Someone calls you at work and says, “Don’t tell anyone, but…” If you are like most people, there are one or two people you will pass it along to with the same admonishment. In fact, they are probably repeating it from someone else, and you are on their list of two people. So for really big secrets, you need a way to spread the secret out so that no one has any real information about the secret, but a certain number of people together can decode it. As [neeaj] explains in a recent post about Shamir’s Secret Sharing, [Adi Shamir] (the S in RSA encryption) devised a way to do this very well in 1979, and the core concept is very easy to understand.
The explanation works with geometry. The equation for a line is y=mx+b, where m is the slope and b is the y-intercept (that is, where the line touches the y-axis when X is 0. An infinite number of lines cross the Y axis at, for example, 10. The line y=3x+10 does, and so does the line y=-1.41x+10. You can’t guess the b value from just the slope, because any slope will satisfy the equation.
