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A Course in Number Theory and Cryptography (Graduate Texts in Mathematics)
Neal Koblitz

Springer, 1994 - 235 pages

average customer review:based on 14 reviews
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   highly recommended  highly recommended





Excellent book for self study

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.


Koblitz's Course in Number Theory and Cryptography

This book is a real gem - very clearly written and covering the subject matter concisely but comprehensively. Particularly welcome are the exercises which are ingenious and extend the subject matter rather than just test knowledge of the chapter. It is extremely helpful too (and rare in a graduate text book) that solutions to all of the problems are provided at the back of the book. Exceptionally, and again very helpfully, there seem to be zero errors/typos in the text.

Strongly recommended as the best introduction to this fascinating and important field


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Pleasant introduction to cryptography

Chapters 1 and 2 give some elementary background material on number theory and finite fields. Chapter 3 discusses some old and naive cryptosystems. Chapter 4 discusses public key cryptosystems. In the RSA system, the receiver chooses two large primes p,q and makes public their product pq=n and some integer e relatively prime to phi(n). The sender then sends his message to the power e reduced mod n. To invert this operation one must know phi(n), i.e. one must know the factorisation n=pq. Since factoring big numbers is hard, only the intended receiver will be able to decipher the message instantly. RSA thus uses the fact that multiplying is easy but inverting it is hard; similarly, one can employ other such "trapdoor functions", such as exponentiation in Z/nZ, to create other public key cryptosystems. In chapter 5 we look at various algorithms and tricks for factorisation and primality testing. As for the cryptosystems, classical number theory that is hundreds of years old still provides the best tools (modulo arithmetic, quadratic residues, continued fractions, etc.), and in chapter 6 we see how another classical theory--elliptic curves--also proves to be fruitful in cryptography. The points of an elliptic curve over a finite field form a finite group, which we can use as the basis for new cryptosystems, analogous to how we made cryptosystems out of Z/nZ for instance. And starting with an integer and constructing corresponding finite field elliptic curves we can employ these groups and elliptic curve techniques to give improved algorithms for primality testing and factorisation.


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Outstanding presentations

This book is an outstanding introduction to cryptographic techniques and algorithms Although it's labelled as a "graduate text in mathematics", most of it should be accessible to anyone who knows a little linear algebra. For readers just interested in the how-to of the algorithms, not even that is needed. Koblitz does a thorough job of leading up to each algorithm and proving its formal properties. He also presents the algorithms themselves, unencumbered by denser material of interest to mathematicians.

The book covers a variety of topics - public-key encryption, primality testing, factoring, and cryptographic protocols. It introduces zero-knowledge proofs and blind transfer, techniques that offer real hope of personal privacy in a world where data transfer is mandatory. I was a little disappointed by the chapters on elliptic cryptography, however. I hoped that Koblitz would bring is explanatory powers to bear on the algorithms. Somehow, I never quite connected with his descriptions of elliptic curves - perhaps I'm just thick, or perhaps a bit more introductory material would have helped.

The rest of the book is a very fine example of clear, readable math writing. Its clarity its range of topics earn it a place with anyone interested in cryptography, factoring, and prime numbers.


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Essential for your secrets

How Neal Koblitz manages to squeeze the amount of material he presents into this slim volume is a miracle of nature. It even includes what most authors of graduate works leave out as a matter of course: answers to exercises. More amazing still is that far from being terse and unreadable the text is a delight.

My advice to anyone interested in this field is to have this book by their side at all times. Then if the need arises to find out what makes an algorithm tick or to refresh one's mind about a well known concept it's just the flick of a page away.



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reviews: page 1, 2, 3



The purpose of this book is to introduce the reader to arithmetic topics, both ancient and modern, that have been at the center of interest in applications of number theory, particularly in cryptography. 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, emphasizing estimates of the efficiency of the techniques that arise from the theory. A special feature is the inclusion of recent application of the theory of elliptic curves. Extensive exercises and careful answers have been included in all of the chapters. Because number theory and cryptography are fast-moving fields, this new edition contains substantial revisions and updated references.


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