Suppose we have a scheme that works so: $\mathcal{P}_k =:\{P_k\colon \{0,1\}^{2n} \to \{0,1\}^{2n} \}$ is a strong PRP family.
Encryption: $\operatorname{Enc}(k,m)=P_k(m\mathbin\Vert r)$, where $\Vert$ indicates the concatenation and $r\leftarrow \\\$ \{0,1\}^n$ ($r$ is random from $\{0,1\}^n$)
Decryption: $m\mathbin\Vert r={P^{-1}}_k (c)$, return $m$;
I remember that we can define $G^{CCA}_{\pi,\mathcal{A}} (\lambda, b)$ as follows:
$k \leftarrow \{0,1\}^\lambda$
The adversary $\mathcal{A}$ chooses two messages $m_0,m_1$.
$c_b \leftarrow Enc(k,m_b)$, where $b \leftarrow \\\$ \{0,1\}$. (I denote with $\\\$$ to say that $b$ is random).
The adversary must say what message has been decrypted. Let's assume that $\mathcal{A}$ has access to both $Enc_k()$ and $Dec_k()$ (CCA-Security) algorithms. So $b' \leftarrow \mathcal{A}(1^{\lambda},c_b)$.
$\mathcal{A}$ succeeds if $b'=b$.
We have CCA-security if $\mathcal{A}$ cannot do better than guessing, that's for all PPT adversaries $A$
$$|Pr[G^{CCA}_{\pi,\mathcal{A}} (\lambda, 1)=1]-Pr[G^{CCA}_{\pi,\mathcal{A}} (\lambda, 0)=1]| \leq \upsilon(\lambda) \in negl(\lambda)$$
How can I proceed by reduction? Can someone help me?
