Elliptische Kurve/Kurze Weierstraßform/Addition/Ohne quadratische Terme/Aufgabe/Lösung

Es ist

${}{\begin{aligned}\alpha ^{2}&=\left({\frac {y_{2}-y_{1}}{x_{2}-x_{1}}}\right)^{2}\\&={\frac {y_{2}^{2}+y_{1}^{2}-2y_{1}y_{2}}{{\left(x_{2}-x_{1}\right)}^{2}}}\\&={\frac {x_{2}^{3}+ax_{2}+b+x_{1}^{2}+ax_{1}+b-2y_{1}y_{2}}{{\left(x_{2}-x_{1}\right)}^{2}}}\\&={\frac {x_{1}^{3}+x_{2}^{3}+a(x_{1}+x_{2})+2b-2y_{1}y_{2}}{{\left(x_{2}-x_{1}\right)}^{2}}}\end{aligned}}$

und

${}{\begin{aligned}&\,\,\,\,\,\,\,\alpha ^{3}\\&={\frac {y_{2}-y_{1}}{x_{2}-x_{1}}}\cdot \alpha ^{2}\\&={\frac {{\left(y_{2}-y_{1}\right)}{\left(x_{1}^{3}+x_{2}^{3}+a(x_{1}+x_{2})+2b-2y_{1}y_{2}\right)}}{{\left(x_{2}-x_{1}\right)}^{3}}}\\&={\frac {y_{2}{\left(x_{1}^{3}+x_{2}^{3}+a(x_{1}+x_{2})+2b\right)}-y_{1}{\left(x_{1}^{3}+x_{2}^{3}+a(x_{1}+x_{2})+2b\right)}-2y_{1}y_{2}^{2}+2y_{1}^{2}y_{2}}{{\left(x_{2}-x_{1}\right)}^{3}}}\\&={\frac {y_{2}{\left(x_{1}^{3}+x_{2}^{3}+a(x_{1}+x_{2})+2b+2(x_{1}^{3}+ax_{1}+b)\right)}-y_{1}{\left(x_{1}^{3}+x_{2}^{3}+a(x_{1}+x_{2})+2b+2(x_{2}^{3}+ax_{2}+b)\right)}}{{\left(x_{2}-x_{1}\right)}^{3}}}\\&={\frac {y_{2}{\left(3x_{1}^{3}+x_{2}^{3}+a(3x_{1}+x_{2})+4b\right)}-y_{1}{\left(x_{1}^{3}+3x_{2}^{3}+a(x_{1}+3x_{2})+4b\right)}}{{\left(x_{2}-x_{1}\right)}^{3}}}.\,\end{aligned}}$

Somit ist die ${}x$-Koordinate der Summe gleich

${}{\begin{aligned}\alpha ^{2}-x_{1}-x_{2}&={\frac {x_{1}^{3}+x_{2}^{3}+a(x_{1}+x_{2})+2b-2y_{1}y_{2}}{{\left(x_{2}-x_{1}\right)}^{2}}}-x_{1}-x_{2}\\&={\frac {x_{1}^{3}+x_{2}^{3}+a(x_{1}+x_{2})+2b-2y_{1}y_{2}-{\left(x_{1}+x_{2}\right)}{\left(x_{2}-x_{1}\right)}^{2}}{{\left(x_{2}-x_{1}\right)}^{2}}}\\&={\frac {x_{1}x_{2}^{2}+x_{1}^{2}x_{2}+a(x_{1}+x_{2})+2b-2y_{1}y_{2}}{{\left(x_{2}-x_{1}\right)}^{2}}}\end{aligned}}$

und die ${}y$-Koordinate der Summe ist gleich

${}{\begin{aligned}&\,\,\,\,\,\,\,-\alpha ^{3}+\alpha (2x_{1}+x_{2})-y_{1}\\&=-\alpha ^{3}+{\frac {y_{2}-y_{1}}{x_{2}-x_{1}}}(2x_{1}+x_{2})-y_{1}\\&=-\alpha ^{3}+{\frac {{\left(y_{2}-y_{1}\right)}{\left(2x_{1}+x_{2}\right)}-y_{1}{\left(x_{2}-x_{1}\right)}}{x_{2}-x_{1}}}\\&=-\alpha ^{3}+{\frac {-x_{1}y_{1}+2x_{1}y_{2}-2x_{2}y_{1}+x_{2}y_{2}}{x_{2}-x_{1}}}\\&=-\alpha ^{3}+{\frac {-y_{1}{\left(x_{1}+2x_{2}\right)}{\left(x_{2}-x_{1}\right)}^{2}+y_{2}{\left(2x_{1}+x_{2}\right)}{\left(x_{2}-x_{1}\right)}^{2}}{{\left(x_{2}-x_{1}\right)}^{3}}}\\&={\frac {y_{1}f+y_{2}g}{{\left(x_{2}-x_{1}\right)}^{3}}}\,\end{aligned}}$

mit

${}{\begin{aligned}f&=x_{1}^{3}+3x_{2}^{3}+a(x_{1}+3x_{2})+4b-{\left(x_{1}+2x_{2}\right)}{\left(x_{2}-x_{1}\right)}^{2}\\&=x_{1}^{3}+3x_{2}^{3}+a(x_{1}+3x_{2})+4b-x_{1}^{3}+3x_{1}x_{2}^{2}-2x_{2}^{3}\\&=x_{2}^{3}+3x_{1}x_{2}^{2}+a(x_{1}+3x_{2})+4b\end{aligned}}$

und

${}{\begin{aligned}g&=-3x_{1}^{3}-x_{2}^{3}-a(3x_{1}+x_{2})-4b+{\left(2x_{1}+x_{2}\right)}{\left(x_{2}-x_{1}\right)}^{2}\\&=-3x_{1}^{3}-x_{2}^{3}-a(3x_{1}+x_{2})-4b+2x_{1}^{3}-3x_{1}^{2}x_{2}+x_{2}^{3}\\&=-x_{1}^{3}-3x_{1}^{2}x_{2}-a(3x_{1}+x_{2})-4b.\end{aligned}}$

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