Imagine that I want to change the key in order to prevent side-channel attacks on key and to protect against big load on the one key. It is desirable that key change procedure would look like random function from key space to itself.
I want to consider the function of the type: $$ F : KeySpace \to KeySpace$$ $$ F(k) = E_k(k) $$ E is my block cipher, for example we can assume that encryption is in ECB mode.
I understand that this is maybe not the best decision: there are numerous of provable secure algos, but now I want to consider this one.
The natural question is: does it look like random function and is it really bijective? If not, then what is the size of this compressed range of the function? Are there any estimates?