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In this paper Patarin says that: "for every $\epsilon > 0$, when $m \ll 2^{n(1 - \epsilon)}$ ... for 7 rounds or more it is secure against all adaptive chosen plaintext attacks" where m is the number of queries that the adversary can evaluate.

What concretely is meant by $m \ll 2^{n(1 - \epsilon)}$?

For instance, to have statistical security $2^{-\sigma}$ (e.g. $\sigma = 40$) concretely how many queries can be evaluated?

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  • $\begingroup$ define statistical security $\endgroup$
    – kodlu
    Apr 25 '20 at 4:15
  • $\begingroup$ How would you even define << concretely? It's sort an asymptotic idea. $\endgroup$ Apr 26 '20 at 1:00
  • $\begingroup$ @kodlu Where statistical security is the advantage of an adversary has in the Chosen Plaintext Attack game, when they are able to make $m$ queries and have unlimited computational resources. $\endgroup$ Apr 26 '20 at 2:24
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    $\begingroup$ ok so you want the advantage to be upper bounded by $2^{-40}.$ $\endgroup$
    – kodlu
    Apr 26 '20 at 3:22
  • $\begingroup$ @kodlu Yes, exactly. $\endgroup$ Apr 26 '20 at 3:32
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The linked paper is missing a lot of proof details. In any case, it seems impossible to say anything concrete for finite $n$ due to the existence of terms in $O(\cdot)$ notation in the bounds. You simply do not know how large those implied constants are.

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  • $\begingroup$ Yeah--the missing proof details is a lot of the problem... In principle if the details of the proof were there I should be able to go through and figure out what the constants are in the $\mathcal{O}()$ notation. But since they are omitted, I was hoping someone out there might know of a place that someone had gone through and calculated the constants. Or if not, a place where I can find the full proof so I can go through and try to figure out the constants myself. $\endgroup$ Apr 26 '20 at 2:28
  • $\begingroup$ Did it not reference where the full version is, in the paper? These kinds of arguments are usually asymptotic. $\endgroup$
    – kodlu
    Apr 26 '20 at 3:21
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    $\begingroup$ see also here crypto.stackexchange.com/questions/43870/… for a related question and the followup comment which has links to more but based on @fgrieu's comment I don't hold high hopes that you'll get what you want. $\endgroup$
    – kodlu
    Apr 26 '20 at 3:24
  • $\begingroup$ Yes, citation 15. "Extended version of this paper. Available from the author." I was hoping to find a source online, but I can contact the author. $\endgroup$ Apr 26 '20 at 3:24

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