This question is about secret sharing. (a) You set up a (3; 37) Shamir threshold scheme, working modulo the prime 227. Three of the shares are (1; 4); (2; 8), and (3; 16). Another share is (5; x), but the part denoted by x is unreadable. Find the correct value of x, the relevant polynomial, and the message. Justify all your steps. (b) In a (4; 41) Shamir threshold scheme working modulo the prime 229, the shares (1; 9); (2; 27); (3; 81), and (4; 243) were given to Alice, Bob, Jerry, and Charles. Calculate the corresponding Lagrange interpolation polynomial p(x) modulo 229; that is, write p(x) = a0 + a1x + a2x2 + a3x3 with a0; a1; a2; a3 2 Z229. Also, identify the secret.