Cryptography is used to hide information. It is not only use by spies but for phone, fax and e-mail communication, bank transactions, bank account security, PINs, passwords and credit card transactions on the web. It is also used for a variety of other information security issues including electronic signatures, which are used to prove who sent a message.

This Master’s program is a brief  introduction to the modern cryptography and its applications.  The course includes  mathematical foundations of cryptography, and various cryptographic protocols. As an applications we consider voting systems, e-banking systems, etc.


  1. Mathematical  background:  Number Theory, Euclidean algorithm, Euler’s  function, Chinese remainder theorem,  finite fields, etc.
  2. A brief historical overview of the development of cryptography.
  3. Block and stream ciphers. A mathematical model of substitution cipher. Attacks on ciphers, perfect ciphers, Shannon's theorem. Modes of block ciphers. GOST 28147-89 and DES encryption standards.
  4. General methods of cryptanalysis.
  5. The principles of public key cryptography.  One-way functions.  Hash Functions. Digital signatures. Diffie-Hellman key exchange.
  6. Cryptosystem RSA. Factorization of  large numbers.
  7. Cryptosystems of ElGamal type. Digital signature DSA. GOST  P34.10-94. Schnorr signature.  Smart-cards.
  8. Blind digital signatures. Electronic moneys. Payment systems.
  9. Discrete logarithms.  Baby-Step/Giant-Step Method. Index Calculus.
  10.  Elliptic Curve Cryptosystems.
  11. The notion of cryptographic protocol. Secret sharing protocols. Voting protocols. Remote coin flip protocols. Playing poker on the phone. Zero-knowledge proofs. Distribution of secret keys protocols.


  1. N. Smart. Cryptography. An Introduction. 3rd edition. McGraw-Hill, 2010.
  2. J. Hoffstein, J. Pipher, and J. H. Silverman. An Introduction to Mathematical Cryptography. Undergraduate Texts in Mathematics. Springer Science+Business Media, LLC, 2008.
  3.  A. J. Menezes, P. C. van Oorschot, and  S. A. Vanstone.  Handbook of Applied Cryptography. CRC Press, 1997.
  4. S.Vaudenay.  A Classical Introductoin to Cryptography.  Applications for Communications Security.  Springer Science+Business Media, Inc., 2006.
  5. T. Baignkres, P. Junod, Yi Lu, J. Monnerat, and S.Vaudenay.  A Classical  Introduction to Cryptography. Exercise Book.  Springer Science+Business Media, Inc., 2006.
  6. D.R.Stinson.  Cryptography. Theory and Practice. Third Edition.  Chapman & Hall/CRC, 2006.
  7. S. Goldwasser and  M. Bellare. Lecture Notes on Cryptography. 2008. URL:
  8. D. Hankerson, A. Menezes, and S. Vanstone.  Guide to elliptic curve cryptography.   Springer-Verlag New York, Inc., 2004.
  9. Handbook of Elliptic and Hyperelliptic Curve Cryptography / H.Cohen, G.Frey, R.Avanzi, C. Doche, T. Lange, Kim Nguyen, and F. Vercauteren.   Taylor & Francis Group, LLC, 2006.
  10.  B.Schneier. Applied Cryptography: Protocols, Algorithms and Source Code. Second Edition. N.Y.: John Wiley & Sons. 1996.
  11.  H.Cohen. A  Course in Computational Algebraic Number Theory.  Third, corrected printing. Springer-Verlag, 1996.  545 pp.