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4

You seem to have some misconception here. Obviously, you are investigating chosen ciphertext attacks (CCAs) on textbook RSA instead of chosen plaintext attacks (CPAs). To help you with your understanding I am discussing CPA on textbook RSA first. To analyse all these kinds of attacks we formally model the attack as a game between an adversary (trying to ...

1

Simple, $r$ just needs to be co-prime to $n$. To test this check that $gcd(r,n)==1$. This is required since you need to "divide" by $r$, i.e., multiply by $r^{-1}$. $r$ is invertible iff $gcd(n,r)==1$.

2

You are correct; whoever put together the above proof typo'ed that point; we have $c \times d \equiv 1 \pmod {\phi(n)}$, or more accurately, $c \times d \equiv 1 \pmod {lcm(p-1, q-1)}$. On the other hand, the attacker isn't expected to be able to compute step 2 (he can't, he doesn't know the value of $lcm(p-1,q-1)$, and hence cannot compute $d$). Instead, ...

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