Suppose that we have a block cipher and want to use it as a hash function. Let X be a specified constant and let M be a message consisting of a single block, where the block size is the size of the key in the block cipher. Define the hash of M as Y $=$ E$($X$,$ M$).$ Note that M is being used in place of the key in the block cipher.

a$.$ Assuming that the underlying block cipher is secure$,$ show that this hash function satisfies the collision resistance and one$-$way properties of a cryptographic hash function.