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6

No, the Runge phenomenon is known not to affect Shamir's scheme. Remember, the point of Shamir's scheme is not actually to form an approximation over an interval; instead, it's to encode a secret in a randomly chosen polynomial, and then divide up clues to that polynomial so that, with enough clues (shares), someone can reconstruct the entire polynomial ...

2

In a short paper "On sharing secrets and Reed-Solomon codes," Communications of the ACM, vol. 24, pp. 583-584, September 1981, Bob McEliece and I described a secret-sharing system that uses no randomness (cf. the second paragraph of the paper). This is most useful for very large secrets (lots of bits) since it divides the secret into $k$ parts, and then ...

5

Any deterministic secret sharing scheme as in the question has the property that any participant can run the deterministic algorithm for a guess of the shared secret, and eliminate the guess if the share that the algorithm deterministically assigns him/her does not match his/her share. This implies that some information about the secret is leaked in his/her ...

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