Clarification for HW1   (posted on 10/1/07)

Question 5 on the homework asks you to show that a certain hash family
is 2-universal. In class, I defined a 2-universal hash family as one
satisfying the following two properties:
1. For all x in U and i in [n], Pr_{h in H} [ h(x) = i ] = 1/n
2. For all x and y in U, Pr_{h in H} [ h(x) = h(y) ] = 1/n

All good hash families don't satisfy these properties exactly. A
somewhat relaxed definition of 2-universal families is as follows:
1. For all x in U and i in [n], Pr_{h in H} [ h(x) = i ] <= (1+o(1)) 1/n
2. For all x and y in U, Pr_{h in H} [ h(x) = h(y) ] <= 1/n

Part 1 still ensures that the probability mass is reasonably well
distributed. The negative correlation in part 2 is only better for
applications where we want to avoid collisions.

For the question on the homework, in order to prove part 1, you should
simply show that the probability approaches 1/n as p increases.