(4) 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,$), but the part denoted by x is unreadable. Find the correct value of :15, 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(flf) modulo 229; that is, write p(m) = a0 + (1135' + @2332 + 0.31173 with a0, a1, a2, a3 6 2229. Also, identify the secret. (c) Verify the solutions of Parts (a) and (b) in sagemath. 24+14+5243 marks Part (a) The student receives 24 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 computing the adjugate matrix of a relevant matrix mod 227, 1 mark for the computation of a relevant determinant, 3 marks for the computation of a relevant inverse mod 227, and 5 marks for giving the correct answers. For different level of correctness the students receives between 23 and 0 marks, Part (b) The student receives 14 marks for each correctly justied step in his/her answer. This includes 12 marks for the computation of the polynomial mod 229. The student needs to justify each step to get the full marks. For different level of correctness the students receives between 13 and 0 marks