3.4 QUIZ 2 ~ Solutions Elliptic Curve Groups over Fp 1. Does the elliptic curve equation y2 = x3 + 10x + 5 define a group over F17? No, since: = 4(10)3 + 27(5)2 mod 17 = 4675 mod 17 = 0 Thus this elliptic curve does not define a group because 4a3 + 27b2 mod p is 0 2. Do the points P(2,0) and Q(6,3) lie on the elliptic curve y2 = x3 + x + 7 over F17? The point P(2,0) is on the elliptic curve since both sides of the equation agree: (0)2 mod 17 = (2)3 + 3 + 7 mod 17 0 mod 17 = 17 mod 17 0 = 0. However, the point Q(6,3) is not on the elliptic curve since the equation is false: (3)2 mod 17 = (6)3 + 6 + 7 mod 17 9 mod 17 = 229 mod 17 9 = 8, does not agree. 3. What are the negatives of the following elliptic curve points over F17? P(5,8) Q(3,0) R(0,6) The negative of a point P = (xP, yP) is the point -P = (xP, -yP mod p). Thus -P(5,9) -Q(3,0) -R(0,11)
	4. In the elliptic curve group defined by y2 = x3 + x + 7 over F17, what is P + Q if P = (2,0) and Q = (1,3)? l = (yP - yQ) / (xP - xQ) mod p = (-3) / 1 mod 17 = -3 mod 17 = 14 xR = l2 - xP - xQ mod p = 196 - 2 - 1 mod 17 = 193 mod 17 = 6 yR = -yP + l(xP - xR) mod p = 0 + 14*(2 - 6) mod 17 = -56 mod 17 = 12 Thus P + Q = (6,12) 6. In the elliptic curve group defined by y2 = x3 + x + 7 over F17, what is 2P if P = (1, 3)? l = (3xP2 + a) / (2yP ) mod p = (3 + 1) * 6 - 1 mod 17 = 4 * 3 mod 17 = 12 xR = l2 - 2xP mod p = 144 - 2 mod 17 = 142 mod 17 = 6 yR = -yP + l(xP - xR) mod p = -3 + 12 * (1 - 6) mod 17 = -63 mod 17 = 5 Thus 2P = (6,5)
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