Math 340 Fall 2002 MWF 12-1

Fall 2002

Math 340, MWF 12-1

Advanced Mathematical Methods in Computer Science

Professor Andre Scedrov

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

Textbooks

Ronald Graham, Oren Patashnik, and Donald Ervin Knuth. "Concrete Mathematics", Second Edition. Addison-Wesley, 1994. ISBN 0-201-55802-5.
J. H. Van Lint and R. M. Wilson. "A Course in Combinatorics". Second Edition, Paperback. Cambridge University Press, 2001. ISBN 0521006015.

Further References

R. P. Grimaldi. "Discrete and Combinatorial Mathematics". Fourth Edition. Addison Wesley Longman, 1999. On reserve in the Math/Physics/Astronomy Library.
H. Wilf. "East Side, West Side". Lecture Notes, 1999.

Topics

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 Repeated Squaring. 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

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).

This is a complete list of assignments due November 4, 2002.

Take-Home Final Exam Due in DRL 4E6 on at 12 noon on Monday, December 16

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.

This is a complete list of assignments due December 16, 2002.