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: