Spring 2008

MATH 341 / LGIC 220, MWF 12-1, DRL 3C2

Discrete Mathematics II

Make-up classes on Friday, February 1, on Monday, February 4, on Friday, April 4, and on Monday, April 14

No class on Friday, February 15, nor on Monday, March 17, nor on Friday, April 25, nor on Monday, April 28

Professor Andre Scedrov

Textbook

• 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, 5, 7, 14, and 17 and Appendix 3.
• 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 ].
• "Handbook of Applied Cryptography" by Menezes, van Oorschot, and Vanstone. CRC Press, Fifth Printing, 2001. ISBN: 0-8493-8523-7.
• "The Rise and Fall of Knapsack Cryptosystems" by A. M. Odlyzko.

Topics

Overview of Probability Theory: Probability Distribution, Random Variable, Conditional Probability, Bayes Theorem, Expected Value. [Grimaldi Chapter 3 and 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. [Grimaldi Chapters 14 and 17 and 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, February 18

• 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 13ab on p. 147 of Grimaldi.
• Exercise 14ac on p. 147 of Grimaldi.
• Exercise 11 on p. 252 of Grimaldi.
• Exercise 12 on p. 252 of Grimaldi.
• 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.

Take-Home Midterm Due in Class in Hardcopy on Wednesday, April 9

• Exercise 2.23.23 on p. 69 of Buchmann.
• Exercise 2.23.25 on p. 69 of Buchmann.
• Exercise 2.23.26 on p. 69 of Buchmann.
• 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.
• Using the Fundamental Theorem of Arithmetic, prove that the product of (1 - 1/p) over all primes p is zero.
• 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 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 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.
• Exercise 4.8.7 on p. 126 of Buchmann.
• Exercise 4.8.8 on p. 126 of Buchmann.

Take-Home Final Exam Due in Hardcopy in DRL 4E6 on Monday, May 5 at 11 a.m.

• Class project: Electronic voting.
• 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.
• Exercise 3.16.16 on p. 113 of Buchmann.
• Exercise 8.7.8 on p. 196 of Buchmann.
• Exercise 11.9.2 on p. 248 of Buchmann.
• Exercise 12.9.6 on p. 274 of Buchmann.
• Exercise 12.9.7 on p. 274 of Buchmann.