Assignments
Introduction to Cryptography
CSCI-462-03, Spring 2025, Semester 2245

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. Problems and exercise numbers refer to the second edition of the textbook from 2024.

Assignment 1, due Tuesday, January 28

Warm-up assignment:

Solve problems 1.2, 1.3 and 1.4 from Chapter 1, pages 28-29.

Assignment 2, due Friday, February 7

  1. Easy mod.
    Solve problems 7, 8, 9 and 10 pages 31-32.
  2. More warm up.
    Solve problems 13, 15 from Chapter 1, pages 33.
  3. Some work.
    Solve problems 4, 7, 8 from Chapter 2, pages 68-69.
  4. Some more work.
    Solve problems 10, 12 from Chapter 2, pages 69-70.

Assignment 3, due Tuesday, February 18

  1. Solve problems 3.1, 3.4, 3.5, 3.9, 3.11 from Chapter 3, pages 106-108.
  2. Solve problems 3.12 and 3.14 from Chapter 3, pages 108-109.
  3. Solve problems 5.4, 5.5, 5.9 from Chapter 5, pages 171-172.
  4. Solve problem 5.17 from Chapter 5, page 174.

Assignment 4, due Saturday, March 1

In all solutions show the details of your work.
  1. Solve problems 4.4, 4.5, 4.6 from Chapter 4, page 143.
  2. Solve problems 4.9.1, 4.10, 4.14 from Chapter 4, pages 144-145.
  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 01101001*01010011 in GF(256), using the AES irreducible polynomial.

Midterm Exam, Tuesday, March 4, 5pm-6:15pm, room 70-2590

Assignment 5, due Monday, March 31

Show the details of your work.
  1. Find the value of the Euler totient function φ(n), for n = 831, 833, 834, 835, 837 and 839. Show the details of computations.
  2. Find all primitive elements (generators) modulo 127. Attach the program which you used to generate them.
  3. Solve problems 6.2, 6.3, 6.4 (skip 6.4.4) from Chapter 6, page 201.
  4. Solve problems 6.5, 6.6 (skip 6.6.4), 6.10 from Chapter 6, pages 201-202.

Assignment 6, due Thursday, April 10

RSA and CRT

  1. Solve problems 7.1, 7.2.1, 7.2.2, 7.4, 7.6, 7.7 from Chapter 7, pages 235-236.
  2. Solve problems 7.12, 7.13, 7.14 from Chapter 7, pages 237-238.

Assignment 7, due Wednesday, April 23

  1. Solve problems 1, 2, 4, 10 from Chapter 8, pages 270-271.
  2. Solve problems 11 and 13 from Chapter 9, page 297.
  3. Solve problems 5, 14 and 15 from Chapter 10, pages 331-334.


Assignment 8, cryptographic hashing

This assignment will not be graded. A question about hashing (Chapter 11) is to be expected in the final exam.

  1. Solve problems 1 (3 out of 12 parts), 3 and 4 from Chapter 11, page 374.
  2. Suppose that you can compute, store, and check for collisions 2000000 instances of SHA-1(x) in one second (this would require significant resources). How long do you have to run such computations to have a probability at least 1/100 of finding a collision?
  3. Compute the probabilities that there is no birthday collision among t people (as on pages 342-344), for 11 <= t <= 38.


Final Exam, Thursday, May 1, 4:15pm - 6:45pm, room 70-2590


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