Introduction to Cryptography with Open-Source Software

Once the privilege of a secret few, cryptography is now taught at universities around the world. Introduction to Cryptography with Open-Source Software illustrates algorithms and cryptosystems using examples and the open-source computer algebra system of Sage. The author, a noted educator in the field, provides a highly practical learning experience by progressing at a gentle pace, keeping mathematics at a manageable level, and including numerous end-of-chapter exercises. Focusing on the cryptosystems themselves rather than the means of breaking them, the book first explores when and how the methods of modern cryptography can be used and misused. It then presents number theory and the algorithms and methods that make up the basis of cryptography today. After a brief review of "classical" cryptography, the book introduces information theory and examines the public-key cryptosystems of RSA and Rabin’s cryptosystem. Other public-key systems studied include the El Gamal cryptosystem, systems based on knapsack problems, and algorithms for creating digital signature schemes. The second half of the text moves on to consider bit-oriented secret-key, or symmetric, systems suitable for encrypting large amounts of data. The author describes block ciphers (including the Data Encryption Standard), cryptographic hash functions, finite fields, the Advanced Encryption Standard, cryptosystems based on elliptical curves, random number generation, and stream ciphers. The book concludes with a look at examples and applications of modern cryptographic systems, such as multi-party computation, zero-knowledge proofs, oblivious transfer, and voting protocols.

Introduction to CryptographyHiding information: confidentiality Some basic definitions Attacks on a cryptosystem Some cryptographic problems Cryptographic protocols Some simple ciphers Cryptography and computer security Basic Number Theory Introduction Some basic definitions Some number theoretic calculations Primality testing Classical Cryptosystems Introduction The Caesar cipher Translation ciphersTransposition ciphers The Vigenère cipher The one-time pad Permutation ciphers Matrix ciphers Introduction to Information TheoryEntropy and uncertainty Perfect secrecy Estimating the entropy of English Unicity distance Public-Key Cryptosystems Based on Factoring Introduction The RSA cryptosystem Attacks against RSA RSA in Sage Rabin’s cryptosystem Rabin’s cryptosystem in Sage Some notes on securityFactoring Public-Key Cryptosystems Based on Logarithms and KnapsacksEl Gamal’s cryptosystem El Gamal in Sage Computing discrete logarithms Diffie-Hellman key exchange Knapsack cryptosystems Breaking the knapsack Digital Signatures Introduction RSA signature scheme Rabin digital signatures The El Gamal digital signature scheme The Digital Signature Standard Block Ciphers and the Data Encryption Standard Block ciphers Some definitions Substitution/permutation ciphersModes of encryption Exploring modes of encryption The Data Encryption Standard (DES) Feistel ciphers Simplified DES: sDES The DES algorithm Security of S-boxes Security of DES Using DES Experimenting with DES Lightweight ciphers Finite Fields Groups and rings Introduction to fields Fundamental algebra of finite fields Polynomials mod 2 A field of order 8 Other fields GF(2n) Multiplication and inversion Multiplication without power tables The Advanced Encryption Standard Introduction and some history Basic structure The layers in detail Decryption Experimenting with AES A simplified Rijndael Security of the AES Hash FunctionsUses of hash functions Security of hash functions Constructing a hash function Provably secure hash functions New hash functions Message authentication codes Using a MAC Elliptic Curves and CryptosystemsBasic definitionsThe group on an elliptic curve Background and history Multiplication Elliptic curve cryptosystems Elliptic curve signature schemes Elliptic curves over binary fields Pairing based cryptography Exploring pairings in Sage Random Numbers and Stream Ciphers Introduction Pseudo-random number generatorsSome cryptographically strong generators The shrinking generator ISAAC and Fortuna Stream ciphers RC4 The Blum-Goldwasser cryptosystem Advanced Applications and ProtocolsSecure multi-party computation Zero knowledge proofs Oblivious transfer Digital cash Voting protocols Appendix A: Introduction to Sage Appendix B: Advanced Computational Number Theory Bibliography Index Exercises appear at the end of each chapter.

Alasdair McAndrew is a senior lecturer in the School of Engineering and Science at Victoria University in Melbourne, Australia.