In this lab we will introduce you to working with polynomials.

Secret Sharing

The points given are , , , :
Compute the 4 delta polynomials that passes the 4 points given, then click "Plot" to plot them and compute their sums, which should be a polynomial passing through all 4 points given. Now read off the secret, which is the value of the polynomial at \(x = 0\). TODO: Remove answers before release Paste the secret here (to 1 decimal place):

Finite Fields?

Recall that when we want to share a secret using a polynomial of degree \(d\), we choose \(d\) other random points each with different \(x\) values and find a polynomial passing through these points and \((0, \text{secret})\). \(d+1\) points of the polynomial are then needed to recover the secret. In this question we investigate what happens when we do secret sharing over the reals, but we have one less point needed to recover the secret. Suppose that in this case besides the secrets, the other points at \( x=-4, -2, 2, 4 \) forming the polynomial are chosen randomly from -5 to 5, and you only have 3 points at \(x=-4, -2, 2 \) whereas 4 points are needed to reconstruct the polynomial. With this information and the interactive graph below to help you, can you give a rough range for the value of the secret?
Would this be a problem if we switched to finite fields? Why or why not?

Error Correcting