$$Let EK$($M$)$ denote four$-$round encryption using the key K$.$ By trying all $29$ keys, show that there are no weak keys for this simplified DES$-$type algorithm. Recall that a weak key is one such that when we encrypt a plaintext twice we get back the plaintext. That is a weak key K satisfies EK$($EK$($M$))=$ M for every possible M$.($Note: For each key K$,$ you need to find some M such that EK$($EK$($M$))$ M$.)$

$($b$)$ Suppose you modify the encryption algorithm EK$($M$)$ to create a new encryption algorithm E$$K$($M$)$ by swapping the left and right halves after the four Feistel rounds. Are there any weak keys for this algorithm?