Secret Sharing - Trivial Secret Sharing

Trivial Secret Sharing

There are several (t, n) secret sharing schemes for t = n, when all shares are necessary to recover the secret:

• Encode the secret as an integer s. Give to each player i (except one) a random integer r_i. Give to the last player the number . The secret is the sum of the players' shares.
• Encode the secret as an arbitrary length binary number s. Give to each player i (except one) a random number p_i with the same length as s. Give to the last player the result of (s XOR p₁ XOR p₂ XOR ... XOR p_i) where XOR is bitwise exclusive or. The secret is the bitwise XOR of all the players' numbers (p).

When space efficiency is not a concern, these schemes can be used to reveal a secret to any desired subsets of the players simply by applying the scheme for each subset. For example, to reveal a secret s to any two of the three players Alice, Bob and Carol, create three different (2,2) secret shares for s, giving the three sets of two shares to Alice and Bob, Alice and Carol, and Bob and Carol. This approach quickly becomes impractical as the number of subsets increases, for example when revealing a secret to any 50 of 100 players, whereas the schemes described below allow secrets to efficiently be shared with a threshold of players.