Recognising one's own messages on an anonymous broadcast channel?

Dear Cryptographers,

We're currently working on a design of a communications systems and
have stumbled upon a problem which we suspect has a solution in the
world of cryptography. If not, perhaps someone can point us in the
right direction?

Problem:

A user sends operations to our server on a private channel. The result
is broadcast to all users (cannot be sent on the private channel).

1. The result must be anonymous, i.e. no other client must be able to
identify which client that sent the operation to the system or that
two operations belongs to the same client.
2. The client that sent the operation must be able to identify the
result as his own.
3. No collisions! I.e. no client should mistakenly identify a result
as his own.

Prerequisites:

o It's ok to add "user id:s", "correlation id:s" etc. to the input
messages.
o It's ok for the server and client to share secrets, if necessary.
The input channel is a persistent connection.
o The algorithms have to be message-based rather than stream-based.
o The algorithms used have to be "fast" since a quite high number
(100k) of messages are processed per second by the common server.

.

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