The final exam question pool is locked up by Prof. Volkovich with a password. Ev

Question

The final exam question pool is locked up by Prof. Volkovich with a password. Every IA has a secret share of the password, but the password can only be reconstructed with 5 or more IAs' shares. Emily and Tom want to change the final questions so that they are less challenging than the midterm. They conspired with Chole, Aditi and Zack to get the full password. Here is a list of their secret shares: (1,16), (2,23), (3,19), (4, 26), and (5,5) over Z31. Now, find out the full password (using Lagrange interpolation)! Here is an example on how to generate the first polynomial, L1(x): IC (x - 2) (x-3) (x-4) (x -5) (1-2 (1-3) (1-4) (1-5) r4-14r3 +71a2 - 154r 120 24 - 22(r4- 14r3 +71a2 - 154120) 22308z3 + 1562x2-3388 2640 22a4 2a3 + 12a2 +22x +5 Hint: you should do this for every share

Explanation / Answer

L1: ((x - 2)*(x - 3)*(x - 4)*(x - 5))/24

L2: -((x - 1)*(x - 3)*(x - 4)*(x - 5))/6

L3: ((x - 1)*(x - 2)*(x - 4)*(x - 5))/4

L4: -((x - 1)*(x - 2)*(x - 3)*(x - 5))/6

L5: ((x - 1)*(x - 2)*(x - 3)*(x - 4))/24

polynomial =(5*(x - 1)*(x - 2)*(x - 3)*(x - 4))/24 - (13*(x - 1)*(x - 2)*(x - 3)*(x - 5))/3 + (19*(x - 1)*(x - 2)*(x - 4)*(x - 5))/4 - (23*(x - 1)*(x - 3)*(x - 4)*(x - 5))/6 + (2*(x - 2)*(x - 3)*(x - 4)*(x - 5))/3

=> polynomial = - (61*x^4)/24 + (349*x^3)/12 - (2795*x^2)/24 + (2291*x)/12 - 85

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