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Here is a method, although it involves both shamir's scheme and XORING, and can only update $(k,n)$ to $(k+t,n+t)$ (but can do so repeatly). There are exactly two sets of agents: original and new. The original agents share a secret $X$ with shamir secret sharing, and the new agents each have a field element, such that $X$ added (which usually means XORED) ...

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For "standard" Shamir secret sharing you use a new random polynomial for each secret you want to share (including when you use Shamir secret sharing in SMC). I.e., you would use different polynomials for each secret. Note, that Shamir secret sharing does not directly work on "words" but rather elements of some field. You would have to translate your data ...

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Let us first consider the problem without involving Shamir secret-sharing at all. Suppose that $n = 140$ and that the secret $\sigma$ is a 140-byte Twitter message. The space is thus restricted considerably, from all possible $256$ byte values to the printable characters permitted to be used in Twitter messages, and the distribution in this restricted space ...

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You are right that if it costs Alice & Bob effort $N$ to agree on a key in this way, then it costs Eve only effort $N^2$ to find it. So the protocol is not secure in the standard sense, and probably not very useful. (Maybe in some highly constrained situation with very short-lived keys?) More generally, this purports to be a key agreement protocol whose ...

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