## Spring 2018

## MATH 341 / LGIC 220, T Th 12-1:30, DRL 4C2

## Discrete Mathematics II

*Office*: Please go to Room DRL 4W1 or 4W5 and the Math Department
staff will conduct you to the Chair's office.

*Office Hours*: Mondays 10:30 - 1:30 starting January 29, or by
appointment. Office hours February 12 rescheduled to Friday, February 9,
11:30 - 2:30.

## Suggested Text

- Jeffrey Hoffstein, Jill Pipher, and Joseph H. Silverman:
"An Introduction to Mathematical Cryptography", Springer.
*
First edition is freely available online.
A hard copy of the 2nd edition
is on reserve in the Mathematics Library on the 3-rd floor of DRL.*

## Further References

## Topics

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.

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

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.

Selected topics from modern cryptography and computer network security.

## Homework #1 Due in Class on Tuesday, February 13

- Exercise 4.23 on pp. 268-269 of Hoffstein
*et al.*, 1st edition.
- There are four biased coins A, B, C, and D which land heads and tails
with the following probabilities: A: heads 1/5, tails 4/5, B: heads 2/5,
tails 3/5, C: heads 3/5, tails 2/5, and D: heads 4/5, tails 1/5. One of the
coins is selected at random and flipped three times, producing one heads
followed by two tails. Which is the most probable coin and what is the
probability it is that coin?
- Exercise 4.34(a) on p. 272 of Hoffstein
*et al.*, 1st edition.
- Exercise 4.42 on pp. 274-275 of Hoffstein
*et al.*, 1st edition.
- Exercise 4.43 on p. 275 of Hoffstein
*et al.*, 1st edition.

This is the complete set of problems for Homework #1 due in class
on Tuesday, February 13, 2018.