Spring 2022

MATH 341 / LGIC 220, T Th 10:15 - 11:35 a.m. EST, still online on Tuesday, January 25.

Starting Thursday, January 27, the class will meet in person in DRL 4C4.

Discrete Mathematics II

Professor Andre Scedrov

Prerequisites

Textbook

• Jeffrey Hoffstein, Jill Pipher, and Joseph H. Silverman: "An Introduction to Mathematical Cryptography", Second edition, Springer, 2014.

Further References

• Johannes A. Buchmann: "Introduction to Cryptography". Springer, Second Edition, 2004. Paperback. ISBN 9780387207568.
• "Handbook of Applied Cryptography" by Menezes, van Oorschot, and Vanstone. CRC Press, Fifth Printing, 2001. ISBN: 0-8493-8523-7.

Topics Covered

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. Finite Fields. Splitting Fields. Quadratic Residues. Legendre Symbol. Jacobi Symbol. Law of Quadratic Reciprocity.

Public-Key Cryptosystems: Diffie-Hellman Key Exchange, Person-in-the Middle Attack. Discrete Logarithm. RSA Public-Key Cryptosystem. Attacks on RSA. ElGamal Public-Key Cryptosystem. Digital Signatures, Selective Forgery, Existential Forgery. Signature Schemes Based on RSA. Signature Schemes Based on Discrete Logarithm: ElGamal Signature Scheme, Digital Signature Algorithm (DSA).

Selected topics from modern cryptography and computer network security, including: Probabilistic Primality Testing, Euler Pseudoprimes, Solovay-Strassen Primality Test, Strong Pseudoprimes, Miller-Rabin Primality Test. Hash Functions.

Basic Course Information

There will be two take-home midterms, the first one due in class on Tuesday, February 15, 2022 and the second one due in class on Tuesday, April 5, 2022. First midterm will be worth 20% of the grade. Second midterm will be worth 25% of the grade. Each midterm will have at least a two-week lead time, during which there will be no homework.

The take-home final exam will be due Tuesday, May 3, 2022 during the final exam period and will be worth 30% of the grade. The take-home final exam will also have at least a two-week lead time, during which there will be no homework.

Most other weeks during the semester there will be homework, each time with at least one week lead time. Total homework will be worth 20% of the grade. One lowest score homework can be dropped.

First homework will be assigned on Tuesday, January 25, 2022 and it will be due in class on Tuesday, February 1, 2022.

First midterm will be assigned on Tuesday, February 1, 2022 and it will be due in class on Tuesday, February 15, 2022.

Homework #1 due in pdf in Canvas by 12 noon on Tuesday, February 1, 2022

Please show all your work and explain your reasoning.

• Exercise 1.1abc on pp. 47-48 of the textbook.
• Exercise 1.4a only on p. 48 of the textbook.
• Exercise 5.10a only on p. 284 of the textbook.
• Exercise 5.11a only on p. 285 of the textbook.
• Exercise 5.24ab only on p. 289 of the textbook.

This is the complete set of problems for Homework #1 due in pdf in Canvas by 12 noon on Tuesday, February 1, 2022.

Midterm #1 due in pdf in Canvas by 12 noon on Tuesday, February 15, 2022

Please show all your work and explain your reasoning.

• Exercise 1.4c only on p. 49 of the textbook.
• Exercise 5.11b only on p. 285 of the textbook.
• Extra Credit: Exercise 5.17 on p. 286 of the textbook.
• Exercise 5.27ab only on p. 290 of the textbook.
• Exercise 5.45ab only on p. 296 of the textbook.

This is the complete set of problems for Midterm #1 due in pdf in Canvas by 12 noon on Tuesday, February 15, 2022.

Homework #2 due in pdf in Canvas by 12 noon on Tuesday, February 22, 2022

Please show all your work and explain your reasoning.

• Consider the LFSR with m = 3, the connection polynomial 1+x+x³, and initial content 011. Find the maximal output sequence with period 7 and verify that the repeating cycle satisfies the three randomness postulates of Golomb. Please see Handbook 5.4.3 on p. 180 and 6.2.1 on pp. 195-197.
• Exercise 1.9a only on p. 50 of the textbook.
• Exercise 1.10 on p. 50 of the textbook regarding 1.9a only.
• Exercise 1.11a only on p. 50 of the textbook.
• Exercise 1.16a only on pp. 51-52 of the textbook.

This is the complete set of problems for Homework #2 due in pdf in Canvas by 12 noon on Tuesday, February 22, 2022.

Homework #3 due in pdf in Canvas by 12 noon on Tuesday, March 1, 2022

Please show all your work and explain your reasoning.

• Exercise 1.16b only on pp. 51-52 of the textbook.
• Exercise 1.22a only on p. 52 of the textbook.
• Exercise 1.26a only on p. 53 of the textbook.
• Exercise 2.18d only on p. 111 of the textbook.

This is the complete set of problems for Homework #3 due in pdf in Canvas by 12 noon on Tuesday, March 1, 2022.

Homework #4 due in pdf in Canvas by 12 noon on Tuesday, March 22, 2022

Please show all your work and explain your reasoning.

• Exercise 1.34c(i) only on p. 55 of the textbook.
• Exercise 1.36b(ii) only on p. 55 of the textbook.
• Let f(x) = 1+x²+x³+x⁴, g(x) = 1+x³, both over F₂[x]. Compute h(x) = g.c.d.(f(x),g(x)) and find u(x) and v(x) over F₂[x] such that h(x) = u(x)f(x) + v(x)g(x).
• Exercise 2.39a only on p. 114 of the textbook.

This is the complete set of problems for Homework #4 due in pdf in Canvas by 12 noon on Tuesday, March 22, 2022.

Midterm #2 due in pdf in Canvas by 12 noon on Tuesday, April 5, 2022

Please show all your work and explain your reasoning.

• Exercise 1.11b only on p. 50 of the textbook.
• Exercise 1.16c only on pp. 51-52 of the textbook.
• Exercise 1.34c(ii) only on p. 55 of the textbook.
• Exercise 1.36b(iii) only on p. 55 of the textbook.
• Exercise 3.6b(i) only on p. 181 of the textbook.
• Exercise 2.38 on p. 114 of the textbook.
• Exercise 3.39b(i) only on p. 190 of the textbook.
• Exercise 2.4b only on p. 108 of the textbook.
• Exercise 2.6 on p. 108 of the textbook.
• Exercise 3.8 on p. 182 of the textbook.

This is the complete set of problems for Midterm #2 due in pdf in Canvas by 12 noon on Tuesday, April 5, 2022.

Topics for a 5-page report due by email by 12 noon on Tuesday, April 12, 2022

A part of the final exam will be a 5-page report on a topic of your choice related to the course. Your choice of the topic for the report must be submitted for my approval by email by 12 noon on Tuesday, April 12, 2022. Please note that for now this is only your choice of the topic. The 5-page report itself will be due together with the final exam on May 3, 2022.

Final Exam due in pdf in Canvas by 12 noon on Tuesday, May 3, 2022

Please show all your work and explain your reasoning.

• 5-page written report on the approved topic.
• Exercise 5.45cd only on p. 296 of the textbook.
• Exercise 1.36c only on p. 55 of the textbook.
• Exercise 3.41b only on p. 190 of the textbook.
• Exercise 2.8abcd on pp. 108-109 of the textbook.
• Exercise 3.15c on p. 184 of the textbook.
• Exercise 5.36a only on p. 293 of the textbook. Please provide the complete computation.

This is the complete set of problems for the Final Exam due in pdf in Canvas by 12 noon on Tuesday, May 3, 2022.