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P (1,22) (10,4) (12,21) (14,21) (5,18) (10,19) (22,7) (15,8) (4,9) (14,2) (15,14) (18,18) (2,11) (15,9) (20,21) (2,12) (12,2) 2P7 ??? ??? ??? ??? ??? ???

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P (1,22) (10,4) (12,21) (14,21) (5,18) (10,19) (22,7) (15,8) (4,9) (14,2) (15,14) (18,18) (2,11) (15,9) (20,21) (2,12) (12,2) 2P7 ??? ??? ??? ??? ??? ??? ??? ??? ??? ??? ??? 3P620,21) (2,11) (414) 10,5) (16,8) (2,12) (10,19) (5,18) (12,21) (0,18) 15,9) (22,7 (18,5) (15,14) (14,2] (18,18) (4.9) 4P (15.9) (15,9) (11,0) 15,14) (11.0) (15,14) (15,9) (11,0) (11.0) (15,9) (15,14) (15,14) |(15,14 (15,9) (11,0) SP05) (22,7) (4.9) (20,21) (16.15) (22.16) (18,5) (5,5) (12,21) (20,2) (15,14)|(2,11) (10,19) (15,9) (1.1) (10,4) (4,14) 6P (12.21) (4.9) (15,9) (12,2) (15.14) (4.14 (12,21) (15,9) (15,9) (12,21) (11,0) ||4,14) (12.2) (11.0) (414) (12,21) (15,14) 7P (14,21) (18,18) (12,2) 11,221 (5,5) (18,5) (2,11) (16,15) (4.14) (1,1) (15,9) (10,4) (22.7) (15,14) (0,5) (22,16) (12,21) SP (11.0) (11,0) 0 (11,00 (11,0) (11,0) 6 0 (11,0) 10 (11,0) (11,0) 0 (11,0 (11,0) 0 9P (142) (18,5) (12,21) 1,1) 15,18) (18,18) (2,12) (16,8) (4.9) (1,22) (15,14) (10,19) (22,16) (15,9) 0,18) (22,7) (12,2 10P (12.1) (4.14) (15,14) ||12,21) (15,9) 49) |(12,2) (15,14) (15,14 (12,2) (11,0 14,9) (12,21) (11,0) 49) (12,2) (15,9) 11P (0.18) (22,16) (4,14) 20,2 |(16,8) (22,7) (18,18 (5,18) 12,2 (20,21) (15,9) ||2,12) (10,4) (1514) |(1,22) (10,19) (4,9) 12P (15,14 (15,14) (11,0) 15,9 (11,0) (15,9) (15,14) (11,0) (11,0) (15,14) 15,9) (15,9) (15,9) (15,14) (11,0) 13P (207) (212) (4,9) 10,18 (16,15) (2,11) (10,4) (5,5) (12,21) (0,5) (15,14) (22,16) (18,18) (15,9) (14,21) (18,5) (4,14) 14P (49) (12,21) (15,9 14,14 15.14) (12,2 14,9) (15,9) (15,9 (4,9) (11,0 112,2 4,14) 11,0) (122) (4,9 (15,14) 15P (1.1) (10,19) (12,2) (14,2 5,5) |(10,4) (22,16) (16,15) (4,14 (14,21) (15,9) [18,5) (2,12) (15,14) (20,2 (2,11) (12,21) Question no 8. Consider the elliptic curve E: y2 = x3- 2x + 2 (mod 23) 1. Verify that the point P = (22, 7) is on E. (1 marks) 2. Calculate the value of 2P. (use extended Euclidian algorithm for calculating modulo inverse). (3+4 marks) 3. Suppose this E and P = (22, 7) are used in an ECC Diffie-Hellman key exchange, where Alice chooses the secret value a = 9 and Bob chooses the secret value b = 12. What value does Alice send to Bob? (1 marks) What does Bob send to Alice? (1 marks)

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