2.4 QUIZ 1 ~ Solutions Elliptic Curve Groups over real numbers 1. Does the elliptic curve equation y2 = x3 - 7x - 6 over real numbers define a group? Yes, since 4a3 + 27b2 = 4(-7)3 + 27(-6)2 = -400 The equation y2 = x3 - 7x - 6 does define an elliptic curve group because 4a3 + 27b2 is not 0. 2. What is the additive identity of regular integers? The additive identity of regular integers is 0, since x + 0 = x for all integers. 3. Is (4,7) a point on the elliptic curve y2 = x3 - 5x + 5 over real numbers? Yes, since the equation holds true for x = 4 and y = 7: (7)2 = (4)3 - 5(4) + 5 49 = 64 - 20 + 5 49 = 49
	4. What are the negatives of the following elliptic curve points over real numbers? P(-4,-6), Q(17,0), R(3,9), S(0,-4) The negative is the point reflected through the x-axis. Thus -P(-4,6), -Q(17,0), -R(3,-9), -S(0,4) 5. In the elliptic curve group defined by y2 = x3 - 17x + 16 over real numbers, what is P + Q if P = (0,-4) and Q = (1,0)? From the Addition formulae: l = (yP - yQ) / (xP - xQ) = (-4 - 0) / (0 - 1) = 4 xR = l2 - xP - xQ = 16 - 0 - 1 = 15 and yR = -yP + l(xP - xR) = 4 + 4(0 - 15) = -56 Thus P + Q = (15, -56) 6. In the elliptic curve group defined by y2 = x3 - 17x + 16 over real numbers, what is 2P if P = (4, 3.464)? From the Doubling formulae: l = (3xP2 + a) / (2yP ) = (3*(4)2 + (-17)) / 2*(3.464) = 31 / 6.928 = 4.475 xR = l2 - 2xP = (4.475)2 - 2(4) = 20.022 - 8 = 12.022 and yR = -yP + l(xP - xR) = -3.464 + 4.475(4 - 12.022) = - 3.464 - 35.898 = -39.362 Thus 2P = (12.022, -39.362)
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