The RSA algorithm is applied for the encryption of a $256$ x $2568-$bit gray$-$scale image on a pixel$-$by$-$pixel basis. Given that p $=13$ and q$=23$ are used in the RSA algorithm.

a$)$ How should n be chosen for the RSA algorithm in this application? What are the

implications when n is set to be too large and too small?

b$)$ Develop a set of public key $\{$e$,$ n$\}$ and private key $\{$d$,$ n$\}.$ Show your step of derivation.

c$)$ Encrypt and decrypt the following three pixel values using the public key developed in

$($b$).$ What observations can be found from the results?

$15,123,350$

d$)$ What is the minimum size $($in bytes$)$ of the image after encryption?