{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, :1], {2,8}, and {3,16}. Another share is [5,:rr), but the part denoted by J: is unreadable. Find the correct value of r, the relevant polynomial, and the message. Justifyr all your steps. {b} In a [4,41] Shamir threshold scheme working modulo the prime 229, the shares (1,9), (2, 2?), {3,81}, and {4, 243) were given to Alioe, Bob, Jerry, and I:lharles. Calculate the corresponding Lagrange interpolation polynomial plat] modulo 229; that is, write p{r] = on + 111.1: + oar? + 0.3::3 with on, o] , oz, :13 E @229. Also, identify the secret. {c} Verify the solutions of Parts (:1) and {b} in sagemath. 24+1a1+5=43 marks Part [a] The student receives 24 marks if all the steps of the computation are correct and hefshe gives an answer. This includes 5 marks for setting up a matrix equation, 1f] 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 oomputation of a relevant inverse mod 22?, and 5 marks for giving the correct answers. For different level of correctness the students receives between 23 and I] marks. Part (b) The student receives 14 marks for each correctly justified step in hisfher answer. This includes 12 marks for the computation of the polynomial mod 22!]. The student needs to justify each step to get the full marks. For different level of correctness the students receives between 13 and fl marks. Part [c] For a) the student receives 3 marks if a correct sagemath code is provided, and for b] the student receives 2 marks if a oorrect sagemath code is provided. For different levels of correctness, the student receives between 4 and marks