number theory and probability theory are included in the appendix. Each chapter closes with a collection of exercises. In the second edition the authors added a complete description of the AES, an extended section on cryptographic hash functions, and new sections on random oracle proofs and public-key encryption schemes that are provably secure against adaptively- chosen-ciphertext attacks. The third edition is a further substantive extension, with new topics added, including: elliptic curve cryptography; Paillier encryption; quantum cryptography; the new SHA-3 standard for cryptographic hash functions; a considerably extended section on electronic elections and Internet voting; mix nets; and zero-knowledge proofs of shuffles. The book is appropriate for undergraduate and graduate students in computer science, mathematics, and engineering.