This is a substantially revised and updated introduction to arithmetic topics, both ancient and modern, that have been at the centre of interest in applications of number theory, particularly in cryptography. As such, no background in algebra or number theory is assumed, and the book begins with a discussion of the basic number theory that is needed. The approach taken is algorithmic, emphasising estimates of the efficiency of the techniques that arise from the theory, and one special feature is the inclusion of recent applications of the theory of elliptic curves. Extensive exercises and careful answers are an integral part all of the chapters.
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58 of 58 people found the following review helpful:
4.0 out of 5 stars
Cryptographer's toolbox,
By A Customer
This review is from: A Course in Number Theory and Cryptography (Graduate Texts in Mathematics) (Hardcover)
Two areas of this book deserve special mention. The first chapter develops a careful treatment of the _exact_ bit complexity of operations on numbers, such as +,-,*,/, modular powering, and gcd. While other books give crude estimates, or leave out such details entirely, Koblitz invests a good deal of time not only in giving the number of operations, but in teaching the reader how to make his own estimates. *Highly* useful.Second, the book contains a concise introduction to modern factoring algorithms. After a discussion of primality testing, it goes on to develop the notion of a "B-smooth" number and then show how this leads to algorithms which use factor bases. Examples are given in the text, and the reasons behind that funny-looking time estimate O(e^(c*sqrt(log n log log n)) are provided. Seriously good stuff. The exercises are also first rate - fun, intriguing, and serve to teach new ideas (not just test knowledge of the chapter). In parts it shows its age (1994); for example, the Chor-Rivest knapsack described on p.115 has been broken by Serge Vaudenay. Much more discussion of randomized cryptography would also have been nice (though perhaps much in an intro book?). The most glaring deficiency is the lack of any real discussion of chosen ciphertext attacks, signature forgery, or padding schemes. You can't use this by itself to develop a new real-world project. Instead, it's more like a "cryptographer's toolbox," which gives you a thorough introduction to the primitives involved, giving you the understanding necessary to start thinking intelligently about how they are used.
8 of 9 people found the following review helpful:
5.0 out of 5 stars
Get your concepts cleared!!,
By qubit (Los Angeles, CA) - See all my reviews
This review is from: A Course in Number Theory and Cryptography (Graduate Texts in Mathematics) (Hardcover)
This is a truly lovely book written by Koblitz. I agree with some of the comments made by earlier reviewers that the content might be outdated, however, it is important to realize that this book is there for building one's foundation in number theory and cryptography. After one is done doing that, one can go and read the current literature in cryptography. I have used this book for a graduate crypto course at USC, and I think it really helped me a lot. This book is a great reference and a great buy.
3 of 3 people found the following review helpful:
5.0 out of 5 stars
Excellent book for self study,
By PST "A Reader from Germany" (Eislingen Deutschland) - See all my reviews
This review is from: A Course in Number Theory and Cryptography (Graduate Texts in Mathematics) (Hardcover)
This is an excellent book fot those, who are interested in the theoretical background of cryptography. It was also my first book in number theory, and I had no trouble following most of the text ( except the chapter on Elliptic curves, which -as I realize now- IS difficult)Highly recommendable! A pleasant surprise is, that there are virtually no typos.
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Inside This Book
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First Sentence:
Most of the topics reviewed in this chapter are probably well known to most readers. Read the first page Key Phrases - Statistically Improbable Phrases (SIPs): (learn more)
affine enciphering transformations, ciphertext digraphs, least absolute residue, plaintext message units, enciphering matrix, plaintext digraphs, ciphertext message units, deciphering matrix, occurring digraphs, superincreasing knapsack problem, repeated squaring method, curve primality test, enciphering key, strong pseudoprime, least nonnegative residue, deciphering transformation, discrete log problem, odd composite integer, rho method, deciphering key, classical cryptosystem, least positive residue, bit operations, oblivious transfer, nontrivial divisor Key Phrases - Capitalized Phrases (CAPs): (learn more)
Hasse's Theorem, Information Theory, Prime Number Theorem, Theory of Computing, Theory of Numbers, Monte Carlo, Scientific American, American Math, Annual Symposium, Foundations of Computer Science, Law of Quadratic Reciprocity, Proceedings of Eurocrypt New!
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Front Cover | Table of Contents | First Pages | Index | Back Cover | Surprise Me!
Most of the topics reviewed in this chapter are probably well known to most readers. Read the first page Key Phrases - Statistically Improbable Phrases (SIPs): (learn more)
affine enciphering transformations, ciphertext digraphs, least absolute residue, plaintext message units, enciphering matrix, plaintext digraphs, ciphertext message units, deciphering matrix, occurring digraphs, superincreasing knapsack problem, repeated squaring method, curve primality test, enciphering key, strong pseudoprime, least nonnegative residue, deciphering transformation, discrete log problem, odd composite integer, rho method, deciphering key, classical cryptosystem, least positive residue, bit operations, oblivious transfer, nontrivial divisor Key Phrases - Capitalized Phrases (CAPs): (learn more)
Hasse's Theorem, Information Theory, Prime Number Theorem, Theory of Computing, Theory of Numbers, Monte Carlo, Scientific American, American Math, Annual Symposium, Foundations of Computer Science, Law of Quadratic Reciprocity, Proceedings of Eurocrypt New!
Books on Related Topics | Concordance | Text Stats Browse Sample Pages:
Front Cover | Table of Contents | First Pages | Index | Back Cover | Surprise Me!
Citations (learn more)
This book cites 27
books:
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100
books
cite this book:
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- Elementary Number Theory and Its Applications (4th Edition) by Kenneth H. Rosen on page 30, and page 82
- Primality and Cryptography (Wiley Teubner on Applicable Theory in Computer Science) by Evangelos Kranakis on page 91, and page 138
- History of the Theory of Numbers (AMS Chelsea Publishing) Volumes I; II; III. THREE VOLUME SET (Vol. 2) by Leonard Eugene Dickson on page 30, and page 154
- Solved and Unsolved Problems in Number Theory (CHEL/297) by Daniel Shanks on page 30, and page 53
- Topics in Algebra by I. N. Herstein on page 52
See all 27 books this book cites
- Identification Numbers and Check Digit Schemes (Classroom Resource Materials) by Joseph Kirtland on page 42, page 93, and Back Matter
- Contemporary Cryptography (Artech House Computer Security Library) by Rolf Oppliger in Front Matter, page 102, and page 193
- The Pleasures of Counting by T. W. Körner on page 398, and Back Matter
- Advances in Cryptology - CRYPTO '98: 18th Annual International Cryptology Conference, Santa Barbara, California, USA, August 23-27, 1998, Proceedings (Lecture Notes in Computer Science) by Hugo Krawczyk on page 256, and page 336
- A Classical Introduction to Cryptography: Applications for Communications Security by Serge Vaudenay in Front Matter, and Back Matter
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