# An Introduction to Mathematical Cryptography

Springer, Sep 11, 2014 - Mathematics - 538 pages

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 DiffieHellmann 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.

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### Contents

 1 An Introduction to Cryptography 1 2 Discrete Logarithms and DiffieHellman 60 3 Integer Factorization and RSA 117 4 Digital Signatures 192 5 Combinatorics Probability and Information Theory 207 6 Elliptic Curves and Cryptography 299
 7 Lattices and Cryptography 372 8 Additional Topics in Cryptography 471 List of Notation 503 References 506 Index 517 Copyright

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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 University 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 since 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. He has authored more than 130 publications and has had more than 20 doctoral students.