Assignments
Foundations of Cryptography
CSCI-662-01, Fall 2024, Semester 2241

In all cases show the details of your work, and give brief reasons for your answers. The homeworks are due in pdf format on myCourses. Convert to single pdf before submission, submit just one pdf file for each assignment. You can use generative AI tools for developing answers to homework assignments, but in all places where such use was significant, it needs to be noted.

Assignment 1, due Friday, September 6

Warm-up assignment:

Solve problems 1.2, 1.3 and 1.4 from Chapter 1.
(pages 24-25 in edition 1, pages 28-29 in edition 2)

Assignment 2, due Wednesday, September 18

  1. Easy mod.
    Solve problems 5, 6, 7, 8 from chapter 1, i.e., 1.5, 1.6, 1.7 and 1.8 pages 25-26
    (1.7, 1.8, 1.9 and 1.10 pages 31-32 in edition 2).
  2. More warm up.
    Solve problems 11, 13 from chapter 1, pages 26-27
    (problems 13, 15 from chapter 1, page 33 in edition 2).
  3. Some work.
    Solve problems 4, 5, 6 from chapter 2, pages 52-53
    (problems 4, 7, 8 from chapter 2, pages 68-69 in edition 2).
  4. Some more work.
    Solve problems 8, 10 from chapter 2, pages 53-54
    (problems 10, 12 from chapter 2, pages 69-70 in edition 2).

Assignment 3, due Wednesday, October 2

  1. Solve problems 1, 3, 4, 7, 9 from chapter 3, pages 83-84
    (problems 1, 4, 5, 9, 11 from chapter 3, pages 106-108 in edition 2).
  2. Solve problems 10, 12 from chapter 3, pages 84-85
    (problems 12, 14 from chapter 3, pages 108-109 in edition 2).
  3. Solve problems 4, 5, 9 from chapter 5, pages 145-146
    (problems 4, 5, 9 from chapter 5, pages 171-172 in edition 2).
  4. Solve problem 16 from chapter 5, page 148
    (problems 17 from chapter 5, page 174 in edition 2).

Assignment 4, due Monday, October 14

In all solutions show the details of your work.
  1. Solve exercises 4, 5, 6 from chapter 4, page 118
    (problems 4, 5, 6 from chapter 4, page 143 in edition 2).
  2. Solve problems 13.1, 14, 17 from chapter 4, pages 120-121
    (problems 9.1, 10, 14 from chapter 4, pages 144-145 in edition 2).
    This is the complete page 121 from the first edition, some places on the web show only one of the problems 16/17.

  3. Find all irreducible polynomials in Z2[x] of degree 5.
    Which of the polynomials x5 + x4 + 1, x5 + x3 + 1, x5 + x4 + x2 + 1 are reducible in Z2[x]? If reducible, then show factors.
  4. Find all irreducible monic polynomials (with the leading coefficient at x2 equal to 1) in Z3[x] of degree 2.
  5. Compute 10011001*01100011 in GF(256), using the AES irreducible polynomial.

Midterm Exam, Wednesday, October 16, class place/time

Assignment 5, due Monday, November 4

Show the details of your work.
  1. Solve problems 2, 3, 4.1, 4.2, 4.3 (skip 4.4) from chapter 6, page 170.
    (problems 2, 3, 4.1, 4.2, 4.3 (skip 4.4) from chapter 6, page 201 in edition 2).
  2. Solve problems 5, 6, 10 from chapter 6, pages 170-171.
    (problems 5 (skip 6.5.3), 6 (skip 6.6.3 and 6.6.4), 10 from chapter 6, pages 201-202 in edition 2).
  3. Trace the execution of the Extended Euclid Algorithm, as in example 6.6 page 163 (193 in ed. 2), for gcd(663,773). Find the mutual multiplicative inverses of 663 and 773 in their respective canonical intervals (663^-1 mod 773, and 773^-1 mod 663 = 110^-1 mod 663).
  4. Find the value of the Euler totient function φ(n), for n = 831, 833, 834, 835, 837 and 839. Show the details of computations.
  5. Find all primitive elements (generators) modulo 131. Attach the program which you used to generate them.


Assignment 6, due Monday, November 18

RSA and CRT

  1. Solve problems 1, 2, 4, 6, 7 from chapter 7, pages 200-201.
    (problems 1, 2.1, 2.2, 4, 6, 7 from chapter 7, pages 235-236 in edition 2).
  2. Solve problems 12, 13, 14 from chapter 7, pages 201-202.
    (problems 12, 13, 14 from chapter 7, pages 237-238 in edition 2).
...


Final Exam, Monday, December 16, 7pm-9:30pm, 70-2590


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