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Newbie here. Not familiar with cryptography; just interested and reading up about it from time to time.

We know that the RSA problem (let's say $c=m^e \bmod N, c^d= m \bmod N$) is about recovering $m$ given $N, c$ and $e$. However, if given $N, m, C$ and $e$ - can I recover $d$?

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No!

RSA is public-key encryption and in public-key encryption, the known plaintexts are free. Given a public key $(n,e)$ we can find many $(p_i,c_i)$ such that $c_i = p_i^e \bmod n$. If this is a weakness, on the first day it will be broken.

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