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The form of equation you expect is not usually applied to Shamir's Sharing Scheme. Instead the equation is usually represented as polynomial: $$f\left(x\right)=a_0+a_1x+a_2x^2+a_3x^3+\cdots+a_{k-1}x^{k-1}\,\! \mod p,$$ where $k$ is number of pieces needed to restore the secret; $a_i$ are random coefficients (modulo p) and $p$ is 13 (in your case). ...

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We have two kind of search while outsourcing: keyword search and text search. In the keyword search you choose some keywords in your text and preprocess them before outsourcing. In the server(s), there is just one instance of each keyword so there is no duplication problem. There is the literature of searchable encryption that mostly covers this kind of ...

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If I understand the question right, what you are looking for is a fair secret sharing protocol. Most secret sharing protocols are unfair. For example, in 3-out-of-4 secret sharing with 2 dishonest parties, once one honest party broadcasts their share, the two dishonest parties can privately collude to reconstruct the secret. Then if they refuse to ...

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Yes we can do better. Secure approaches exist. We implemented regular expression matching in the ShareMonad, which is secure in the semi-honest setting. IIRC, the paper touches on our DFA construction and selected algorithm - which is the hardest part really. Once you know the algorithm it's just a matter of turning the crank.

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there are some special constructions that could be used. for example see this paper of florain kerschbaum (freely available version). also, you can use some secure DFA evaluation protocols, as any regex can be represented as a DFA. what you have proposed is not secure and the servers can learn lots of information.

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