Secret sharing:-

Suppose you have a secret message m (a number) that you wish to divide among n people, so that the message can be determined by any t members of the group, but not by any subgroup of less than t members. This can be done as follows:

(a) Select a prime number p larger than any m that might be used.

(b) Select t - 1 coefficients s₁,s₂, ... ,s_t-1 and form the polynomial

(c) Index people from 1 to n, and tell them all the number p. Tell person with index k the value s(k) mod p. Any group of t people can reconstruct the (t -1)th order polynomial from their information by using the Lagrange interpolation formula. Let T be the set of indices of t people. That group can determine m as

Where T /k denotes every index in T except k. For example, for determination by three out of five people, suppose the message is 17, p = 43, and s(x) = 17 + 24x + 36x². Then the five individuals would be told p, and each would be given a result according to this list:

Suppose that persons 2, 3, and 5 share their information. They compute

Suppose on another day, the group is told p = 57 and persons 1, 3, and 4 are told the values 54, 48, 39 respectively. What is the message?