Suppose that $H(m)$ is a secure hash function that generates a $12-$bit output.

A$)$ How many collisions would you expect to find if you hash $1024$ randomly selected

messages?

B$)$ What is the expected number of hashes that must be computed to find $25$ collisions?

That is$,$ what is the expected number of hashes that must be computed to find pairs

$(x_i,y_i),x_i{y}_i,$ with $H(x_i)=H(y_i),$ for $i=1,2,$dots,$25?$