Grader: Ms. Xinxuan (Jennifer) Wang

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

- 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.

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.

The take-home final exam will be due **Tuesday, May 2, 2023** 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. The take-home final exam will also include a 5-page written
report on a topic of your choice related to the course and approved in
advance by the instructor.

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 24, 2023 and it will be due in class on Tuesday, January 31, 2023.

First midterm will be assigned on Tuesday, January 31, 2023 and it will be due in class on Tuesday, February 14, 2023.

Please bear in mind that everyone is expected to submit written solutions individually. That is, even if in some cases your work may be a result of group discussions, each person is responsible to write up the solutions in their own words by themselves. Please take the time to show all your work and provide a detailed explanation of your reasoning in your own words.

- 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, January 31, 2023.

Please bear in mind that everyone is expected to submit written solutions individually. That is, even if in some cases your work may be a result of group discussions, each person is responsible to write up the solutions in their own words by themselves. Please take the time to show all your work and provide a detailed explanation of your reasoning in your own words.

- Exercise 1.4c only on p. 49 of the textbook.
- Exercise 5.11b only on p. 285 of the textbook.
*Small project*: 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 14, 2023.

Please bear in mind that everyone is expected to submit written solutions individually. That is, even if in some cases your work may be a result of group discussions, each person is responsible to write up the solutions in their own words by themselves. Please take the time to show all your work and provide a detailed explanation of your reasoning in your own words.

- Consider the LFSR with m = 3, the connection
polynomial 1+x+x
^{3}, 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 21, 2023.

- 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, February 28, 2023.

- Exercise 1.34c(i) only on p. 55 of the textbook.
- Exercise 1.36b(ii) only on p. 55 of the textbook.
- Exercise 2.35b only on p. 114 of the texbook.
- Exercise 2.36b only on p. 114 of the texbook.
- 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 21, 2023.

- 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 4, 2023.

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 to me by 12 noon on Tuesday, April 11, 2023.
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 2, 2023.

*Please take the time to show
all your work and provide a detailed explanation of your reasoning in your
own words.*

- 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 2, 2023.