Split a secret into pieces so that, say, any 3 of 5 pieces rebuild it, while 2 pieces reveal nothing at all. Like a vault that only opens when enough keyholders show up.
The idea: Two points fix a straight line. Hide the secret as the spot where a random line crosses the vertical axis, and give each person one point on the line. Any two people can draw the line and read off the secret. One person alone holds a point that lies on endless possible lines, each crossing at a different secret, so they learn nothing. For "any 3 of 5", use a curve that needs three points (a parabola), and so on.
Why it matters: Used to back up important keys and wallets without trusting any single person.
Split a secret into n shares so that any t of them rebuild it, while t - 1 shares reveal nothing at all[1].
The trick: hide the secret as the starting value of a random polynomial of degree t - 1 and hand out points on it. Two points fix a line, three fix a parabola, and with fewer points every secret is equally possible. Security is unconditional.
Where you meet it: backing up wallet seeds and root keys across several people or places, threshold signing, and inside secure multiparty computation.
If any of these breaks, this breaks too.
If this breaks, these break with it.
Nothing on the map depends on it: it is an end point, something people use directly.