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I have the elliptic curve equation $E(\mathbb F_{5^2}): y^2=x^3+10x+17$, and I have that the points $(3,7)$ and $(8,3)$ belong to $E$. According to the addition law, the slope $\lambda=(y_2-y_1)/(x_2-x_1)$ exists when the $\gcd(x_2-x_1,p)=1$, in this case is not equal to $1$. What should I do? Should I say, in this case, $(3,7)+(8,3)= \infty$? I appreciate the help.

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Since $8\equiv 3\pmod{5}$, the points have the same $x$-coordinate; you have a "vertical" line and the third point of intersection with the curve is the point at infinity. – Arturo Magidin May 12 '12 at 2:58
Thank you a lot Arturo. – megjoh May 12 '12 at 3:19
up vote 3 down vote accepted

As explained by Arturo, in the field $F_{25}$ we have $(3,7)=(3,2)$ and $(8,3)=(3,3)=(3,-2)$. Therefore the two points that you want to add are negatives of each other (on the same vertical line), so their sum is the point at infinity.

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