Which one of the following is a correct statement regarding a cryptographic Hash function?

a$.$

Changing a single bit in the input of a cryptographically strong hash function should lead to a change of exactly half of the bits in its output.

b$.$

The pre$-$image resistance $($also known as the one$-$way property$)$ means that given a message digest $($hash output$),$ it should be computationally difficult to find the message $($the input$)$ that produced it$.$

c$.$

According to the birthday attack $($birthday paradox$),$ if the output length of a cryptographic hash function is n bits, which means there are $2^$n possible hash outputs, after trying about half of this many random inputs, you will find a collision with probability of about $50\%.$

d$.$

The "second pre$-$image resistance" property is neither stronger nor weaker than the "first pre$-$image resistance" property. That is$,$ we cannot conclude first pre$-$image resistance property from second pre$-$image resistance property, or vice versa.

e$.$

No two inputs should lead to the same output.