**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,
- Determine, by trial and error,
the value of a so that a
^{-1} t mod p is a permutation
of a superincreasing list.
- 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 Z_{181}, i.e.,
solve for x : 2^{x} = 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 t^{2} - 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 y^{2} + y = x^{3} 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.