Combined scheme security proof
No, it's not so simple. In particular, you have to be much more precise about what do you mean by combine.
if you want a good example about how to combine insecure things could become secure you can read :
Let's consider the following one-time-signature scheme $\Sigma_1, \Sigma_2$ both which are equal to
https://eprint.iacr.org/2010/446.pdf . with message space $\{0, 1\}^4$.
Remark that these two schemes are proven secure (the fact they are the same is not relevant in the hypothesis you made).
Now let's consider the Signature scheme $\Sigma_3$ over $\{0, 1\}^8$, which is the concatenation of $\Sigma_1$ and $\Sigma_2$. i.e :
$\Sigma_3.\texttt{KeyGen} = \Sigma_1.\texttt{KeyGen}$, and $\Sigma_3.\texttt{Sign} ( sk, m_1 || m_2) =\Sigma_1.\texttt{Sign} (sk, m_1)||\Sigma_2.\texttt{Sign} (sk, m_2)$. $\Sigma_3.\texttt{Verify} ( vk, \sigma_1 || \sigma_2) =\Sigma_1.\texttt{Verify} (vk, \sigma_1)\wedge\Sigma_2.\texttt{Verify} (vk, \sigma_2)$.
Then, if you obtain the signature of $00001111$, you can sign every message in $\{0, 1\}^8$ which contradicts the one-time-unforgeability.QED
To &quot;combine schemes&quot; people are using universable-composability :
https://en.wikipedia.org/wiki/Universal_composability
What is universal composability guaranteeing, specifically? Where does it apply, and where does it not?
But :
1)It's not enough for your schemes to be secure, they have to be UC-secure.
2)You have to be sure your combination is captured by the formalism of UC.
To conclude, it's a little bit like cooking. It's not because two meals are good separately, that combining both would create a good meal.
About the error in your reasoning &quot; Suppose an adversary A is able to break the security of CS by breaking the security of subscheme&quot;, you don't have to suppose how the adversary breaks your scheme, when you are doing a security proof.