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

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?

• @Reppiz stackexchange did not let me reply to your comment or upvote it. Shamir's Secret Sharing is great, but I don't believe one can implement it with informational security. I like this method because it can be implemented with informational security. I just don't understand how to algorithmically implement it. Thank you for the suggestion. – Nicholas Iun Jul 23 at 17:25
• What makes you say that you can't implement Shamir's Secret Sharing with information-theoretic security? – Aman Grewal yesterday