When we use a secret sharing scheme we usually want to reconstruct the polynomial function $p(x)\in\mathbb{Z}_q[X]$ with the Lagrange interpolation method and then compute $s=p(0)=a_0$. However, the secret $s$ is just a number and usually what we have as a secret could represent a private information that is a whole statement. For example, let's suppose that player $i$ knows a secret that every other player $j=-i$ does not know, that is "The price of the stock of Amazon is expected to have payoff $v\sim N(\mu,\sigma^2)$". This is a whole statement, but maybe it is enough to report to the other players what is informational enough to understand what the secret is. The key words are Amazon stock, payoff, mean $\mu$ and variance $\sigma^2$ and let's suppose that both $\mu$ and $\sigma^2$ are positive integer numbers. Let's say that we can translate the words Amazon stock and payoff with the help of a cipher $C$ in numbers, positive integers say $x_1,x_2$ respectively. The mean and the variance are positive integers and for tractability let's say that these integers are $x_3$ and $x_4$ respectively. Then player $i$ has a secret that is a vector $x=(x_1,x_2,x_3,x_4)$, my questions are the following
- how will she share $x$ with the rest of the players $j$? Suppose that there are $N$ players.
- after reconstructing the secret $x$ the integers $x_1$ and $x_2$ are meaningless if we do not translate them in words again. What is this function that I need to define so as $x_1$ and $x_2$ will be translated as Amazon stock and payoff respectively? Recall, that we used a cipher that takes words and translates them to positive integers
