2.4. The group structure on E(Q) 25 P Q R P+Q Figure 1. Addition of points on an elliptic curve E(Q). Let us find P + Q. First, we find the equation of the line L = PQ. The slope must be m = 0 − 6 5 − (−4) = − 6 9 = − 2 3 and the line is L : y = − 2 3 (x − 5). Now we find the third point of intersection of L and E by solving y = − 2 3 (x − 5) y2 = x3 − 25x. Plugging the first equation into the second one, we obtain an equation x3 − 4 9 x2 − 185 9 x − 100 9 = 0, which factors as (x − 5)(x + 4)(9x + 5) = 0. The first two factors are expected, since we already knew that P = (5,0) and Q = (−4,6) are in L ∩ E. The third point of intersection must have x = − 5 9 , y = − 2 3 (x − 5) = 100 27 and, indeed, R = (− 5 9 , 100 27 ) is a point in
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