Spring 2014
MATH 341 / LGIC 220, T Th 121:30, DRL 4C8
Discrete Mathematics II
Office: Room DRL 4E6
Office Hours: Mondays and Tuesdays 1:30  2:30
or by appointment. No office hours on Mondays February 10, February 17,
March 31, and April 21. Those weeks the Tuesday office hours
February 11, February 18, April 1, and April 22 will be held 1:30  3:30.
Textbook
Further References

Ralph P. Grimaldi. "Discrete and Combinatorial Mathematics".
Fifth Edition. Addison Wesley, 2003. ISBN 0201726343,
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 3rd floor of DRL.
 Yiannis N. Moschovakis. "Notes on Set Theory".
Undergraduate Texts in Mathematics, SpringerVerlag, Second Edition, 2005.
ISBN 9780387287232, especially Chapters 1 and 2.
A copy of this book is on reserve in the Mathematics Library
on the 3rd floor of DRL.

"Handbook of Applied Cryptography" by
Menezes, van Oorschot, and Vanstone.
CRC Press, Fifth Printing, 2001. ISBN: 0849385237.
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.
[Grimaldi Chapters 3, 5, and 7, and Appendix 3 and
Moschovakis Chapters 1 and 2].
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, 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.
[Grimaldi Chapters 14 and 17 and 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.
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 twoelement 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. A32 from Appendix 3 of Grimaldi.
 Exercise 7 on p. A32 from Appendix 3 of Grimaldi.
 Problem x2.1 from Moschovakis.
 Problem x2.2 from Moschovakis.
This is the complete set of problems for Homework #1 due in class
on Thursday, February 27, 2014.
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.
This is the complete set of problems for Homework #2 due in class
on Tuesday, April 8, 2014.
TakeHome 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.
This is a complete list of assignments due May 9, 2014.