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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?

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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!

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