A 2 The solving of LGS also plays a role in polynomial interpolation and is performed at-

used in the secret-sharing procedure of Shamir: We

want to divide a secret given as a real number ao among n persons. The

secret should only be able to be reconstructed if all n persons have their

Part of the mystery. To do this, proceed as follows:

One chooses a polynomial p (x) = a0 + a1x ++ an-1x" of degree n - 1, where the

The coefficient of the secret is. The coefficients a1,..., an-1 can be chosen arbitrarily

be. Each of the n persons now receives a pair (xk, yk) consisting of x-value xk and

Function value uk = p(xk), k € {1,...,n).

By means of interpolation (solving a suitable LGS, see tasks below), the po can be-

lynom p(x) and thus the secret a0 from the n pairs of values (xk, yk), k € {1,...,n),

reconstructed.

In this task, we specifically consider a secret sharing procedure for three people.

To split the secret ag € R, a polynomial was

1

p: R-> R, p(x) = a0 + a1x + a2x²/nof degree less than or equal to 2 with arbitrarily chosen coefficients a1 and a2. The three

Pairs of values (-1, 17), (2,32), (3, -3) will be distributed to the three people.

(a) Show that the secret a0 cannot be determined if the third person

is not present and only the value pairs (-1, 17), (2,32) are known: enter all

possible polynomials of degree less than or equal to 2, which at the point x

= -1 take the value 17 and the value 32 at the position x = 2. To do this, create a

suitable LGS and determine the amount of solution.

(b) Show that the secret a0 can be determined if all three people

bringing your information together. Create a suitable LGS again and

solve it.

(4 + 1 Points)