  We consider the communication scenario where K transmitters are each
connected to a common receiver with an orthogonal noiseless link. One of the
transmitters has a message for the receiver, who is prohibited from learning
anything in the information theoretic sense about which transmitter sends the
message (transmitter anonymity is guaranteed). The capacity of anonymous
communications is the maximum number of bits of desired information that can be
anonymously communicated per bit of total communication. For this anonymous
communication problem over a parallel channel with K transmitters and 1
receiver, we show that the capacity is 1/K, i.e., to communicate 1 bit
anonymously, each transmitter must send a 1 bit signal. Further, it is required
that each transmitter has at least 1 bit correlated randomness (that is
independent of the messages) per message bit and the size of correlated
randomness at all K transmitters is at least K-1 bits per message bit.
