# Sharing information scheme of cryptography - operations in modular arithmetic

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 $$f(0)=s$$?