Please answer all parts as fully as possible

[10 marks] In this question, UW publishes the point P = (2,17) on a certain elliptic curve over Zoz. Pairs of people will use P and this elliptic curve to establish shared secrets, similar to Examples 5c.11 to 5c.14. The following table shows the result of some addiltions of points over this elliptic curve. You can use these without proof. + (11,20) | (19,6) | (2,17) | (17,0) | (19,17) (1120 [ &17) | G11) | (5) | (18) | (518) (2,17) (5,5) | (6,14) | (19,6) | (6,9) (2,6) (5,5) (19,6) | (1,15) | (9,11) | (9,12) | (11,3) (19,6) (9,11) | (17,0) | (6,14) | (19,17) O (a) [4 marks| Determine with proof the smallest positive integer k& such that kP = O. (b) [4 marks| Quara is exactly three years younger than her sibling Clinton. Every year starting on Quaras first birthday, Clinton chooses as his private key his age, c. Similarly, Quara chooses as her private key her age, g. They use these values to establish a shared secret, Kcg. Determine with justification Quaras age the first time their shared secret is (19,17). (c) [2 marks| Re-using the setup from (b), determine with justification Quaras age the first time their shared secret is (6,9), or prove that will never happen