(1) This question is about secret sharing. (a) You set up a (3, 29) Shamir threshold scheme, working modulo the prime 211. Four of the shares are (1, 4), (2, 5), and (3, 6). Another share is (4,13), but the part denoted by :1: is unreadable. Find the correct value of 1', the relevant polynomial, and the message. Justify all your steps. (b) In a (4, 31) Shamir threshold scheme, working mod the prime 223. The shares (1,8), (2,16), (3,32), and (4, 64) were given to Alice, Bob, Jerry, and Charles. Calculate the corresponding interpolation polynomial p(:r,) modulo 223; that is, write 33(3) = a0 +a1$+a21t2 +a31'3 with (10,111, (12, a; 6 Egg. Also, identify the secret. Part (a) The student receives 26 marks if all the steps of the computation are correct and he/ she gives an answer. This includes 5 marks for setting up a matrix equation, 10 marks for giving the adjugate matrix of a relevant matrix mod 211, 2 marks for the computation of a relevant determinant, 3 marks for the computation of a relevant inverse mod 211, and 6 marks for giving the correct answers. For different level of correctness the students receives between 25 and 0 marks. Part (b) The student receives 20 marks for each correctly justied step in his/her answer. This includes 18 marks for the computation of the polynomial mod 223. The student needs to justify each step to get the full marks. For different level of correctness the students receives between 19 and 0 marks