- 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.

- 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.

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.

- 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).

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.