Threshold Scheme

Let with .
An -threshold scheme is a method of
sharing a message among a set of participants
such that any subset of participants
can reconstruct
but no subset of smaller size can.

Shamir’s method

Let be the secret which can be chosen at random
by the Leader.
The Leader chooses a prime .
The Leader chooses independent random coefficients
with where ,
and distinct integers with
Define:

The member receives a shadow pair

Now suppose members are together.
They have the data
where and
and can solve:

because the Vandermonde matrix has nonzero determinant

Then so

However, if only members are present then:

Which can be solved for any
so we have no idea which one is correct.