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I want to know if the below algorithm , secure against quantum computing attack, and how I can compute the running time for the original algorithm and the proposed attack

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Source: Yan Zhu, HuaiXi Wang, ZeXing Hu, Gail-Joon Ahn & HongXin Hu, Zero-knowledge proofs of retrievability, in Sci. China Inf. Sci. 54, 1608 (2011).

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I want to know if the below algorithm , secure against quantum computing attack

No, it's not secure against Quantum Computers. To quote the text: the secret key is $sk = x \in_R \mathbb{Z}_p$ and the public key is $pk = (g, v = g^x)$.

Shor's algorithm will directly recover the secret key from the public key; that runs in polynomial time.

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  • $\begingroup$ even that p and g are secret sand unknown to the adversary? $\endgroup$
    – Shi Li
    Oct 5, 2021 at 13:56
  • $\begingroup$ @Shima: $g$ is not unknown to the adversary - it's right there in the public key. As for $p$, well, that can be recovered (or at least, a multiple of it) by doing point counting on the curve (and Shor's can also directly recover it, but that'd be more effort...) $\endgroup$
    – poncho
    Oct 5, 2021 at 14:03
  • $\begingroup$ how is can be modified to post-quantum secure algorithm, any clue! $\endgroup$
    – Shi Li
    Oct 5, 2021 at 14:42
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    $\begingroup$ @Shima: doesn't look likely - it uses a pairing operation - I can't think of a post-quantum pairing operation off the top (don't the idea of a 'pairing operation' imply a group; discrete logs in a group are not postquantum), and revising it not to use a pairing operation isn't a "modification", but closer to a complete redesign... $\endgroup$
    – poncho
    Oct 5, 2021 at 15:15

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