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Yes, the problem of finding unknown random $x\in\mathbb Z_N$ given $N\in\mathbb N$, $u\in\mathbb N$ with $1<u\le\lceil\log_2N\rceil$, and $x^u$ computed in $\mathbb Z_N$, is believed hard unless the factorization of $N$ can be determined, which itself is believed hard for appropriately constructed RSA moduli $N$. Moreover, depending on $u$ and $N$, there ...


I'm not sure if this is what you meant, but computing arbitrary roots modulo a composite number IS the RSA-problem, which is considered hard. I'm pretty sure that squaring the modulus won't make a difference in the hardness, as you still don't know the prime-factors, but don't think that it's "more" secure than with normal N, and it will certainly be slower ...

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