Elementary Number Theory, Cryptography and Codes

Springer Science & Business Media
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In this volume one finds basic techniques from algebra and number theory (e.g. congruences, unique factorization domains, finite fields, quadratic residues, primality tests, continued fractions, etc.) which in recent years have proven to be extremely useful for applications to cryptography and coding theory. Both cryptography and codes have crucial applications in our daily lives, and they are described here, while the complexity problems that arise in implementing the related numerical algorithms are also taken into due account. Cryptography has been developed in great detail, both in its classical and more recent aspects. In particular public key cryptography is extensively discussed, the use of algebraic geometry, specifically of elliptic curves over finite fields, is illustrated, and a final chapter is devoted to quantum cryptography, which is the new frontier of the field. Coding theory is not discussed in full; however a chapter, sufficient for a good introduction to the subject, has been devoted to linear codes. Each chapter ends with several complements and with an extensive list of exercises, the solutions to most of which are included in the last chapter.

Though the book contains advanced material, such as cryptography on elliptic curves, Goppa codes using algebraic curves over finite fields, and the recent AKS polynomial primality test, the authors' objective has been to keep the exposition as self-contained and elementary as possible. Therefore the book will be useful to students and researchers, both in theoretical (e.g. mathematicians) and in applied sciences (e.g. physicists, engineers, computer scientists, etc.) seeking a friendly introduction to the important subjects treated here. The book will also be useful for teachers who intend to give courses on these topics.

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Springer Science & Business Media
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Published on
Nov 28, 2008
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Computers / Information Theory
Computers / Programming / Algorithms
Mathematics / Algebra / General
Mathematics / Combinatorics
Mathematics / Geometry / General
Mathematics / Number Theory
Mathematics / Probability & Statistics / Stochastic Processes
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This self-contained introduction to modern cryptography emphasizes the mathematics behind the theory of public key cryptosystems and digital signature schemes. The book focuses on these key topics while developing the mathematical tools needed for the construction and security analysis of diverse cryptosystems. Only basic linear algebra is required of the reader; techniques from algebra, number theory, and probability are introduced and developed as required. This text provides an ideal introduction for mathematics and computer science students to the mathematical foundations of modern cryptography. The book includes an extensive bibliography and index; supplementary materials are available online.

The book covers a variety of topics that are considered central to mathematical cryptography. Key topics include:

classical cryptographic constructions, such as Diffie–Hellmann key exchange, discrete logarithm-based cryptosystems, the RSA cryptosystem, and digital signatures;

fundamental mathematical tools for cryptography, including primality testing, factorization algorithms, probability theory, information theory, and collision algorithms;

an in-depth treatment of important cryptographic innovations, such as elliptic curves, elliptic curve and pairing-based cryptography, lattices, lattice-based cryptography, and the NTRU cryptosystem.

The second edition of An Introduction

to Mathematical Cryptography includes a significant revision of the material on digital signatures, including an earlier introduction to RSA, Elgamal, and DSA signatures, and new material on lattice-based signatures and rejection sampling. Many sections have been rewritten or expanded for clarity, especially in the chapters on information theory, elliptic curves, and lattices, and the chapter of additional topics has been expanded to include sections on digital cash and homomorphic encryption. Numerous new exercises have been included.
"You think too much! You mother F@$#%&* think too much! You're nothing but an arrogant, pointy-headed intellectual — I want you out of my classroom and off the premises in five minutes or I'm calling the police and having you arrested for trespassing." — Hollywood acting teacher to Randy Olson, former scientist

After nearly a decade on the defensive, the world of science is about to be restored to its rightful place. But is the American public really ready for science? And is the world of science ready for the American public?

Scientists wear ragged clothes, forget to comb their hair, and speak in a language that even they don't understand. Or so people think. Most scientists don't care how they are perceived, but in our media-dominated age, style points count.

Enter Randy Olson. Fifteen years ago, Olson bid farewell to the science world and shipped off to Hollywood ready to change the world. With films like Flock of Dodos: The Evolution-Intelligent Design Circus (Tribeca '06, Showtime) and Sizzle: A Global Warming Comedy (Outfest '08), he has tried to bridge the cultural divide that has too often left science on the outside looking in.

Now, in his first book, Olson, with a Harvard Ph.D. and formerly a tenured professor of marine biology at the University of New Hampshire, recounts the lessons from his own hilarious-and at times humiliating-evolution from science professor to Hollywood filmmaker. In Don't Be Such a Scientist, he shares the secrets of talking substance in an age of style. The key, he argues, is to stay true to the facts while tapping into something more primordial, more irrational, and ultimately more human.

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