Spring 2006

MATH 341 / LGIC 220, MW 1-2, DRL 3C2 and W 2-3 DRL A7

Discrete Mathematics II

Professor Andre Scedrov

Textbooks

• Johannes A. Buchmann: "Introduction to Cryptography". Springer, Second Edition, 2004. ISBN 0-387-21156-X.

Further References

• Ralph P. Grimaldi. "Discrete and Combinatorial Mathematics". Fifth Edition. Addison Wesley, 2003. ISBN 0-201-72634-3, especially Chapters 3 and 5.
• Yiannis N. Moschovakis. "Notes on Set Theory". Undergraduate Texts in Mathematics, Springer-Verlag, 1994. ISBN 0387941800, especially Chapters 1 and 2. Errors in this book [ ps , pdf ].

Topics

Overview of Probability Theory: Probability Distribution, Random Variable, Conditional Probability, Bayes Theorem, Expected Value. [Buchmann Chapter 4].

Basic Concepts of Cryptology: Substitution Ciphers, Permutation Ciphers, Vigenere Cipher, Rotor Machines, Attack Models. 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. [Buchmann Chapters 3 and 4].

Introduction to Number Theory: Congruences, Chinese Remainder Theorem, Fermat's Little Theorem, Euler's Theorem, Modular Exponentiation by Repeated Squaring. [Buchmann Chapters 1 and 2].

Public-Key Cryptosystems: Diffie-Hellman Key Exchange, Person-in-the Middle Attack. Discrete Logarithm, Giant-Step Baby-Step Algorithm, Pohlig-Hellman Algorithm, ElGamal Public-Key Cryptosystem. RSA Public-Key Cryptosystem. Digital Signatures, Selective Forgery, Existential Forgery, Signature Schemes Based on RSA, Signature Schemes Based on Discrete Logarithm: ElGamal Signature Scheme.

Homework #1 Due in Class on Monday, January 30

• Let R be a binary relation on a set A. Show that the union of R and the identity relation I on A is the least reflexive relation that includes R. That is, show that:
  • a) The union of R and I is itself reflexive and that it includes R, and that
  • b) For any binary relation S on A, if S is reflexive and S includes R, then S also includes the union of R and I.
• Let R be a binary relation on a set A. Show that the union of R and its opposite relation R^o is the least symmetric relation that includes R. That is, show that:
  • a) The union of R and the R^o is itself symmetric and that it includes R, and that
  • b) For any binary relation S on A, if S is symmetric and S includes R, then S also includes the union of R and R^o.
• Exercise 4ab on p. 146 of Grimaldi.
• Exercise 5abcdefghi on p. 147 of Grimaldi.
• Exercise 13ab on p. 147 of Grimaldi.
• Exercise 14ac on p. 147 of Grimaldi.

Homework #2 Due in Class on Monday, February 6

• Exercise 1.3 on p. 5 of Moschovakis.
• Exercise 1.4 on p. 5 of Moschovakis.
• Exercise 16abcdef on p. 289 of Grimaldi.
• Exercise 17abcdefghi on p. 289 of Grimaldi.
• Exercise 18abcde on p. 289 of Grimaldi.
• Exercise 20ab on p. 289 of Grimaldi.
• Exercise 21ab on p. 289 of Grimaldi.
• Exercise 23ab on p. 289 of Grimaldi.
• Exercise 28abc on p. 307 of Grimaldi.

Homework #3 Due in Class on Monday, February 20

• Exercise 14ab on p. 156 of Grimaldi.
• Exercise 16ab on p. 156 of Grimaldi.
• Exercise 14 on p. 165 of Grimaldi.
• Exercise 15 on p. 165 of Grimaldi.
• Exercise 3.16.1 on p. 111 of Buchmann.
• Exercise 3.16.3 on p. 111 of Buchmann.
• Exercise 4.8.2 parts 1 and 2 on p. 125 of Buchmann.
• Exercise 4.8.3 on p. 125 of Buchmann.
• Exercise 4.8.5 on p. 125 of Buchmann.

Take-Home Midterm Due in Class on Monday, March 20

• Exercise 18ab on p. 174 of Grimaldi.
• Exercise 22 on p. 174 of Grimaldi.
• Exercise 24ab on p. 174 of Grimaldi.
• Exercise 20abcd on p. 186 of Grimaldi.
• Exercise 21 on p. 186 of Grimaldi.
• Exercise 4.8.7 on p. 126 of Buchmann.
• Exercise 4.8.8 on p. 126 of Buchmann.
• Consider an affine cipher mod 26. Do a chosen plaintext attack using hahaha. The ciphertext is NONONO. Determine the encryption function.
• The ciphertext CRWWZ was encrypted using an affine cipher mod 26. The plaintext starts with ha. Decrypt the message.
• Suppose we have a language with only three letters a, b, c, and occur with frequencies .7, .2, and .1, respectively. The ciphertext ABCBABBBAC was encrypted by the Vigenere method using shifts mod 3 instead of mod 26. If we are told that the key length is 1, 2, or 3, show that the key length is probably 2 and determine the most probable key.

Take-Home Final Exam Due in DRL 4E6 at 10 a.m. on Friday, April 28

• Prove that if (2^n) - 1 is a prime, then n is a prime, and if (2^n) + 1 is a prime, then n is a power of 2. The first type of prime is called a Mersenne prime, and the second type is called a Fermat prime.
• Exercise 2.23.16 on p. 69 of Buchmann.
• Exercise 2.23.25 on p. 69 of Buchmann.
• Using the Fundamental Theorem of Arithmetic, prove that the product of (1 - 1/p) over all primes p is zero.
• Exercise 8.7.8 on p. 196 of Buchmann.
• Exercise 12.9.7 on p. 274 of Buchmann.
• Joint project: Attacks on MD5 and SHA-1 hash functions.