MATH 340 / LGIC 210 Fall 2003 MWF 12-1

Fall 2003

MATH 340 / LGIC 210, MWF 12-1, DRL 3C8

Advanced Mathematical Methods in Computer Science

Professor Andre Scedrov

We will also sometimes meet for extra classes on Fridays 1-2 in DRL 2C6.

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

R. P. Grimaldi. "Discrete and Combinatorial Mathematics". Fourth Edition. Addison Wesley Longman, 1999. ISBN 0-201-19912-2.
Johannes A. Buchmann: "Introduction to Cryptography". Springer, 2001. ISBN 0-387-95034-6.

Further References

Yiannis N. Moschovakis. "Notes on Set Theory". Undergraduate Texts in Mathematics, Springer-Verlag, 1994. ISBN 0387941800, especially Chapters 1 and 2. On reserve in the Math/Physics/Astronomy Library. Errors in this book [ ps , pdf ].
Ronald Graham, Oren Patashnik, and Donald Ervin Knuth. "Concrete Mathematics", Second Edition. Addison-Wesley, 1994. ISBN 0-201-55802-5. On reserve in the Math/Physics/Astronomy Library.
J. H. Van Lint and R. M. Wilson. "A Course in Combinatorics". Second Edition, Paperback. Cambridge University Press, 2001. ISBN 0521006015. 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.

Counting, Permutations, and Combinations, Binomial Theorem, Multinomial Theorem, Combinations with Repetition.

Recurrences, sums, and integer functions: Towers of Hanoi, Quicksort recurrence, floor and ceiling functions.

Introduction to Number Theory: Congruences, Chinese Remainder Theorem, Fermat's Little Theorem, Euler's Theorem, Modular Exponentiation by Repeated Squaring.

Asymptotic functions, Stirling's Approximation Formula, Wallis's Formula.

Overview of Probability Theory: Probability Distribution, Random Variable, Conditional Probability, Bayes Theorem, Expected Value.

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.

Public-Key Cryptology: 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.

Take-Home Midterm Due in Class on Monday, November 10

Exercise 22 on p. 249 of Grimaldi.
Exercises 20a,b on p. 260 of Grimaldi.
Exercises 23a,b on p. 261 of Grimaldi.
Exercise 7 on p. 273 of Grimaldi.
Exercise 8 on p. 273 of Grimaldi.
Exercise 13 on p. 318 of Grimaldi.
Exercises 16a,b,c on p. 318 of Grimaldi.
Exercises 1c,d on p. 444 of Grimaldi.
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 midterm assignments due November 10, 2003.

Take-Home Final Exam Due in DRL 4E6 on at 12 noon on Monday, December 15.
Please slide your exam solutions under my office door.
Please show all your work, not just the final result.

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.
Exercise 2.22.10 on p. 65 of Buchmann.
Exercise 2.22.16 on p. 66 of Buchmann.
Exercise 2.22.19 on p. 66 of Buchmann.
Exercise 2.22.25 on p. 66 of Buchmann.
Exercise 7.6.8 on p. 168 of Buchmann.
Exercise 11.6.1 on p. 231 of Buchmann.
Exercise 11.6.7 on p. 232 of Buchmann.

This is a complete list of assignments due December 15, 2003.