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1

This is really an extended comment on @cygnusv's excellent answer. Each share in Shamir's secret-sharing scheme really has two parts: the share $s_i=P(x_i)$ of the secret (where $P(x) = P_0 + P_1x +\cdots + P_{k-1}x^{k-1}$ is the polynomial of degree $k-1$ whose coefficient $P_0$ is the secret while the other $k-1$ coefficients are chosen at random) and ...


7

With Shamir's Secret Sharing you can add as many shares as you want, as long as the threshold and the secret is unchanged. You can see that neither the generation of the polynomial that produces the shares nor the reconstruction of the secret by polynomial interpolation require the parameter $N$ (number of shares), but only the threshold $k$. Of course ...


1

If the ring is multiplicatively commutative, the obvious way to do this is: Have each pair $a, b$ create a secure connection, and jointly select two random invertible elements $r_{ab}$ and $r_{ba}$ with $r_{ab}r_{ba} = 1$ (that is, they're inverses of each other). Then, $$r_a = \prod_{b \ne a} r_{ab}$$ This is secure (in that any group of parties know ...



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