Suppose that we have a multi-secret sharing scheme and let $I$ be the a set of agents. Say that $S$ is the space of the (uniform) random variables $s=(s_1,s_2,\cdots,s_I)\in S$ such that the share $s_1$ is known to $P_1$, $s_2$ is known to $P_2$ and so on.

According to Shamir's secret sharing, suppose that $\mathbb{F}$ is a finite field. This scheme makes use of the following general fact about polynomial interpolation: a polynomial of degree at most $t$ is completely determined by $t+1(<I)$ points on the polynomial. For example, two points determine a line, and three points determine a parabola. This general fact not only holds for the real numbers and complex numbers, but over any algebraic domain in which all non-zero elements have a multiplicative inverse (such a domain is called a field as it is defined above in its most generic situation).

$\textbf{Question:}$ Could anyone give the explicit mathematical structure and a proof of an appropriate multi secret sharing scheme where the requirements of correctness and information-theoretic privacy are fellfield?

Any references of the literature are very welcome! Also instead of using the polynomial formulation for the proofs I would appreciate it if you use the $+$ - modulo $q$ or the $\times$ - modulo $q$, where $q$ is the cardinal of the finite field $\mathbb{F}$.


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