Prove RSA scheme is insecure [closed]

Question

Im reviewing my crypto and I would like to prove the following: RSA experiment where the adversary is given N, e, and y. The adversary knows p, q.

I know the RSA scheme is built on the idea that it is hard to find 2 large primes that are factors of N, and we already have this so it's easy to solve for d now. In order to prove this we need to solve for d which is the secret because [$c^d$ mod N] is the decryption of a cypher text. I just don't know how to formalize this into an adversary and show the steps to solve for d which would give A a success prob of 1.

also d is equivalent to

d = [e^{-1} mod 𝜆(𝑛)]

We know everything in that equation except for 𝜆(𝑛). Is the equation for 𝜆(𝑛)=𝑙𝑐𝑚(𝑝−1,𝑞−1)? I'm almost sure this is the correct equation, and if it is then we can use it to solve for d and the question is solved.

Answer 1

The security of the RSA cryptosystem requires that p,q are secret. If an adversary knows prime pair $(p,q)$, it can compute $\lambda(n)$ as follows:

$\lambda(n)=lcm(p-1,q-1)$.

For a properly chosen public key part $e$, the private key, $d$, is then computed as follows:

$d \equiv e^{-1} \pmod{\lambda(n)}$

Thus, an attacker who knows $(p,q)$ can recover the private key from the public key.