UDP... how are connections differentiated from behind NAT routers?

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http://serverfault.com – I have a very simple UDP chat server written in PHP, running on a Ubuntu VPS. It seems to be working, but only as long as there is no more than one client per site, The information I know how to extract is the sender I.P., the sender port, and the message... To the point: What methods might I use to differentiate between multiple clients behind NAT routers? My current idea is to use some kind of a unique key for each client, but it'd be great if that could be avoided.... So, thanks in advance for any help! (HowTos)