Assume a common RSA public key $(N,e)$ with $N$ the product of two unknown distinct randomly chosen large primes $p$ and $q$ of about the same size.
Does revealing $p^e\bmod N$ (in addition to the public key) allow factorization of $N$, or otherwise makes it markedly easier to solve the RSA problem? What about also revealing $q^e\bmod N$?
Does revealing $p^e\bmod N$ (in addition to the public key) allow factorization of $N$
Yes, with that extra information, $N$ is easy to factor. A computation of $\gcd( N, p^e \bmod N )$ gives the factor $p$ (because $p$ is the only prime the two integers $N$ and $p^e \bmod N$ have in common, it is a factor of both integers, while $q$ is not a factor of $p^e \bmod N$)
