Spring 2010
MATH 341 / LGIC 220, MWF 10-11, DRL 3C6
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
E-mail: lastname at math
Office Hours: M W 1:15 - 2:15 or by appointment
Textbook
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
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, 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 Due in Class on Wednesday, February 17
- 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.
- 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.
This is the complete set of problems for homework due in class
on Wednesday, February 17.
Take-Home Midterm Due in Class in Hardcopy on Wednesday, April 7
- 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.
This is the complete set of problems for take-home midterm due in class
on Wednesday, April 7, 2010.
Take-Home Final Exam Due in Hardcopy in DRL 4E6 on Monday,
May 10 at 11 a.m.
- Class project: Hash functions: Why the current competition?
- 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.
This is a complete list of assignments due May 10, 2010.