Spring 2018

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

Discrete Mathematics II

Professor Andre Scedrov

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

• Johannes A. Buchmann: "Introduction to Cryptography". Springer, Second Edition, 2004. Paperback. ISBN 9780387207568. A copy of this book is on reserve in the Mathematics Library on the 3-rd floor of DRL.
• "Handbook of Applied Cryptography" by Menezes, van Oorschot, and Vanstone. CRC Press, Fifth Printing, 2001. ISBN: 0-8493-8523-7.

Topics

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.

Take-Home Midterm Due in Class on Thursday, March 22

• Exercise 1.1 on p. 47 of Hoffstein et al., 1st edition.
• Exercise 1.2 on p. 47 of Hoffstein et al., 1st edition.
• Exercise 1.3 on p. 47 of Hoffstein et al., 1st edition.
• Exercise 1.8 on p. 49 of Hoffstein et al., 1st edition.
• Exercise 1.9 on p. 49 of Hoffstein et al., 1st edition.
• Exercise 1.12 on p. 50 of Hoffstein et al., 1st edition.
• Exercise 1.15 on pp. 50-51 of Hoffstein et al., 1st edition.
• Exercise 1.20 on p. 51 of Hoffstein et al., 1st edition.
• Exercise 1.23 on p. 52 of Hoffstein et al., 1st edition.

Take-Home Final Due in DRL 4E6 on or before Friday, May 4, 2018 at 12 noon

• 5 page report on a topic of your choice relevant to the class topics. Please note that I need to approve your selected topic first. If you have not already done so, please let me know your choice of the report topic in class on Tuesday, April 24 or by email on or before Friday, April 27 at 12 noon.
• Exercise 2.5 on p. 106 of Hoffstein et al., 1st edition.
• Exercise 2.6 on p. 106 of Hoffstein et al., 1st edition.
• Exercise 2.8 on p. 106 of Hoffstein et al., 1st edition.
• Exercise 3.7 on p. 177 of Hoffstein et al., 1st edition.
• Exercise 3.38 on p. 185 of Hoffstein et al., 1st edition.
• Exercise 7.1 on p. 458 of Hoffstein et al., 1st edition.
• Exercise 7.4 on p. 459 of Hoffstein et al., 1st edition.