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Consider a RSA group $Z_N$ for $N=pq$, where $p,q$ are large prime numbers. Under strong RSA assumption, can an adversary efficiently compute the inverse of a random element $z$ from $Z_N$ without access to $p,q, \phi(N)$? Mathematically does there exists efficient algorithm $\mathcal{A}$ s.t

$$ \mathbb{P}\left( z.u=1,\quad z \leftarrow Z_N,\; u=\mathcal{A}(x,N) \right) \geq \text{neglible}? $$

PS: I am learning cryptography so my notations are quite shaky.

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    $\begingroup$ Welcome to crypto.stackexchange - How does the reference-request tag fit into your question? Are you hoping to find some kind of paper with the solution or ? If so then you may want to state that explicitly in your answer. $\endgroup$
    – Ella Rose
    Commented Feb 10, 2019 at 20:43
  • $\begingroup$ @EllaRose: I was hoping to find a reference to lecture notes. $\endgroup$
    – Vivek Bagaria
    Commented Feb 10, 2019 at 21:57

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Mathematically does there exists efficient algorithm $\mathcal{A}$

Yes; the Extended Euclidean algorithm can be used to efficiently compute multiplicative inverses modulo $N$, without knowledge of the factorization of $N$.

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  • $\begingroup$ Addition: this works by using the Extended Euclidean algorithm to find the $(u,v)$ in Bézout's identity $u\,z+v\,N=1$. We do not need $v$, and that allows to reduce the cost by a small factor, see this. There's also a binary variant, see this. $\endgroup$
    – fgrieu ♦
    Commented Feb 12, 2019 at 10:51

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