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In traditional secret sharing schemes, like Shamir's Secret Sharing, first a secret S is generated. Then it is divided into n parts, such that a subset k of the n parts are sufficient to reconstruct S.

My problem is that I already have n individual and independent secrets. Is there a way to come up with a shared secret S from these pre-existing n secrets, such that a subset k of the n parts are sufficient to reconstruct S?

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marked as duplicate by Ilmari Karonen, Geoffroy Couteau, tylo, otus, SEJPM Mar 10 '17 at 15:35

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    $\begingroup$ Questions like this are always difficult. I'm not aware of any way to do it, but I'm also not aware of a proof that no such thing exists. The simple case is when $k=n$, in which case use just use the XOR of the secrets. For $k<n$, my gut feeling is that this is impossible. $\endgroup$ – mikeazo Mar 9 '17 at 16:38

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