Taking into account my previous question here and the answer about the proposed encryption-decryption scheme. I am trying to understand how to make possible operations in modular arithmetic for a secret sharing scheme as proposed there

Suppose that $\mathbb{F}$ is a finite field such that $x\in\mathbb{F}$. We consider a network of five agents denoting with $i$ the generic agent and every player knows her own coordinate from $x$, namely player $1$ knows $x_1$, player $2$ knows $x_2$ and so on. Each of them wants to share her secret with the other players in such a way that she does not want to reveal her information with a secret sharing scheme as in Shamir's scheme. For example player $i$ shares her secret $x_1$ with the otehr palerys as it follows

$\tau_{12}=z_{12}(x_{1})+\beta_{12} (mod{n_1})$ $\tau_{13}=z_{13}(x_{1})+\beta_{13} (mod{n_2})$ and hence $\tau_{ij}=z_{ij}(x_{i})+\beta_{ij} (mod{n_i})$

such that $z_{ij}(x_{i})=\alpha_{ij}\cdot x_{i}=w_{ij}$, where $j=-i$

$\textbf{Question 1:}$ Player $1$ for example has given four different shares of her information $s_1$, so if we sum the four parts $\alpha_{12}\cdot x_{1}+\alpha_{13}\cdot x_{1}+\alpha_{14}\cdot x_{1}+\alpha_{15}\cdot x_{1}=(\alpha_{12}+\alpha_{13}+\alpha_{14}+\alpha_{15})\cdot x_{1}=a_1\cdot x_1$ could we make the following operations $t_1=t_{12}+t_{13}+t_{14}+t_{15}=w_{12}+\beta_{12}(mod{n}_1)+w_{13}+\beta_{13}(mod{n}_1)+w_{14}+\beta_{14}(mod{n}_1)+w_{15}+\beta_{15}(mod{n}_1)=\alpha_1\cdot x_1+\beta_1(mod{n}_1)=w_1\bigoplus_{n_1}\beta_1$. Are these calculations of summations right, where $t_i=w_i\bigoplus_{n_i}\beta_i$, $\forall i$?

$\textbf{Question 2:}$ All these schemes refer to polynomials, so are $\tau_i-w_i-\beta_i$ is a multiple of $n_i$. By summing all these $\tau_i-w_i-\beta_i$ of the five players, do we obtain the polynomial $f(x)$ of the secret sharing scheme such that $f(0)=s$?


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