You should be able to solve this.

A group of jealous professors is locked up in
a room. There is nothing else in the room but pencils
and one tiny scrap of paper per person. The profes-
sors want to determine their average (mean, not me-
dian) salary so that each one can gloat or grieve over
his or her personal situation compared to their peers.
However, they are secretive people, and do not want to
give away any personal salary information to anyone
else. Can they determine the average salary in such a
way that no professor can discover any fact about the
salary of anyone but herself? For example, even facts
such as "three people earn more than $40,000" or "no
one earns more than $90,000" are not allowed.

This means one person cannot tabulate and average anonymously collected salary data?

>>2
Right, 'cuz then that person would know what everyone's salaries are (even if they didn't know which salary belonged to which person).

How many people are in the group?
If there is enough, divy up the piece of paper between them; have them put just the number on the paper, fold it in half, place it into a hat or garbage bin to be scrambled up and then place them all back on the table face up. This will give everyone a chance to see what he or she makes compared to the rest; only one of them will KNOW for sure that he/she is the highest paid. Everyone else will feel like shit.

>>1
>Can they determine the average salary in such a
>way that no professor can discover any fact about the
>salary of anyone but herself?

Women professors? Surely you jest. Women are not supposed be educated, this question is invalid due to its impossible assumptions.

Not in general, no.
For instance, if the group consists of just two professors, the average itself would give each one perfect information on the other one's salary, given that he knew his own.
Similarly, in any group size, the average by its nature gives away information like "No one earns more than n\bar{x}", and "At least one person earns more than \bar{x}"

Why am I doing your discrete homework for you? Ah well.

Let's rephrase the requirement to be no n-2 or less professors can know the salaries of any other professors in a group of n.

The setup:
Have the professors decide on a number m that the average of the salaries will be less than. If they bitch about how discussing that m might reveal too much, take an assload of primes, repeat the following procedure a bunch of times, and then use Chinese Remainder Theorem.

The procedure:
Everyone but one dude takes their piece of paper and writes down a number between 1 and m. This one dude adds up all of the numbers mod m, and puts the additive inverse of their sum on his paper. The papers go in a hat. Everyone takes a number, adds their salary to it, and then rips off and eats the original number. They add all of their numbers together, and they will get the sum of their salary mod m. Given appropriate assumptions about the total sum of every salary, their average salary should be fairly simple to find.

>>7
They add all of their numbers together (mod m, right?), and they will get the sum of their salary mod m

Assuming there's an eraser at the end of the pencils, there's a simpler, though less secure method: everybody writes down a random number and passes the paper to the person next to them.  The next person adds his salary to the number, and erases the original number.  This continues until the paper returns to the original profs.  They add their own salary to the resulting number, subtract the original random number, divide by the # of profs in the room, and we're done.

The first one adds a random number (like 2450) to his salary, writes the sum down on the paper and passes it to the next person. The next one adds his salary and another random number, and so on.
Then the paper returns to the first guy and he'll substract his random number from the sum. The next guy substracts his random number. So the paper passes everyone two times.
At the end they have the sum of all salaries from which we can get the average.

>>10. I assume you are refining the method in >>9. For the nth guy, the n-1th guy and the n+1th guy can work together to figure out his salary. The n-1th guy and n+1th guy have enough information from the first go-round to determine (nth guys salary + random number) and enough information from the second go-round to determine (random number), which combined give you the nth guy's salary.

>>10
There's a puzzle involving sending a diamond through a corrupt postal system. Any unlocked boxes sent through the mail will be opened and their contents stolen, any keys sent through the mail will be stolen as well, just for kicks. How do you send your friend this diamond?

The answer is you put a lock on a box containing the diamond, send that to your friend, he puts his own lock on, sends it back, you unlock your lock, and send the box back to him. The box now only has his lock on it, he unlocks it and now has the diamond.

Your method is trying to do the same with encryption, each professor puts his lock on, stores his diamond, and then they all remove the locks in turn. However, because the process is (apply encryption1, apply encryption2, apply decryption1, apply decryption2), the encryption is necessarily commutative. I believe that for all commutative encryptions, seeing the result before and after decryption reveals the decryption key, and ruins the encryption, so the locked box method cannot be applied to cryptography.

I might be wrong about that though. Anyone care to go more in depth?

>>12
Oh crap, I forgot that they can collude to learn about the other profs.  I assumed that all the profs were reclusive, and would use only what they know to learn about the other profs.  I guess my plan fails.

>>13
In a semirealistic scenario, you still have the problems
a) (authentication?) are you sure it was your friend's lock on that diamond, and not somebody who intercepted the box, and guessed your algorithm?
b) (corruption) malicious 3rd parties can add their own locks for the shits & giggles, thus depriving everybody of the diamond

Each professor tears an arbitrary number of small pieces off the scrap of paper and writes on them numbers summing up to the professor's salary.
They can then each covertly put their pieces of paper into a covered box, and sum up the results.

>>14
RSA has these problems too. We're talking encryption, not the overall process of secure communication.