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If they don't trust the server they sure shouldn't send any money. The "trusted" third party is used to solve the problem of participants who don't trust each other. So by definition, your problem can only be somewhat mitigated, not solved completely. I'm not sure what you mean by "provably fair". If the server can't prove he cannot cheat, it's not provably ...


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The original question was: Alice knows: $a,b$ and $x$ such that $a^{(x\cdot x)} = b$ Bob knows: $a,b$ and DrLecter referenced this paper (fixed the link), which covers the question. Now, the question was changed to Alice knows: $b$ and $x$ such that $x^x=b$; Bob knows: $b$. The given structure was: ... multiplicative group $G$ ...


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Below is one possibility, but for a large set of values not a really efficient one as the work and the size is linear in the number of values. Say we are working in a cyclic multiplicative group $G$ of prime order $p$ with generators $g$ and $h$ (such that the discrete logarithm between $g$ and $h$ is unknown to the users) and I assume that the values on ...


1

Your approach makes getting information other than count of cards in possession of each player at least as hard as breaking the PRFness of HMAC. To make it information-theoretically impossible "for all the ... each player", $\;\;$ if different card_values have different lengths then use $\;\;$ SHA256(commonly_agreed_public_salt:card_value) $\;\;$ ...



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