CS456/CS656 Cryptography
Spring 2000

Instructor: Christino Tamon
Lecture: Science Center 340 MWF 15:00
Office hours: Science Center 373
Pre-requisites: MA211 or MA346, good programming skills, and a healthy mathematical curiosity.
Syllabus: Cryptography is the study of secure communication over insecure channels. We will study the basic methods and concepts in theoretical cryptography along with their applications. Concepts such as one-way functions and trapdoor permutations (functions that are easy to compute but computationally hard to invert), pseudorandom sequence generators (devices that produces sequences that are computationally random), public-key cryptosystems (secure systems that require no secret agreement), one-way hash functions (tools to authenticate messages and to verify data integrity), digital signatures (mechanisms for signing documents), and zero-knowledge proofs (convincing a party of a fact without revealing its proof). Most of the topics require background in number theory and probability theory. The first part of the course will be spent on developing the necessary background in these areas, mainly number theory. The second part of the course is spent on the applications of these to building cryptographic tools.
Grading scheme
  • Assignments and Quizzes 40%
  • Midterm 20%
  • Final exam 40%
Required texts:
  • Undergraduate: Schneier, Bruce. Applied Cryptography. John Wiley and Sons, Second edition, 1996.
  • Graduate: Goldreich, Oded. Modern Cryptography, Probabilistic Proofs, and Pseudorandomness. Springer-Verlag, 1999.
Recommended references
  • Cormen, Leiserson, and Rivest. An Introduction to Algorithms. MIT Press. (highly recommended)
  • Kernighan and Ritchie. The C Programming Language. Second edition. Prentice-Hall, 1988.

Topics covered (outline from Spring 1999)

  1. General framework of cryptographic system: Alice, Bob, and Eve. One-time XOR pad (unconditionally secure). Problems with reusing the XOR pad. Types of attacks on cryptosystems: ciphertext-only, known-plaintext, chosen-plaintext, adaptive chosen-plaintext, chosen-ciphertext.
  2. The RSA cryptosystem:
    • Original reference: Ronald L. Rivest, Adi Shamir, Leonard M. Adleman. A Method of Obtaining Digital Signatures and Public-Key Cryptosystem. Communications of the ACM, vol. 21, no. 2, 1978, pp 120-126.
    • Number-theoretic background to understand RSA: see Chapter 33 of Cormen, Leiserson, and Rivest's text.
    • Assumptions and weaknesses of RSA: deterministic property. multiplicative property. hardness of Factoring and computing the Euler-Totient function.
  3. The Goldwasser-Micali probabilistic Public-Key cryptosystem:
    • Original reference: Shafi Goldwasser and Silvio Micali. Probabilistic Encryption. Journal of Computer and System Sciences, vol. 28, no. 2, 1984, pp 270-299.
    • Number-theoretic background: quadratic residues, Blum integers. computing square roots modulo a Blum integer. Legendre and Jacobi symbols.
    • Assumptions and inefficiencies: hardness of the Quadratic Residuosity Problem. sending 1-bit messages as large random pseudosquares.
    • Security: semantically secure (not covered).
  4. The Blum-Blum-Shub pseudorandom bit generator:
    • Original reference: Lenore Blum, Manuel Blum, Michael Shub. A Simple Unpredictable Pseudo-Random Number Generator. SIAM Journal of Computing, vol. 15, no.2, 1993, pp 364-383.
    • Universal tests: concept defined by Blum, Micali, and Yao.
    • Assumptions: hardness of extracting square roots modulo Blum integers.
    • Period and security: coming up
  5. The Blum-Goldwasser probabilistic Public-Key cryptosystem:
    • Original reference: Manuel Blum and Shafi Goldwasser. An Efficient Probabilistic Public-Key Encryption Scheme Which Hides All Partial Information. Advances in Cryptology: Proceedings of CRYPTO'84, Springer-Verlag, 1985, pp 289-299.
    • An improvement on the Goldwasser-Micali cryptosystem based on a one-time pad using the BBS pseudorandom generator.
    • Number-theoretic background: principal square roots modulo Blum integers. the squaring function is a permutation over the quadratic residues.
Number theory and notes.


  • Assignment 1: cryptogram (only in paper form).
    Out: 1/14/00. Due: 1/21/00.