Universal composability

For an optimal ideal authenticated channel, the ideal functionality ${\displaystyle {\mathcal {F}}_{\mathsf {Auth}}}$ takes a message ${\displaystyle m}$ from a party with identity ${\displaystyle P}$ as input, and outputs the same message together with the identity ${\displaystyle P}$ to the recipient and the adversary. To model the power of the adversary to delay asynchronous communication the functionality ${\displaystyle {\mathcal {F}}_{\mathsf {Auth}}}$ may first send a message to the adversary ${\displaystyle {\mathcal {A}}}$ and would only deliver the message ${\displaystyle m,P}$ once it receives the command to do so as a reply.

In an ideal secure channel, the ideal functionality ${\displaystyle {\mathcal {F}}_{\mathsf {Sec}}}$ only outputs the identity of the sender to both the recipient and the adversary, while the message is only revealed to the recipient. This models the requirement that a secure channel is both authenticated and private. To model some leakage about the information that is being transferred, ${\displaystyle {\mathcal {F}}_{\mathsf {Sec}}}$ may reveal information about the message to the adversary, e.g. the length of the message. Asynchronous communication is modeled through the same delay mechanism as for ${\displaystyle {\mathcal {F}}_{\mathsf {Auth}}}$.

While the technical means, and the physical assumptions behind anonymous and pseudonymous communication are very different,[2] the modeling of such channels using ideal functionalities is analogous. See also onion routing and Anonymous P2P. Similar functionalities can be defined for broadcast communication, or synchronous communication.

In an ideal anonymous channel, the ideal functionality, ${\displaystyle {\mathcal {F}}_{\mathsf {Anon}}}$ takes a message ${\displaystyle m}$ from a party with identity ${\displaystyle P}$ as input, and outputs the same message but without disclosing the identity ${\displaystyle P}$ to the recipient and the adversary.

In an ideal pseudonymous channel, the participating parties first register unique pseudonyms with the ideal functionality ${\displaystyle {\mathcal {F}}_{\mathsf {Pseu}}}$. To do a transfer ${\displaystyle {\mathcal {F}}_{\mathsf {Pseu}}}$ takes a message ${\displaystyle m}$ and the pseudonym ${\displaystyle nym}$ of the recipient as input. The ideal functionality looks up the owner of the pseudonym and transfers the message ${\displaystyle m,nym}$ without revealing the identity of the sender.

These formalisations abstract from the implementation details of the concrete systems that implement such channels. In their pure form an ideal functionality may be found to be unrealizable. It may be necessary to relax the functionality by leaking more information to the adversary (degree of anonymity). On the other hand communication channels can be physical,[3][4] e.g. a mobile device can achieve an anonymous channel by constantly changing its location before transmitting messages that do not contain identifiers.