Spring 2009

MATH 341 / LGIC 220, MWF 10-11, DRL 3C2

Discrete Mathematics II

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. A copy of this book is on reserve in the Mathematics Library on the 3-rd floor of DRL.
• 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 2

• There are no integers k and n, with n not equal to 0, such that 2 * (n^2) = k^2. That is, square root of 2 is not a fraction.
• Let A and B be sets. Show that the intersection of A and B is the largest set which is a subset of A and a subset of B, that is, show the following two facts:
  • a) The intersection of A and B is a subset of A and the intersection of A and B is also a subset of B.
  • b) Let S be a set such that S is a subset of A and S is a subset of B. Then S is a subset of the intersection of A and B.
• Let R be a relation from a set A to a set B. Let I be the identity relation on A and let I' be the identity relation on B. Show that IR = R = RI'.
• Show that the composition of relations is associative. That is, let A, B, C, and D be sets, let R be a relation from A to B, S a relation from B to C, and T a relation from C to D. Then (RS)T = R(ST).
• 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.

Homework #2 Due in Class on Wednesday, February 18

• Exercise 13a on p. 147 of Grimaldi.
• Exercise 16abcdef on p. 289 of Grimaldi.
• Exercise 17abcdefghi on p. 289 of Grimaldi.
• Exercise 20ab on p. 289 of Grimaldi.
• Exercise 21ab on p. 289 of Grimaldi.
• Exercise 28abc on p. 307 of Grimaldi.

Take-Home Midterm Due in Class in Hardcopy on Wednesday, April 8

• 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 20abcd on p. 186 of Grimaldi.
• Exercise 21 on p. 186 of Grimaldi.
• Exercise 3.16.1 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 11 at 11 a.m.

• Class project: Set up, implement, and use a PGP public-key cryptosystem for secure communication among the students in the class. Report on the experience and on the mathematics behind the cryptosystem.
• Exercise 2.23.23 on p. 69 of Buchmann.
• Exercise 2.23.26 on p. 69 of Buchmann.
• Exercise 2.23.28 on p. 70 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.
• The ciphertext CRWWZ was encrypted using an affine cipher mod 26. The plaintext starts with ha. Decrypt the message.
• 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.