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.

Additional topics, including hash functions, pseudorandom number generators, zero-knowledge proofs, digital cash and DES/AES, are briefly described in the final chapter. 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.

**About the Author**
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. He has authored more than120 publications and has had more than 20 doctoral students.