Perfect Secrecy $[12$ Marks$].$

A private$-$key encryption scheme is said to be $2-$perfectly secure if two ciphertexts encrypted using the same

key provide no information about the corresponding plaintexts.

$($a$)(7$ marks$)$ Propose a formal definition, in terms of a suitably defined experiment, of a $2-$perfectly secure

private$-$key encryption scheme.

$($b$)(5$ marks$)$ Is the One$-$Time Pad scheme $2-$perfectly secure$?$ Prove your answer using the experiment de$-$

fined in part $($a$).$