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
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.
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.