Math 340, MWF 12-1
Advanced Mathematical Methods in Computer Science
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: By appointment
Algebra of sets, power set, cartesian product, binary relations,
closure properties, equivalence relations, functions, Cantor
Theorem, Schroeder-Bernstein Theorem, countable sets, equinumeruous
sets, Cantor space, uncountability of the set of reals, algebraic
numbers, existence of transcendental numbers.
Recurrences, sums, and integer functions: Towers of Hanoi, Quicksort
recurrence, floor and ceiling functions.
Basic Concepts of Cryptology, Substitution Ciphers, Permutation Ciphers,
Vigenere Cipher, Rotor Machines, Attack Models. Overview of Probability
Theory: Probability Distribution, Random
Variable, Conditional Probability, Bayes Theorem, Expected Value.
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. Public-Key Cryptography Overview.
Introduction to Number Theory: Congruences, Chinese Remainder Theorem,
Fermat's Little Theorem, Euler's Theorem, Modular Exponentiation by
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.
Counting, Permutations, and Combinations, Binomial Theorem, Multinomial
Theorem, Combinations with Repetition.
Asymptotic functions, Stirling's Approximation Formula, Wallis's Formula.
Take-Home Midterm Due in Class on Monday, November 4
This is a complete list of assignments due November 4, 2002.
- Solve the recurrence a_(n+1) - 2a_n = 5, a_0 = 1.
- Solve the recurrence a_(n+1) - 2a_n = 2^n , a_0 = 1.
- Exercise 3.9 on p. 95 of "Concrete Mathematics".
- Exercise 3.25 on p. 97 of "Concrete Mathematics".
- Exercise 3.34 on p. 98 of "Concrete Mathematics".
- Let f(n) be any polynomial of degree d with integer coefficients
such that the leading coefficient is positive. Prove that
f(n) = O(n^d).
Take-Home Final Exam Due in DRL 4E6 on at 12 noon on
Monday, December 16
This is a complete list of assignments due December 16, 2002.
- 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.
- Implement in Maple the Silver-Pohlig-Hellman algorithm for computing
discrete logarithms in (Z_q)*. How large is a prime q for which you could
compute all discrete logarithms base b when b
is a generator of (Z_q)*. What are the results of your computations?
- Use exhaustive key search to decrypt the following ciphertext, which
was encrypted using a shift cipher: JBCRCLQRWCRVNBJENBWRWN .
- Consider the following linear recurrence over Z_2 of degree four:
z_(i+4) = (z_i + z_(i+3)) mod 2, for i greater or equal to 0.
For each of the 16 possible initialization vectors (z_0 , z_1 , z_2 , z_3)
determine the period of the resulting keystream.