How can I enrich this mechanism of communication to become more efficient and secure?

Suppose that we have a Bayesian game, where $$t_i\in T_i$$ denotes the type of player $$i$$. Say that we have a communication game (communication equilibrium). The players do send each other an encrypted message about their type. If $$L_i$$ is an isomorphic space of $$T_i$$ and $$\phi_i:T_i\to L_i$$ is an permutation (injection + surjection= bijection), then every player $$i$$ instead of sending their type to each other player they can send the message $$\phi_i(t_i)=l_i$$. Also, in order to protect herself from cheating, let $$\rho_i:L_i\times Y_i\to X_i$$ be a cipher, that encodes the private information of player $$i$$, that is, $$y_i\in Y_i$$ is the key and $$x_i\in X_i$$ is the code, where again $$\rho_i(\cdot,y_i)$$ is a bijective so that the pair $$(x_i,y_i)$$ is associated with exactly one $$l_i$$. For technical reasons we made the asumption $$|Y_i|\geq|T_i|$$ (but why? Is this Shannon's property?).

In our game we have $$I$$ players, with the above representation we can use a lemma from probability theory, that is:

$$\textbf{Lemma:}$$ If $$\phi_i$$ 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 $$\phi_i$$, then the random variable $$x_i$$ defined as $$x_i=\phi_i\ominus_{n_i}y_i$$ (where $$\phi_i\ominus_{n_i}y_i=\phi_i-y_i(mod{n}_i)$$) is also uniformly distributed over $$\{1,2,\dots,n_i\}$$.

In other words $$l_i=\phi_i(t_i)=x_i\oplus_{n_i}y_i$$. Then every player $$i$$ instead of sending $$l_i$$ to the other agents as a message, she sends in the half of them $$x_i$$ and the rest of them (we do not know if $$I=2k$$ or $$I=2k+1$$, with $$k\neq 0$$ a potitive integer) $$y_i$$. Then in a subsequent phase they communicate each one to unite the pieces and verify that they only learn $$l_i=\phi_i(t_i)=x_i\oplus_{n_i}y_i$$ (however they still haven't learned $$t_i$$ but $$\phi_i(t_i)=l_i$$.

My questions are the following

1. Is this mechanism of information transmission secure? If not, how can I make it?
2. Could I use a scheme as in secret sharing, where every player $$i$$ could distribute shares of the key $$y_i$$ to all of the other players $$j\in I-\{i\}$$? For example could I further assume that $$y_i$$ is written as a linear combination of some $$w_i$$ that all these $$w_i$$ are non-zero and independent such that $$y_i=\sum_{j=1}^{I-1}w_jy_j$$? Is this right or wrong? Could anybody provide some help-references-guidance or either show some maths that could make it possible to make such a construction?

In general, how can I enrich this mechanism of communication to become more efficient and secure?