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Let $p,q$ be odd primes, $n:=pq$ and $a,x \in \mathbb{Z}/n\mathbb{Z}$ be quadratic residues such that $x^2 \equiv a \pmod n$. I have understood the proof that calculating the least significant bit of $x$ is computationally as hard as deciding the Quadratic Residuosity Problem and I am aware of the fact that it is even as hard as factoring $n$. Now the "Handbook of Applied Cryptography" states in Fact 3.89 (chapter 3, page 117) that computing the $k = \mathcal{O}(\log(\log(n)))$ least significant bits of $x$ is also as hard as factoring $n$. (The key word is "simultaneous security".)

Could you please tell me how to prove this? How can I relate this to the established result that the least bit of $x$ is secure?

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    $\begingroup$ A little note on the style of the book; one has to go Notes and Further references of the chapter to find the reference. page 127 section 3.5 $\endgroup$
    – kelalaka
    Oct 8, 2020 at 14:38

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