I recently started studying cryptography but I am not sure I quite understand the concept of proof by reduction. Question I am trying to solve is as follow:
Suppose $\Pi$ is a symmetric encryption scheme which $ C \subseteq M$ ($M$ is message space and $C$ is ciphertext space). then we have $\Pi'$ with same key generation and decryption algorithm as $\Pi$ ($K' = K , D' = D$) with encryption algorithm as follow
$$E_k'(m) = E_k(E_k(m)).$$ I am trying to proof or reject:
a ) if $\Pi$ is indistinguishable in presence of an eavesdropper (simplest case when attacker can only see a ciphertext) then $\Pi'$ is indistinguishable.
b ) if $\Pi$ is CPA-secure then $\Pi'$ is CPA-scure.
$ C \subseteq M $ implies a bijection between $M$ and $C$ so whenever $A'$ guess the chosen bit correctly $A$ will do as well so we have
$$Advantage\: of\: A \geq Advantage\: of\: A'$$
So if $A'$ be a attacker with non negligible advantage $A$ will be as well so a is true.
Am I using reduction correctly? What bout part b? Can we use almost same reasoning or there is an attacker for this case to prove $\Pi'$ is not CPA-secure?
part a : wrong . and one time pad is an counterexample and first picture is totally wrong.
part b : right . and proof is by reduction(picture two). for this reduction we have
$$Advantage\: of\: A = Advantage\: of\: A'$$
so if advantage of $A'$ be non-negligible advantage of $A$ will be too.