# Could you provide the proof of a secure multi - secret sharing scheme that fulfils the requirements of correctness and information-theoretic privacy?

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( 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 $$\mathbb{F}$$.