Let denote the space of atomic messages. The set of all messages over some set of atomic messages is defined inductively as follows:
Because keys have inverses, we take this space modulo the equivalence . It is also important to note that we make the following perfect encryption assumption. The only way to generate is from m and k . In other words, there do not exist messages and and key k such that , and implies m = m' and k = k' .
Let be a subset of messages. The closure of B (denoted ), representing the set of everything that can be derived from B , is defined by the following rules: