How can any 3 of 5 people unlock a secret with a curve drawn through points?

Shamir's secret sharing hides a secret in a curve. A straight line is pinned down by two points, a parabola by three, a cubic curve by four, and so on: t points are enough to define a polynomial of degree t − 1. So to share a secret among n people so that any t of them can open it, you build a polynomial of degree t − 1 whose first coefficient is the secret and whose other coefficients are picked at random. Then you find n points on the curve and give one to each person.

When at least t people reveal their points, there is enough information to fit the curve through them, and its first coefficient is the secret. For 3 of 5, the curve is a parabola, and any three of the five points pin it down. With only two points, plenty of parabolas fit, and each would point to a different secret. That is why the scheme has information-theoretic security: even if an attacker steals some shares, it is impossible to reconstruct the secret unless they have stolen enough of them.

Adi Shamir first developed the scheme in 1979. He is also the S in RSA, the famous algorithm he co-invented with Ron Rivest and Len Adleman. The classic picture is a company vault. If one person knows the code, it might be lost or unavailable when the vault must be opened. If several people know it, they may not trust each other to always act honestly. Handing out shares fixes both problems: the vault opens only when enough shares are presented.

The scheme can also treat people differently. With a threshold of 3, a president given three shares could unlock the safe alone, while three secretaries holding one share each would have to combine theirs.