## Math 5410 Final Exam Spring 2004

Answer all questions (provide partial solutions if you can not solve a question). Your answers should be clear, concise, and correct. As this is an exam, hand in only your own work. Exams are due in my mailbox (UCD -Building 6th floor) by 4:00, Tuesday May 11.

1. Give generator and parity check matrices for the binary code consisting of all even weight vectors of length 6.

2. Suppose the Merkle-Hellman Knapsack Cryptosystem has as its public list of sizes the vector

t = (1394, 1256, 1987, 439, 650, 724, 339, 2303, 810).
If Oscar has discovered that the prime used to set up the system is 2503,
1. Determine, by trial and error, the value of a so that a-1 t mod p is a permutation of a superincreasing list.
2. How would Oscar decrypt the ciphertext 3155? (Answer is a binary string)
3. Use the Pohlig-Hellman algorithm to find the discrete logarithm of 125 to the base 2 in Z181, i.e., solve for x : 2x = 125 mod (181). [Note: I expect to see the details of the Pohlig-Hellman algorithm, the answer alone is not sufficient, nor is any other method for obtaining it.]

4. Suppose there are four people in a room, exactly one of whom is a foreign agent. The other three people have been given pairs corresponding to a Shamir secret sharing scheme in which any two people can determine the secret. The foreign agent has randomly chosen a pair of numbers for himself. The people and pairs are as follows. All the numbers are mod 11.

A: (1,4) B: (3,7) C: (5,1) D: (7,2)

Determine who the foreign agent is and what the secret is.

5. a) Prove that if k º 2 mod 4 then we can not factor a large odd integer n using generalized Fermat factorization with this choice of k.

b) Prove that if k = 4, and if generalized Fermat factorization works for a certain t, then simple Fermat factorization (with k = 1) would have worked equally well.

(Note: In simple Fermat factorization one checks the sequence of integers t2 - n, where t = [sqrt n] +1, [sqrt n] + 2, ...., for squares. In the generalized Fermat factorization, for a small integer k, one checks the sequence where t = [sqrt (kn)]+1, [sqrt (kn)]+2, ... .)

6. Let E be the elliptic curve with equation y2 + y = x3 over the field GF(16).

a) Show that every point P on E is a point of order 3. (P different from O)

b) Show that any point of E actually has coordinates in GF(4) (as a subfield of GF(16)). Then use Hasse's Theorem with q = 4 and 16 to determine the number of points on the curve.