A proposed secret sharing scheme: Suppose that $p:S\times Y\to X$, with $|Y|\geq|S|$ is a cipher where, $y\in Y$ is the key and $x\in X$ the code, $p$ is bijective, namely $(x,y)$ is associated with only one $s$. Hence the decrypted message $s=x\oplus y$ and it is easy to proove it.
$\textbf{Proof:}$ Suppose that we have a mechanism of communication $\mathcal{M}=(p,d)$ such that $\mathcal{M}$ is defined over $(Y,S,X)$, where $Y$ is the key, $S$ the message and $X$ the cipher spaces respectively. To simplify the problem even more I assume that $Y=M=L=G$ where $G$ is an arbitrary finite field.
$$p(y,s)=x,\quad\text{is the encrypted message, which by definition equals $x$}$$
$$h(y,x)=s,\quad\text{is the decrypted message, which by definition equals $s$}$$
So, indeed $(y,x)$ is defined to be associated with only one $s$ and hence $p(y,\cdot)$ is bijective by definition. To answer the question how are they associated, when someone knows both $x$ and $y$, then indeed $x\oplus_{G} y=s$
In order to decrypt the message we have that
$$d(y,x)=d(y,g(y,s))=y\oplus_G x=s$$
where $\oplus_{G}$ is the operation of $+$ as it is defined in the finite field $G$. And hence we have show that the calculation that you ask for, it holds by definition.
$\textbf{Proposed scheme:}$ Could I use the following sharing scheme here: Instead of sharing the secret $s$ I divide the key of the encrypted message by generating a cipher with $k$ keys and only if someone knows all the keys and the one code that is generated, then she will learn the secret $s$ - Let y be shared in $k$ shares such that $y=\sum_{i=1}^k y_i$ where as in Shamir's scheme every $y_i$ is random variable and all of them are independent and define another cipher
$$p:S\times(\Pi_{i\in K}Y)\to X$$ such that the $k+1$-vector $\left(i.e. (s,y_1,y_2,\cdots,y_k)\right)$ is associated with one $s$ and hence the message is decrypted (namely reconstructed) only if all the players communicate and add their $k+1$ shares, namely $s=x\oplus\sum_{i=1}^ky_i=x\oplus y$
Is this scheme some well known scheme?
