Spring 2006
MATH 341 / LGIC 220, MW 12, DRL 3C2 and W 23 DRL A7
Discrete Mathematics II
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
Email: lastname at math
Office Hours: By appointment
Textbooks
Further References

Ralph P. Grimaldi. "Discrete and Combinatorial Mathematics".
Fifth Edition. Addison Wesley, 2003. ISBN 0201726343,
especially Chapters 3 and 5.
 Yiannis N. Moschovakis. "Notes on Set Theory".
Undergraduate Texts in Mathematics, SpringerVerlag, 1994.
ISBN 0387941800, especially Chapters 1 and 2.
Errors in this book [
ps ,
pdf ].
Topics
Algebra of sets, power set, cartesian product, binary relations,
closure properties, equivalence relations, functions, Cantor
Theorem, countable sets, equinumeruous
sets, uncountability of the set of reals.
[Moschovakis Chapters 1 and 2].
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, OneTime Pad, InformationTheoretic
Properties of OneTime Pad, Perfect Secrecy, Misuses of OneTime Pad,
Malleability. Stream Ciphers, Linear Feedback Shift Register, Golomb's
Randomness Postulates, Linear Complexity, Nonlinear 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].
PublicKey Cryptosystems:
DiffieHellman Key Exchange, Personinthe Middle Attack. Discrete
Logarithm, GiantStep BabyStep Algorithm,
PohligHellman Algorithm, ElGamal PublicKey Cryptosystem.
RSA PublicKey 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.
This is the complete set of problems for Homework #1 due in class
on Monday, January 30.
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.
This is the complete set of problems for Homework #2 due in class
on Monday, February 6.
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.
This is the complete set of problems for Homework #3 due in class
on Monday, February 20.
TakeHome 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.
This is the complete set of problems for takehome midterm due in class
on Monday, March 20.
TakeHome 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 SHA1 hash functions.
This is the complete set of problems for takehome final exam due in
My office DRL 4E6 at 10 a.m. on Friday, April 28.