$2\mathrm{.}18.$ Complete the proof of Theorem $2\mathrm{.}16.$

$($a$)$ Formally prove $($using the hybrid technique$)$ that the scheme in that theorem satisfies

one$-$time secrecy$.$

$($b$)$ Give a distinguishing calling program to show that the scheme doesn't satisfy one$-$time

uniform ciphertexts.