3.4 The Adversary

The adversary \(\Adv \) has the same shape: a single per-\(\id \) interface \(\id .\fopl {A}\) guarding and silencing the core \(\id .\op {A}\). What separates it from the environment is who may reach it. Where \(\Zenv .\admits \) is only \(\Apid \cup \Zpid \), the field \(\Adv .\admits = \Stdpid \cup \Apid \cup \Zpid \) holds every process id that ever places a call. Only \(\Corr \)’s is missing, and \(\Corr \) places none. That width is a requirement rather than a convenience: the mediation hooks of Chapter 2, and the calls ideal functionalities place through the abbreviation \(\Adv (\cdot )\), are both calls placed on the adversary by a machine acting for some party.

These interfaces are what a backdoor looks like in this notation. The adversary holds no token of its own and cannot begin a call, so every attack it mounts starts from another machine handing it control: a party’s interface through \(\opl {Mediate}\), or the environment directly, which is how an environment instructs an adversary at all. What it may say in return is narrow: the silenced set is \(\{\id '' : \id '' \ne \id \}\), so an interface of \(\Adv \) may claim only the identity it runs at.

Inside an interface with identity \(\id \) we write \(\Adv (in)\) for the call \((A,\id .P).\fopl {A}(in)\) as \(\id \), and \(\Zenv (in)\) likewise. The claim must be \(\id \) and no other, since line 5 silences a core on everything but its own identity; the adversary therefore learns the caller as the claimed id, and needs no separate field for it. Its core is entered as \(\id .\op {A}(in)\) from \(\id '\), the shape \(\opl {Silence}\) declares and the shape every other core is entered in. The target’s party component is \(\id .P\), so the call also records on whose behalf it is made.