#### CSCI 7000 - Cryptography - Spring 2003

### Problem Set #1

#### Due: Feb 4th, 2001 at 11am

1. Assume Pr[E] > 0 and Pr[F] > 0. Prove Pr[E | F] = Pr[E] iff
Pr[F | E] = Pr[F].
2. Recall the definition of **mutually independent events**. In class,
we mentioned the defintion of **pairwise independence** as well, but
didn't write it on the board. Here it is again: Let X_{1}, ...,
X_{n} be n events (from the same sample space). We say that the
set of events {X_{1}, ... , X_{n}} is
**pairwise independent** iff for any distinct i,j in {1, ..., n},
X_{i} is independent of X_{j}.

Give an example of a collection of events which are pairwise independent,
but not mutually independent. Clearly explain why your example exhibits
these properties.

3. "The Drunken Professor"

Your professor shows up drunk to class (typical, eh?). And proceeds to
hand back n corrected homework assignments to the n students in the class.
But he's so intoxicated that instead of doing this correctly, he hands them
back to random people (ie, the mapping of papers to students is a uniform
random permutation). What is the expected number of students who will
receive his own homework back?

(Hint: consider breaking into smaller RVs and using the linearity of
expectation to help you.)

4. A Dealer shuffles a deck of cards perfectly (ie, each of the possible 52!
orderings of the deck are equally likely). Then he deals 26 of these cards
to Alice and the remaining 26 cards to Bob. Eavesdropper Eve is listening in,
but cannot see any of the cards.

- Alice has a message M of 48 bits. Give a method whereby Alice can
send M to Bob by saying something out loud (Eve can hear this!) such that
Eve obtains
*zero* information about M.
- Now suppose Alice's message M is 49 bits instead of 48 in the
preceeding scenario. Prove that no protocol exists which allows
Alice to communicate M to Bob in such a way that Eve obtains no
information about M.
(To say that Eve learns nothing means that the probability
of Alice saying S is independent of the message. More precisely,
for all messages M1, M2, we have that
Pr[ Alice says S | M = M1 ] = Pr[ Alice says S | M = M2]

where the probability is taken over the shuffles of the cards.)