Given an encryption scheme that takes as input a key $k$ and converts it to an encryption operation $E$ via the function $M(k) = E$, is it possible that more than one $k$ can give the same $E$?
For example, let $k$ and $k’$ be two distinct keys. Is it possible that $M(k) = M(k’) = E$. In this sense, the mapping from the ‘key space’ to the ‘encryption operation space’ is surjective.
It doesn’t seem like it would be a vulnerability or flaw me given the number of different possible encryption operators given by the scheme is large enough, and that each distinct encryption operator is mapped onto by roughly the same number of keys. For instance, let’s say the key space for a given key of length $n$ is $2^n$, but the corresponding encryption operation space is of size $2^{frac{n}{2}}$. For large values of $n$, although a given key would have many ‘collisions’ (other keys that give the same encryption operator), the protocol could still be secure.
