# 椭圆曲线

${\displaystyle y^{2}=x^{3}+ax+b\,}$

${\displaystyle y^{2}=P(x)\,}$，其中P为任一没有重根的三次或四次多项式，然后可得到一亏格1的无奇点平面曲线，其通常亦被称为椭圆曲线。更一般化地，一亏格1的代数曲线，如两个三维二次曲面相交，即称为椭圆曲线。

## 实数域上的椭圆曲线

${\displaystyle y^{2}=x^{3}+ax+b}$

${\displaystyle \Delta =-16(4a^{3}+27b^{2})}$

## 群

${\displaystyle (sx+d)^{2}=x^{3}-px-q\,\Rightarrow 0=x^{3}-s^{2}x^{2}-2xd-px-q-d^{2}}$

${\displaystyle P,Q,R}$是两线的交点，即方程的解。有：

${\displaystyle 0=(x-x_{P})(x-x_{Q})(x-x_{R})=x^{3}+x^{2}(-x_{P}-x_{Q}-x_{R})+x(x_{P}x_{Q}+x_{P}x_{R}+x_{Q}x_{R})-x_{P}x_{Q}x_{R}}$

${\displaystyle x_{R}=s^{2}-x_{P}-x_{Q}\,}$
${\displaystyle y_{R}=-y_{P}+s(x_{P}-x_{R})\,}$

${\displaystyle x_{P}=x_{Q}\,}$

• ${\displaystyle y_{P}=-y_{Q}\,}$${\displaystyle P+Q=0\,}$
• ${\displaystyle y_{P}=y_{Q}\,}$${\displaystyle R=2P\,}$，其值为：
${\displaystyle s={\frac {3{x_{P}}^{2}-p}{2y_{P}}}\,}$
${\displaystyle x_{R}=s^{2}-2x_{P}\,}$
${\displaystyle y_{R}=-y_{P}+s(x_{P}-x_{R})\,}$

## 一般域上的椭圆曲线

${\displaystyle y^{2}=x^{3}-px-q}$

${\displaystyle y^{2}=4x^{3}+b_{2}x^{2}+2b_{4}x+b_{6}}$

${\displaystyle y^{2}+a_{1}xy+a_{3}y=x^{3}+a_{2}x^{2}+a_{4}x+a_{6}}$

## 参考文献

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