The ssss unix utility does this, it's fun to use.

Docs at http://point-at-infinity.org/ssss/, demo at http://point-at-infinity.org/ssss/demo.html and it can be installed on ubuntu through the 'ssss' package, listed as ssss - Shamir's secret sharing scheme implementation.

Here's one I wrote that wraps the algorithm with passwords https://github.com/ryancdotorg/threshcrypt and you can set the same password multiple times to accomplish the second 'advanced scenario'.

It explains how a password can be cut up in pieces and distributed so that each piece individualy can be used to reveal the secret.

Skip to >>Shamir’s Algorithm<< section to get to the most interesting part.

hmm.. the article says: "If we encoded our secret with a cubic function and distributed coordinate sub-passwords it would require any combination of four points to determine the intercept and the secret."

Maybe I'm missing something here, but, taking any of the cubic function diagrams as example, what if all four coordinates have their "x" between -10 and 0? Those four coordinates will not be enough to generate the whole curve, no?

No.

As soon as you have 4 points, these 4 points uniquely define a cubic. Even if those 4 points are "right next" to eachother.

If this is difficult to imagine, it is the same with a line. Wherever you place 2 points, they always uniquely define a single line.

Unless, of course, you place 2 points exactly on top of eachother.

really? that's amazing! I'd love to see a formal proof of this.

edit: I suppose this counts as proof http://www.had2know.com/academics/cubic-through-4-points.htm..., but atm I don't remember why matrixes are connected with equations.

There's a more general proof at https://docs.google.com/viewer?a=v&q=cache:euFgQLUCTfwJ:... that looks reasonable at a skim.

Awesome!

This is an extremely interesting and well written article. More like this please!