4.4 QUIZ 3 ~ Solutions Elliptic Curve Groups over F2m 1. Does the elliptic curve equation y2 + xy = x3 + g5x2 + 6 define a group over F23? Since the parameter b = 6 is not zero, the equation y2 + xy = x3 + g5x2 + 6 does define an elliptic curve group over F23 2. Do the points P(g3, g6) and Q(g5, g2) lie on the elliptic curve y2 + xy = x3 + g2x2 + g6 over F23? The point P(g3,g6) is on the elliptic curve y2 + xy = x3 + g2x2 + g6 over F23 since that equation holds true: (g6)2 + (g3)(g6) = (g3)3 + g2(g3)2 + g6 g5 + g2 = g2 + g + g6 (111) + (100) = (100) + (010) + (101) (011) = (011) g3 = g3 However, the point Q(g5)(g2) is not on the elliptic curve, since the equation disagrees: (g2)2 + (g5)(g2) = (g5)3 + g2(g5)2 + g6 g4 + 1 = g + g5 + g6 (110) + (001) = (001) + (111) + (101) (111) = (000) g5 = 0 which is false. 3. What are the negatives of the following elliptic curve points over F23? P(g3,g6) Q(g,0) R(0,g3) The negatives of the points are defined by (xP, xP + yP) -P = (g3, g3 + g6) = (g3, g4) -Q = (g, g + 0) = (g, g) -R = (0, 0 + g3) = (0, g3)
	4. In the elliptic curve group defined by y2 + xy = x3 + g2x2 + g6 over F23, what is P + Q if P = (g2,g6) and Q = (g5,g5)? P + Q = R where: l = (yP - yQ) / (xP + xQ) = (g6 + g5) / (g2 + g5) = g / g3 = g-2 = g5 xR = l2 + l + xP + xQ + a = g3 + g5 +g2 + g5 + g2 = g3 yR = l(xP + xR) + xR + yP = g5 * (g2 + g3) + g5 + g6 = g5 * g5 + g3 + g6 = g3 + g3 + g6 = g6 Thus P + Q = (g3, g6) 5. In the elliptic curve group defined by y2 + xy = x3 + g2x2 + g6 over F23, what is 2P if P = (g3, g4)? 2P = R where: l = xP + yP / 2xP = g3 + g4 / g3 = g3 + g = 1 xR = l2 + l + a = 1 + 1 + g2 = g2 yR = xP + (l + 1) * xR = g6 + 0 * g2 = g6 Thus 2P = (g2, g6)
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