Devise a scheme so that a message M can be shared among X, Y and Z in such a say

Question

Devise a scheme so that a message M can be shared among X, Y and Z in such a say that the only way of recovering the message is when X is present with either Y or Z. When X isn't present or each is alone, they will not be able to recover the shared secret.

Additionally, generalize the above to devise a scheme so that a M can be shared among 5 entities such that the only way of recovering the secret is when either X and Y are both present or when C, D, E are all present together.

I have no idea how to create a hierarchy such as what is being proposed. Any ideas?

Explanation / Answer

This is actually a fairly trivial case of secret sharing:

In the first case, we select a random value R, we give X the value R?M, and we give both Y and Z the value R. Obviously, X alone, nor Y and Z together cannot reconstruct R.

In the second case, we select random values R1, R2 and R3, we give:

R1?M to X

R1 to Y

R2?R3?M to C

R2 to D

R3 to E

Shamir Secret Sharing is a neat trick; however sometimes problems can be solved by simpler methods.

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