Cryptography I
VCSG-705/VCSS-482
Winter 2012/2013

Assignment 1, due Tuesday, December 11

1. Find the value of the Euler totient function phi(n), for n = 928, 930, 931, 932, 933, 934 and 935. Show the details of computations.
2. Compute 37³⁷ modulo 521, using the modular exponentiation algorithm. Show the details of computations.
3. Compute order of a modulo n, for all possible a, for n = 17, 21 and 25. If you wish, you may write a simple program. For all cases, identify all primitive roots.
4. Use extended Euclid algorithm to find the multiplicative inverse of 55 modulo n, if it exists, for n = 1033, 1034, 1035 and 1036. Show the details of computations.

5. For each of the following compute the value of or argue that it is not defined. For at least two of the five cases below, do the computations without using the program from the previous question, and describe briefly how you have done it.

   discrete logarithm of 2 base 3 mod 11
   discrete logarithm of 2 base 3 mod 19
   discrete logarithm of 3 base 3 mod 97
   discrete logarithm of 3 base 4 mod 97
   discrete logarithm of 3 base 5 mod 97

6. Solve exercise 8 page 226. Show the details of your work.

Assignment 2, due Thursday, December 20, 2012

1. Solve exercise 5 page 226. Show the details of your work.
2. Solve exercise 6 page 226.
3. Solve exercise 7 page 226.
4. Solve exercise 12 page 227. Show the details of your work.
   The ciphertexts from Tables 5.1 and 5.2 are available.

Assignment 3, due Tuesday, January 8, 2013

Perform cryptanalysis of four ciphertexts which were obtained using substitution cipher, affine cipher, and the Vigenere cipher. You have to determine which cipher was used in each case. The frequencies of characters, digrams and trigrams can be obtained using the program freq.c, written by Adam Kropelin. The header contains information about the usage.

Submit hardcopy of all your work (including source of your code), and explain in detail how you arrived to the solution. Just recovering the plaintexts is not sufficient.

Hints

• The general substitution cipher was applied to a plaintext in English.
• In Polish, the letter "o" is very common, while the letters "x", "v" and "q" are used rarely.
• In Spanish, the letters "e", "a" and "o" are very common, while the letters "k", "w" and "x" are used rarely.

Midterm Exam, Tuesday, January 15, 6pm-7:50pm, 70-3560

Assignment 4, due Thursday, January 31, 2013

1. Solve exercise 3 page 114 (DES complements).
2. Find all irreducible polynomials in Z₃[x] of degree 3 with leading coefficient 1. Show the details of your work.
3. The polynomial x⁴ + x + 1 is irreducible in Z₂[x]. Using square-and-multiply algorithm compute x¹¹ mod x⁴ + x + 1 in Z₂[x], i.e. in the Galois field 2⁴. Show the details of computations.
4. Find orders of all elements of the field GF(9) represented modulo irreducible x² + 1 over Z₃ (given the table of powers of g in the handout, you actually don't need to perform any multiplications in GF(9)). Explain how you obtained the results.
5. Compute 10101001*11010011 in GF(256), using AES field representation as in section 3.6.1. Perform your calculations by hand, show the details of work.

Assignment 5, due Thursday, February 14, 2013

Primality Testing

A note on page 187 says that the error probability of the Miller-Rabin primality test is at most 1/4. You will now test this claim experimentally.

• Find the largest error probability of Miller-Rabin primality testing algorithm for odd integers between 85,000 and 87,000. It is ok to brute-force. In your answer, you should give this largest error probability (with decimals of precision) and 10 odd integers between 85,000 and 87,000 for which the highest error probability occurs.
• Submit a commented hardcopy of your code, and a clear description of what you did. You are encouraged to compare your error probabilities with your classmates.
• If you want more of a challenge, find the largest error probability of Miller-Rabin for odd integers from intervals bounded by larger numbers. Your answer should have the same format as the case above.

Back to the course page