An Introduction to Mathematical Cryptography

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.

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 recent cryptographic innovations, such as elliptic curves, elliptic curve and pairing-based cryptography, lattices, lattice-based cryptography, and the NTRU cryptosystem.

This book is 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.

Auteur

Dr. Jeffrey Hoffstein has been a professor at Brown University since 1989 and has been a visiting professor and tenured professor at several other universities since 1978. His research areas are number theory, automorphic forms, and cryptography. He has authored more than 50 publications.

Dr. Jill Pipher has been a professor at Brown Univesity since 1989. She has been an invited lecturer and has received numerous awards and honors. Her research areas are harmonic analysis, elliptic PDE, and cryptography. She has authored over 40 publications.

Dr. Joseph Silverman has been a professor at Brown University 1988. He served as the Chair of the Brown Mathematics department from 2001-2004. He has received numerous fellowships, grants and awards and is a frequently invited lecturer. His research areas are number theory, arithmetic geometry, elliptic curves, dynamical systems and cryptography. Hehas authored more than120 publications and has had more than 20 doctoral students.

Résumé
ThecreationofpublickeycryptographybyDi?eandHellmanin1976andthe subsequent invention of the RSA public key cryptosystem by Rivest, Shamir, and Adleman in 1978 are watershed events in the long history of secret c- munications. It is hard to overestimate the importance of public key cr- tosystems and their associated digital signature schemes in the modern world of computers and the Internet. This book provides an introduction to the theory of public key cryptography and to the mathematical ideas underlying that theory. Public key cryptography draws on many areas of mathematics, including number theory, abstract algebra, probability, and information theory. Each of these topics is introduced and developed in su?cient detail so that this book provides a self-contained course for the beginning student. The only prerequisite is a ?rst course in linear algebra. On the other hand, students with stronger mathematical backgrounds can move directly to cryptographic applications and still have time for advanced topics such as elliptic curve pairings and lattice-reduction algorithms. Amongthemanyfacetsofmoderncryptography,thisbookchoosestoc- centrate primarily on public key cryptosystems and digital signature schemes. This allows for an in-depth development of the necessary mathematics - quired for both the construction of these schemes and an analysis of their security. The reader who masters the material in this book will not only be well prepared for further study in cryptography, but will have acquired a real understanding of the underlying mathematical principles on which modern cryptography is based.

Contenu

An Introduction to Cryptography.- Discrete Logarithms and Diffie Hellman.- Integer Factorization and RSA.- Combinatorics, Probability and Information Theory.- Elliptic Curves and Cryptography.- Lattices and Cryptography.- Digital Signatures.- Additional Topics in Cryptography.