Introduction to Cryptography
Spring 2004
Homework Assignments

• To be completed before Wednesday, May 5 (but does not need to be turned in):
  • Read Chapter 17 of the text.
  • Do Problems 1-4 of Section 17.4
• Due Wednesday, April 28:
  • Read Chapter 13 of the text.
  • Do Problems 1 through 6 of Section 13.5
• Due Wednesday, April 28:
  • Read Chapter 8 of the text.
  • Do Problems 2, 3, 4, 6, 8, 9, and 10 of Section 8.6
  • Do Problems 1, 2, 3, and 4 of Section 8.7
• Due Monday, April 12:
  • Read Chapter 7 of the text.
  • Do Problems 2, 3, 4, and 5 of Section 7.5
  • Do Problems 3 and 4 of Section 7.6
• Due Wednesday, April 7:
  • Read Sections 6.4 and 6.5 of the text.
  • Do Problems 2, 14, 16, and 17 of Section 6.8
  • Do Problems 4, 7, 9, 10, 11, 12, and 14 of Section 6.9
• Due Wednesday, March 31:
  • Read Sections 6.1 through 6.3 and 6.6 through 6.7 of the text.
  • Do Problems 1, 3, 6, 7, 9, 10, 11, and 13 of Section 6.8
  • Do Problems 1, 2, 3, and 13 of Section 6.9
• Due Wednesday, March 24:
  • Read Sections 3.5 through 3.9 of the text.
  • Do Problems 2, 7, 8, 9, 10, 11, 12, 15, and 16 of Section 3.11
  • Do Problems 3, 7, 8, 9, and 11 of Section 3.12
• Due Monday, March 8:
  • Read Chapter 14 of the text.
  • Do Problems 2, 3, 4, 5, 6, 8, 9, and 14 of Section 14.6
• Due Friday, February 27:
  • Read Chapter 5 of the text.
  • Do Problems 18 and 19 of Section 3.11.
  • Use row reduction to compute the inverse of the matrix used in the Mix Columns layer of Rijndael. (Recall from Math 215 that you can compute the inverse of a matrix using row reduction as follows: adjoin an identity matrix to the right hand side of your matrix. Row reduce. When you are finished, the left hand side of your matrix will be the identity and the right hand side will be the inverse of the original matrix.) Show your work. The answer is given on page 133 of the text.
  • Use the description of the S-box on pages 132-133 of the text to verify the lower right and lower left entries of the S-box given on page 130.
  • Compute the round constant r(i) (bottom of page 131) for i=4, 8, ..., 40.
  • Here is an alternative description of the Mix Columns layer: for each column

    a0

    a1

    a2

    a3

    of data, form a polynomial with coefficients in the field of 256 elements: a₀+a₁T+a₂T²+a₃T³. Multiply this polynomial by X+T+T²+(X+1)T³ and then reduce the result modulo T⁴+1. Prove that this gives the same result as the version of Mix Column described in class.
  • Compute round key 1 if round key 0 is 101010...1010 (64 repetitions of 10).
• Due Friday, February 20:
  • Read Chapter 4 of the text.
  • Do Problems 1, 2, 4, and 5 of Section 4.8.
  • Consider the simplified DES-type algorithm described in Section 4.2 of the book that we discussed in the class. Let "100011101011" be our message and let "001110001" be the key. Using the same expander function and the same S-boxes as in the book, find the encrypted message after two rounds.
• Due Friday, February 13:
  • Read Sections 3.1 to 3.4 and Section 3.10 up to the part labeled “LFSR Sequences” on page 90.
  • Do Problems 1, 3, 4, 6, 13, 14, and 17 from 3.11 and Problems 2, 4, 5, and 6 from 3.12.
• Due Friday, February 6:
  • Read Sections 2.6 through 2.10 of the text.
  • Do Problems 10, 11, and 12 of Section 2.13 and Problems 6 and 10 of Section 2.14. For 2.14.6, your "code" can be a function in Maple, Mathematica, or Matlab.
  • Suppose that in the course of breaking an affine cipher you arrive at the following system of equations (mod 26): 2a + 4b = 22 and 6a + 10b = 8. Explain precisely what, if anything, you can conclude about a and b from these equations.
• Due Friday, January 30:
  • Read Sections 2.4, 2.5, and 2.3 of the text.
  • Do Problems 8, 15, and 16 of Section 2.13 and Problem 9 of Section 2.14.
  • Try to decrypt this substitution cipher and these substitution and Vigenère ciphers.
  • Extra credit: another substitution cipher.
• Due Friday, January 23:
  • Read Chapter 1 and Sections 2.1 and 2.2 of the text.
  • Do Problems 2, 4, 5, 6, and 7 of Section 2.13 and Problems 2 and 3 of Section 2.14.