# Proof that secret sharing based scheme is CPA secure as long as one of the scheme is CPA secure

I want to construct a CPA-secure schemes using two given schemes $$\prod_1$$ and $$\prod_2$$ if only one of them is CPA secure.

Taking suggestions from this answer, I am able construct a scheme as follows.

Given a message $$m$$, Generate a random string $$r$$, of same lenght as $$m$$.

Compute $$r_2 = r \oplus m$$

Encrypt $$r$$ using $$\prod_1$$ to get $$c_1$$ and $$r_2$$ using $$\prod_2$$ to get $$c_2$$.

Send both $$c1$$ and $$c2$$

The scheme must be CPA secure because both of the ciphertexts need to be decrypted to get back the message.

However, I am not able to prove that the resultant scheme is CPA secure other than intuition.

How do I prove it rigorously?