Suppose that there are $5$ players and each of them learns a secret that is a coordinate of the random vector $s=(s_1,s_2,\cdots,s_5)$, such that $s$ is a uniformly distributed over the field $V$. Each of them wants to share their secret by using a multiparty computation scheme with the other players. For example say player $i$ (who is the generic player form the set of the $5$ players) wants to share his secret $s_i$. A pre-play communication phase takes place where the players communicate with a device that gives them some information about how to encrypt their messages. How could I generate a procedure where each player uses a scheme s.t.
$$s_i=\sum_{j=1}^2a_j^ib_j^i=a_1^ib_1^i\oplus a_2^ib_2^i$$
in the sense that player $i$ gives player $j=-i_1$, $(a_1^i,b_1^i)$ and to player $k=-i_2$ the pair of $(a_1^i,b_2^i)$. There are two groups of players that receive a message by $i$, divided in groups of 2 people each of them, so the index $j$ refers to the one group of two players and the index $k$ refers to the other. This means that all the players need to contribute in the multiparty computation process so as to extract the information $s_i$. This procedure is going to be followed by all players to share their secrets
In this point i want to make some questions.
- What assumptions need to be done about $(a_j^i,b_j^i)_{j=1}^5$? Do they need to be uniformly distributed in $V$ as well?
- Do we need to assume that the operations of $\oplus$ and $\otimes$ are defined in the field $V$?
- And more precisely is this scheme secure? Could someone provide something better?
