Spring 2014

MATH 341 / LGIC 220, T Th 12-1:30, DRL 4C8

Discrete Mathematics II

Professor Andre Scedrov

Textbook

• Johannes A. Buchmann: "Introduction to Cryptography". Springer, Second Edition, 2004. Paperback. ISBN 9780387207568. A copy of this book is on reserve in the Mathematics Library on the 3-rd floor of DRL.

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, Second Edition, 2005. ISBN 978-0387287232, especially Chapters 1 and 2. A copy of this book is on reserve in the Mathematics Library on the 3-rd floor of DRL.
• "Handbook of Applied Cryptography" by Menezes, van Oorschot, and Vanstone. CRC Press, Fifth Printing, 2001. ISBN: 0-8493-8523-7.

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.

Selected topics from modern cryptography and computer network security.

Homework #1 Due in Class on Thursday, February 27

• 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.
• Let 2 be the two-element set {0,1}. Let A be any set, let P(A) be the power set of A and let 2^A be the set of all functions from A to 2. Prove that P(A) and 2^A are equinumerous.
• Exercise 3 on p. A-32 from Appendix 3 of Grimaldi.
• Exercise 7 on p. A-32 from Appendix 3 of Grimaldi.
• Problem x2.1 from Moschovakis.
• Problem x2.2 from Moschovakis.

Homework #2 Due in Class on Tuesday, April 8

• Exercise 3.16.1 on p. 111 of Buchmann.
• Exercise 3.16.16 on p. 113 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.
• Exercise 2.23.10 on p. 68 of Buchmann.
• Exercise 2.23.13 on p. 68 of Buchmann.

Take-Home Final Exam Due in Hardcopy in DRL 4E6 on Friday, May 9, 2014 at 11 a.m.

• Class project: The vulnerability in the OpenSSL protocol and the Heartbleed Bug.
• 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.
• Using the Fundamental Theorem of Arithmetic, prove that the product of (1 - 1/p) over all primes p is zero.
• 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 with p = 131 and not 130.
• Exercise 12.9.7 on p. 274 of Buchmann.