I want to know if there is an efficient secret sharing scheme based on XOR such that a share of the message is given to three people and at least two must combine their share in order to construct the message?
What I have tried so far:
Suppose I got a message $M$, and three people A, B and C. Then what I do is
- A gets $a_1 = M \oplus \alpha$ and $a_2 = \beta$
- B gets $b_1 = M \oplus \beta$ and $b_2 = \alpha$
- C gets $c_1 = \alpha$ and $c_2 = \beta$
Where $\alpha$ and $\beta$ are two randomly generated bits the size of the message.
This works fine (with some modification), but the problem is that each person have to store twice the size of the message. Is there a more efficient one?