How to create (n, t) secret splitting from (n, n) secret splitting?

Question

Given a secret splitting scheme $(n ,n)$ that creates $n$ shares from secret $s$. In this scheme all shares must be combined to create $s$.

How do you create a secret splitting scheme $(n, t)$? Of $n$ parts at least $t$ parts must be combined to determine secret $s$?

$n =$ # of Parts

$s =$ Secret

$t =$ Threshold of parts needed to create the secrets

$s_1, s_2, s_3, ... =$ Shares in a $(n, n)$ secret splitting scheme

$P_2, P_2, P_3, ... =$ Shares in a $(n, t)$ secret splitting scheme

$l =$ Intermediary value to determine the n needed in $(n, n)$ secret splitting scheme

Example 1 $(4, 3)$:

$l$ = $4 \choose 3-1$ = $6$

$s \rightarrow (6, 6) = [s_1, s_2, s_3, s_4, s_5, s_6]$

$s \rightarrow (4, 3) = [P_1= [s_3, s_4, s_5], P_2= [s_1, s_4, s_6], P_3= [s_1, s_2, s_5], P_4= [s_2, s_3, s_6]]$

Example 2 $(4, 2)$:

$l$ = $4 \choose 2-1$ = $4$

$s \rightarrow (4, 4) = [s_1, s_2, s_3, s_4]$

$s \rightarrow (4, 2) = [P_1 = [s_1, s_2, s_3], P_2 = [s_1, s_2, s_4], P_3 = [s_1, s_3, s_4], P_4 = [s_2, s_3, s_4]]$

What is a methodology to determine an arbitrary $(n, t)$ scheme? For example, what would $(6,3)$ look like?

Answer 1

If you are just looking for one methodology on how to create an (n,t)-scheme, you may take a look at Shami'r Secret Sharing. It basically uses the fact, that you need at least t points to fit a polynomial of degree t-1.