No, the scheme does not have CPA security either.
Distinguisher (identical to CPA$): The same distinguisher from (a) can be used to attack CPA security as well. The distinguisher can query the encryption oracle for any message and check for divisibility by 10 to learn information about the encrypted message.
Advantage: The distinguisher's advantage remains significantly greater than 0 .
If c is divisible by 10, it implies the padding scheme successfully created a prime number with a trailing zero. This has a non-negligible probability of occurring, especially for smaller message lengths.
If c is not divisible by 10, it means the padding scheme failed to produce a prime number with a trailing zero, suggesting a different message was encrypted.
Key Point:
The fundamental flaw lies in the predictability of the padding scheme. Knowledge of the padding scheme's structure allows an attacker to glean information about the message by examining ciphertext properties (divisibility by 10 in this case).
1st step:
The encryption scheme is not CPA secure because an attacker can determine if a message is encrypted by checking if the ciphertext is divisible by 10. This is because the padding scheme always adds a trailing zero to the message before encrypting it. If the ciphertext is divisible by 10, then the attacker knows that the message was encrypted with the padding scheme. Otherwise, the attacker knows that the message was encrypted with a different scheme.