computational number theory or algebra and especially applications of the latter to cryptography. I would not hesitate, though, to recommend this book even to students 'only' interested in the algebra itself (and not the computational aspects thereof); especially for computer science majors, this book is one of the best available introductions to that subject."
SIGACT News
"As computer science students have very likely not previously mastered probability theory, linear algebra, or basic abstract algebra, Shoup packages crash courses in each. Despite taking time for so many basics, Shoup climaxes with careful treatment of the late-breaking, ingenious, polynomial-time deterministic primality test of Agrawal, Kayal, and Saxena. Apart from number theory, one could easily build a fine discrete mathematics course on this book. Highly recommended."
Choice
TABLE OF CONTENT:
Chapter 01 - Basic properties of the integers
Chapter 02 - Congruences
Chapter 03 - Computing with large integers
Chapter 04 - Euclid's algorithm
Chapter 05 - The distribution of primes
Chapter 06 - Finite and discrete probability distributions
Chapter 07 - Probabilistic algorithms
Chapter 08 - Abelian groups
Chapter 09 - Rings
Chapter 10 - Probabilistic primality testing
Chapter 11 - Finding generators and discrete logarithms in Z*p
Chapter 12 - Quadratic residues and quadratic reciprocity
Chapter 13 - Computational problems related to quadratic residues
Chapter 14 - Modules and vector spaces
Chapter 15 - Matrices
Chapter 16 - Subexponential-time discrete logarithms and factoring
Chapter 17 - More rings
Chapter 18 - Polynomial arithmetic and applications
Chapter 19 - Linearly generated sequences and applications
Chapter 20 - Finite fields
Chapter 21 - Algorithms for finite fields
Chapter 22 - Deterministic primality testing
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