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I recently started studying about cryptography and I tried to search the internet but I can't find much about these subjects. I want to see the proof of Shamir's Secret Sharing being Proactive and dealer-free .

Later Edit: I found the proof of it being proactive, however I still struggle with the 2nd thing. Basically, if the users know (k,n,p) , how can they compute a secret with none of them knowing it ?

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    $\begingroup$ Does this paper help? $\endgroup$
    – Ilmari Karonen
    May 21, 2017 at 19:37
  • $\begingroup$ I suspect what you are trying to prove is that the parties can make adjustments to defend the secret without the dealer not that there was never a dealer to begin with. $\endgroup$
    – Meir Maor
    May 21, 2017 at 19:54
  • $\begingroup$ Are you asking how any party of $k$ people amongst $n$ knowing each a different $p_i$, for $i \in \{1, \ldots, n\}$ can recover a secret value shared using Shamir's Secret Sharing scheme? $\endgroup$
    – Lery
    May 22, 2017 at 10:48
  • $\begingroup$ As @MeirMaor notes, the usual definition of "dealer-free" seems to be that a trusted dealer is not required to proactively update the shares, even if such a dealer might be needed to generate the initial shares to begin with. Anyway, if you're asking for a scheme to let n semi-honest parties collaboratively generate a (random) $k$-out-of-$n$ shared secret, this answer describes one way to do it: basically, each party generates a random secret and shares it with the others using Shamir's scheme, and then everybody sums up the shares they received. $\endgroup$
    – Ilmari Karonen
    May 22, 2017 at 17:07
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    $\begingroup$ Anyway, could you please edit your question further to clarify exactly what you're asking? I feel like I could probably answer your question, if I just knew exactly what it was. Hence all these comments. $\endgroup$
    – Ilmari Karonen
    May 22, 2017 at 17:09

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