Why are in asymmetric cryptography as many keys as people involved needed? The question: "10 friends wish to establish secure communications between themselves using asymmetric-key cryptography. How many keys need to be known by each user?" was posed by my teacher. He says 10 keys are needed. I say only 2 keys are needed. As long as they have copies of the same keypair (keypair=2 keys, a public and a private), they don't need more than two. Why in the WORLD would you need more than a public key and a private key?

Edit:

I have already seen a few questions of this type by googling. Turns out this kind of question never specifies that the people of the group must be able to talk individually to each other. Nevertheless, the student must always assume they want to talk individually and secretly. That makes no sense to me, but that's reality. Ie: In public key cryptography, how many keys are required for secure communication between n parties? 

In other words, in criptography, when they say communication between N entities, it is not actually communication between N entities, it's more like N private communications between each of the N entities.
Actually you are both wrong, assuming that each user wants to authenticate themselves individually and / or establish private conversations between pairs. Unfortunately this is assumption is missing from the question. If everybody is using (EC)DH then they need a key pair each to setup communication by establishing a session key. That means 10 times 2 keys = 20 keys. However, quite often key and key pair are confused, so the professor is really talking about the latter, only counting the private keys / key pairs, making 10 the correct answer.

If you establish a "master" user that you trust, and no entity authentication is required then you are correct. You need just one private key pair known to all, initially distributed by the master setting up the group. Then you can e.g. encrypt a session key with the public key and only those that have the private key can decrypt. Arguably this is not a good example of asymmetric cryptography though: the private key is now a shared secret, something that is common for symmetric cryptography rather than asymmetric cryptography. The sharing of the private key is required in this scheme, otherwise an adversary can join by identifying as one of your friends.

Note that just having 10 key pairs is not enough to authenticate all the users and setup 1:1 communication. It is also required to trust the public keys. Authentication is about proving you have (access to) a private key that belongs to a public key known to belong to you by the other user.



Note that I've talked about (EC)DH rather than RSA in the first section. It is common to setup (multiple) session keys in transport protocols rather than encrypting directly with the public key and decrypt with the private key. Of course, this requires a set of temporary keys in addition to the asymmetric keys.

It is possible to perform encryption / signature generation with just 10 key pairs, so session keys are not necessary. However, that's really not following good cryptographic practice. So I decided simply not to count the symmetric session keys.