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I've read this scribe that provides a public coin interactive proof for graph non-isomorphism.

In the proof, the verifier samples both a pairwise-independent hash function and a target $y$. Then it continues to show bounds for the probability that there's a pre-image of $y$ w.r.t. the hash function that's also inside a given subdomain $S$.

In all calculations carried out in page 5 (lemma 3), I couldn't find a single instance that used the fact that $y$ is random. It seems as if randomizing the hash function is enough.

This is also the case for any other source I found that shows a specific public-coin interactive proof for GNI.

Why do we need to randomize $y$?

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  • $\begingroup$ I encourage you to edit your question to make it self-contained by providing relevant background and defining the notation. Also it would help to identify which part of that 7-page document you are referring to (e.g., Section number, page number, which lemma/theorem/proof). Also it might helpful to provide a proper reference in addition to the link (e.g., title of the document, author, class, semester), so we can find the document in the future if the link stops working. $\endgroup$
    – D.W.
    Commented Oct 27, 2023 at 19:53
  • $\begingroup$ Of course, thank you $\endgroup$
    – AmirD
    Commented Oct 27, 2023 at 21:16

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I agree. I believe you are correct. As far as I can see, $y$ does not need to be random.

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