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Chinese Remainder Theorem, CRT, is one of the jewels ofmathematics. It is a perfect combination of beauty and utility or, inthe words of Horace, omne tulit punctum qui miscuit utile dulci. Knownalready for ages, CRT continues to present itself in new contexts andopen vistas for new types of applications. So far, its usefulness hasbeen obvious within the realm of "three C's". Computing was itsoriginal field of application, and continues to be important asregards various aspects of algorithmics and modularcomputations. Theory of codes and cryptography are two more recentfields of application.This book tells about CRT, its background andphilosophy, history, generalizations and, most importantly, itsapplications. The book is self-contained. This means that no factualknowledge is assumed on the part of the reader. We even provide brieftutorials on relevant subjects, algebra and informationtheory. However, some mathematical maturity is surely a prerequisite,as our presentation is at an advanced undergraduate or beginninggraduate level. We have tried to make the exposition innovative, manyof the individual results being new. We will return to this matter, aswell as to the interdependence of the various parts of the book, atthe end of the Introduction.A special course about CRT can be basedon the book. The individual chapters are largely independent and,consequently, the book can be used as supplementary material forcourses in algorithmics, coding theory, cryptography or theory ofcomputing. Of course, the book is also a reference for matters dealingwith CRT. ... Read more

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High Quality Content by WIKIPEDIA articles! Secret sharing consists of recovering a secret S from a set of shares, each containing partial information about the secret. Secret sharing can thus use the CRT to produce the shares presented in the congruence equations and the secret could be recovered by solving the system of congruences to get the unique solution, which will be the secret to recover. There are several types of secret sharing schemes. The most basic types are the so-called threshold schemes, where only the cardinality of the set of shares matters. In other words, given a secret S, and n shares, any set of t shares is a set with the smallest cardinality from which the secret can be recovered, in the sense that any set of t-1 shares is not enough to give S. This is known as a threshold access structure. We call such schemes (t,n) threshold secret sharing schemes, or t-out-of-n scheme. Threshold secret sharing schemes differ from one another by the method of generating the shares, starting from a certain secret. ... Read more

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High Quality Content by WIKIPEDIA articles! In arithmetic, when the result of the division of two integers cannot be expressed with an integer quotient, the remainder is the amount "left over." If a and d are natural numbers, with d non-zero, it can be proven that there exist unique integers q and r, such that a = qd + r and 0 ? r < d. The number q is called the quotient, while r is called the remainder. The division algorithm provides a proof of this result and also an algorithm describing how to calculate the remainder. ... Read more

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Beginning with the arithmetic of the rational integers and proceeding to an introduction of algebraic number theory via quadratic orders, Fundamental Number Theory with Applications reveals intriguing new applications of number theory. This text details aspects of computer science related to ocryptographyofactoringoprimality testingocomplexity analysisocomputer arithmeticocomputational number theoryFundamental Number Theory with Applications also covers:oCarmichael numbersoDirichlet productsoJacobsthal sumsoMersenne primesoperfect numbersopowerful numbersoself-contained numbersNumerous exercises are included, testing the reader's knowledge of the concepts covered, introducing new and interesting topics, and providing a venue to learn background material.Written by a professor and author who is an accomplished scholar in this field, this book provides the material essential for an introduction to the fundamentals of number theory. ... Read more

Customer Reviews (3)

Terrible Book, and Terrible Author
I had the luxory of using this book, and also the great honor of being taught by the "great man" himself, who has written this 120 dollar peace of trash. The book is terrible, when trying to read through the proofs in each section, halfway through the proof it says "this part is left as an exercise"
and an even excersise none the less, I mean if your gonna go through a proof make it COMPLETE, don't tell me its an easy exercise, thats BS, now thats the book, now ill leave you to imagine how terrible of a proffesor he was, let me just say this, i love mathematics, and im planning on concentrating in crpytography, but that SOB Mollin, has me reconsidering the hole thing all together, i'd rather quit the concentration than to have to even open that textbook again and sit through one of his lectures. Whatever you do, don't buy this book, and stay away from Mollin, and if you have to take a class with Mollin, i would deffinitely buy a secondary textbook

The basis of cryptography
A very solid, rigorous introduction to its subject. Scads of exercises, which is utterly essential for you to gain fluency. As a nice touch, the author also provides answers to half the questions. Alternatively, if you are a lecturer casting around for a suitable text for your class, then this may be useful. You can assign many problems for which your students cannot blithely look up the answers.

The book covers topics like Euler's Totient, Quadratic residues, Chinese Remainder Theorem and Diophantine equations. The breadth of number theory.

The book has another merit. Pure number theory texts are suitable for those majoring in the field. But even amongst mathematicians, this is a small set. These days, number theory forms the foundations of cryptography. The continued rise of the Internet, especially for commercial applications, including the still incipient web services, has spawned a critical need for better encryption. A major theme of the book is its usefulness to cryptographers. To this end, public keys, elliptic curves and other related topics are prominently described.

Unimpressed
This book is a confusing mess.If your considering this for an introductory number theory course, stay away.There are several big problems I see with this book.
The first being references.Many of the proofs contain references of the essential parts to exersizes (earlier or later) in the book.Add to the fact that none of the exersizes have full solutions, this book is in fact full of incomplete proofs.Also, there are many errors and exersizes which are plain wrong ( I can think of a few exersizes that we used to find infinite counterexamples to)
The second problem is notation.The author seems to take full advantage of making everything as short and notation heavy as possible.Unless you really know all your mathematical and logical symbols, this becomes a tedious read.
Finally, it's much more of annoyance, the footnotes are huge.Sometimes filling up half the page.I do enjoy reading about mathematical history, but I feel a footnote does not do a biography justice.Either have an appendix, or follow the style in Nicholson's "Abstract Algebra" with a page or two near the endo of every chapter dedicated to the biographies.
If you can put up with all these problems, you may find this text useful.It does provide a good basis for number theory and a good introduction to cryptography using the material in the book. ... Read more

Product Description
An update of the most accessible introductory number theory text available, Fundamental Number Theory with Applications, Second Edition presents a mathematically rigorous yet easy-to-follow treatment of the fundamentals and applications of the subject. The substantial amount of reorganizing makes this edition clearer and more elementary in its coverage.

New to the Second Edition

•          Removal of all advanced material to be even more accessible in scope

•          New fundamental material, including partition theory, generating functions, and combinatorial number theory

•          Expanded coverage of random number generation, Diophantine analysis, and additive number theory

•          More applications to cryptography, primality testing, and factoring

•          An appendix on the recently discovered unconditional deterministic polynomial-time algorithm for primality testing

Taking a truly elementary approach to number theory, this text supplies the essential material for a first course on the subject. Placed in highlighted boxes to reduce distraction from the main text, nearly 70 biographies focus on major contributors to the field. The presentation of over 1,300 entries in the index maximizes cross-referencing so students can find data with ease. ... Read more