CMPSCI 601: Theory of Computation

Homework Assignment #7

Due in discussion class Friday 23 April 2010, or to the CMPSCI main office by 4:00 p.m. that day (due date changed 15 April).

Most of these questions are from the textbook, Computational Complexity: A Modern Approach by Arora and Barak. There are seven questions for 50 points plus 10 extra credit.

Question 8.3 uses the definition of BP · NP and hence the definition of ≤r on page 138 of [AB]. The latter definition (7.16) contains a serious typo -- the last probability should be "Pr[C(M(x)) = B(x)]" rather than "Pr[B(M(x)) = C(x)]".

• Exercise 8.1 (20)

• Exercise 8.3 (10) (see green note above)

• Exercise 8.4 (5)

• Problem G-1 (5) Define a family H of hash functions (as in Definition 8.14 on page 152 of [AB]) to be triplewise independent if for any three distinct strings x, x', and x'' in {0,1}n and any three strings y, y', and y'' in {0,1}k, and for h uniformly randomly chosen from H, the probability that h(x) = y, h(x') = y', and h(x'') = y'' is the product of the probabilities of these three individual events. Prove that the hash function family of Theorem 8.15 is not triplewise independent. Define a hash function family that is triplewise independent.

• Exercise 8.5 (10XC)

• Exercise 9.2 (5)

• Exercise 9.5 (5)