Math 690 Fall 2000, MW 12-1:30 DRL 4C8

Mathematical Foundations of Computer Security

Professor Andre Scedrov

Office: Room 4E6 in David Rittenhouse Laboratory
Telephone: eight five nine eight three ( Math. Dept. Office: eight eight one seven eight )
Fax: three four zero six three
E-mail: lastname at math
Office Hours: By appointment


About This Course

"What is to distinguish a digital dollar when it is as easily reproducible as the spoken word? How do we converse privately when every syllable is bounced off a satellite and smeared over an entire continent? How should a bank know that it really is Bill Gates requesting from his laptop in Fiji a transfer of $100,000,.....,000 to another bank? Fortunately, the mathematics of cryptography can help. Cryptography provides techniques for keeping information secret, for determining that information has not been tampered with, and for determing who authored pieces of information." (From the Foreword by R. Rivest to the "Handbook of Applied Cryptography" by Menezes, van Oorschot, and Vanstone.)

This course will be followed by a course on Advanced Topics in Mathematical Foundations of Computer Security, Math 691, in Spring 2001.


Basic Concepts of Cryptology, Substitution Ciphers, Permutation Ciphers, Vigenere Cipher, Rotor Machines, Attack Models, Needham-Schroeder Key Exchange Protocol.Overview of Probability Theory: Probability Distribution, Random Variable, Conditional Probability, Bayes Theorem, Expected Value. Symmetric Ciphers, Block Ciphers, One-Time Pad, Information-Theoretic Properties of One-Time Pad, Perfect Secrecy, Misuses of One-Time Pad, Malleability. Stream Ciphers, Linear Feedback Shift Register, Golomb's Randomness Postulates, Linear Complexity, Non-linear Filters, Knapsack Keystream Generator. Feistel Networks, Data Encryption Standard, S-boxes, Key Schedule, DES Properties, DES Modes of Operation, Message Authentication Code, Exhaustive Search Attack, Triple DES, DESX, Matsui's Linear Cryptanalysis of DES. Public-Key Cryptography Overview, Merkle Puzzles. Introduction to Number Theory: Modular Exponentiation by Repeated Squaring, Prime Factors of (b^n) - 1, Finite Fields, Roots of Unity, Quadratic Residues, Legendre Symbol, Jacobi Symbol, Law of Quadratic Reciprocity, Computation of Square Roots Modulo p, Probabilistic Tests for Primality: Solovay-Strassen Test, Miller-Rabin Test. Diffie-Hellman Key Exchange, Person-in-the Middle Attack. Discrete Logarithm, Random Self-Reduction, Giant-Step Baby-Step Algorithm, Pohlig-Hellman Algorithm, ElGamal Public-Key Cryptosystem. RSA Public-Key Cryptosystem, Attacks on RSA: Pollard's p - 1 Algorithm, Low Private Exponent, Low Public Exponent. Digital Signatures, Selective Forgery, Existential Forgery, Signature Schemes Based on RSA: PKCS #1, Signature Schemes Based on Discrete Logarithm: ElGamal Signature Scheme, Digital Signature Standard. Hash Functions, Preimage Resistance, Second Preimage Resistance, Collision Resistance, Compression Functions, Merkle-Damgard Iteration Construction, Cryptographic Message Authentication Code, Information-Theoretic Message Authentication Code. Key Distribution and Authentication Protocols: TMN Protocol, Kerberos, Wide-Mouthed Frog, Woo-Lam, Yahalom. Formal Methods in the Analysis of Cryptographic Protocols.

Further References

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Take-Home Midterm Due in DRL 4E6 on Monday, November 6 at 4 p.m.

This is a complete list of assignments due November 6, 2000.

Take-Home Final Exam Due in DRL 4E6 Tuesday, December 19 at 4 p.m.

This is a complete list of assignments due December 19, 2000.