Suppose that we have a game with $I$ players and each of them has a private secret say $e_i$. Every player wants to share her secret with the rest of the players but in such a way that she will not be cheated. We have the following formulation
$$p_i:E_i\times Y_i\to X_i$$
where $|Y_i|\geq|E_i|$ and $p_i(\cdot,y_i)$ is bijective so that every pair $(x_i,y_i)$ is associated with exactly one $e_i$. More precisely, $p_i$ is a cipher mapping, $x_i$ is a code and $y_i$ is a private key uniformly distributed over $Y_i$. Let us further assume that $z_i(e_i)$ is a permutation of the information $e_i$. With the help of the following lemma we have
$\textbf{Lemma:}$ If $z_i$ is a random variable with support on $\{1,2,\dots,n_i\}$, and $y_i$ is uniformly distributed over $\{1,2,\dots,n_i\}$ indepedent of $z_i$, then the random variable $x_i$ defined as $x_i=z_i\ominus_{n_i}y_i$ (where $z_i\ominus_{n_i}y_i=z_i-y_i(mod{n}_i)$) is also uniformly distributed over $\{1,2,\dots,n_i\}$.
Could I use a secret sharing scheme based on this encryption-decryption scheme, that could be multiparty in the sense that player $i$ could somehow share the key $y_i$ splitting it in parts and how could I formulate this? Suppose that we want to share the key $y_i$ in a way such that after all the players will communicate each other will obtain $y_i$. Namely, player $i$ will only say a part of the key $y_i$, for example, player $j=-i$ learns $\tau_{ij}=a_{ij}y_j$ and if for any $j\in I-\{i\}$ we take the sum of $\tau_{ij}$ we learn $y_i=\sum_{j\in I-\{i\}}\tau_{ij}$ (in other words $x_i=z_i\ominus_{n_i}\sum_{j\in I-\{i\}}\tau_{ij}$).
How could I do this? Should I define $p_i$ differently and what should be the conditions to find a set that is copy of $Y_i$ such that $\tau_{ij}=a_{ij}y_j$, where $j=-i$?
$\textbf{The goal is the following:}$ There are $I$ players and each of them has a secret say $e_i$. Instead of sharing $e_i$, every player uses a cipher which is defined as $p_i$ and $x_i$ is the code that is generated from the encryption scheme. Also $y_i$ denotes the key. Let as assume that $z_i(e_i)$ is a permutation of $e_i$ such that $z_i(e_i)=x_i\oplus_{n_i}y_i$. I want each player when she shares her secret to split her key $y_i$ to all the other players $j∈I−{i}$ so as to prevent from cheating, in such a way that every player will take $x_i$, but only a part of $y_i$. In essence, $y_i$ is splitted in $|I|−1$ parts, with the other players taking each of them one part. Hence, they will need to further communicate to obtain $y_i$ and hence learn the information $z_i(e_i)$
