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# 椭圆曲线加法

## 椭圆曲线在$$\mathbb{R}$$上的加法

$$K$$为$$\mathbb{R}$$，$$K$$的特征不为$$2,3$$时： $$y^2=x^3+a\_4x+a\_6$$ $$\Delta=-16\left(4a\_4^3+27a\_6^2\right)\neq0$$

求逆运算$$-P=\left(x\_1,-y\_1\right)$$

> * $$P\left(x\_1,y\_1\right)\neq Q\left(x\_2,y\_2\right)$$，$$P,Q$$不互逆
>
>   $$\left{\begin{array}{**lr**} x\_3=\lambda^2-x\_1-x\_2\ y\_3=\lambda\left(x\_1-x\_3\right)-y\_1\ \lambda=\left(y\_2-y\_1\right)/\left(x\_2-x\_1\right) \end{array} \right.$$
> * $$P\left(x\_1,y\_1\right)=Q\left(x\_2,y\_2\right)=2P\left(x\_1,y\_1\right)$$
>
>   $$\left{\begin{array}{**lr**} x\_3=\lambda^2-2x\_1\ y\_3=\lambda\left(x\_1-x\_3\right)-y\_1\ \lambda=\left(3x\_1^2+a\_4\right)/\left(2y\_1\right) \end{array} \right.$$

## 椭圆曲线在$$F\_p$$上的加法

$$K$$为$$F\_p$$，$$p$$为大于$$3$$的素数，$$K$$的特征不为$$2,3$$时： $$y^2=x^3+a\_4x+a\_6\left(\mod p\right)$$ $$\Delta=-16\left(4a\_4^3+27a\_6^2\right)\neq0\left(\mod p\right)$$

求逆运算$$-P=\left(x\_1,p-y\_1\right)$$

> * $$P\left(x\_1,y\_1\right)\neq Q\left(x\_2,y\_2\right)$$，$$P,Q$$不互逆
>
>   $$\left{\begin{array}{**lr**} x\_3=\lambda^2-x\_1-x\_2\left(\mod p\right)\ y\_3=\lambda\left(x\_1-x\_3\right)-y\_1\left(\mod p\right)\ \lambda=\left(y\_2-y\_1\right)\cdot{\left(x\_2-x\_1\right)}^{-1}\left(\mod p\right)\end{array} \right.$$
> * $$P\left(x\_1,y\_1\right)=Q\left(x\_2,y\_2\right)=2P\left(x\_1,y\_1\right)$$
>
>   $$\left{\begin{array}{**lr**} x\_3=\lambda^2-2x\_1\left(\mod p\right)\ y\_3=\lambda\left(x\_1-x\_3\right)-y\_1\left(\mod p\right)\ \lambda=\left(3x\_1^2+a\_4\right)\cdot{\left(2y\_1\right)}^{-1}\left(\mod p\right)\end{array} \right.$$

* $$F\_p$$上$$E$$的阶为$$#\left(E\left(F\_p\right)\right)=p+1+\sum\_\limits{x=0}^{p-1}{\left(\frac{x^3+a\_4x+a\_6}{p}\right)}$$，括号为勒让德符号
* 当循环群$$E$$的阶$$n$$是足够大的素数时，这个循环群中的离散对数问题是困难的

## 椭圆曲线在$$F\_{2^n}$$上的加法

$$K$$为$$F\_{2^n}$$，$$K$$的特征为$$2$$时： $$y^2+xy=x^3+a\_2x^2+a\_6$$ $$\Delta=a\_6\neq0$$ 求逆运算$$-P=\left(x\_1,x\_1+y\_1\right)$$

> * $$P\left(x\_1,y\_1\right)\neq Q\left(x\_2,y\_2\right)$$，$$P,Q$$不互逆
>
>   $$\left{\begin{array}{**lr**} x\_3=\lambda^2+\lambda+x\_1+x\_2+a\_2\ y\_3=\lambda\left(x\_1+x\_3\right)+x\_3+y\_1\ \lambda=\left(y\_2+y\_1\right)/\left(x\_2+x\_1\right) \end{array} \right.$$
> * $$P\left(x\_1,y\_1\right)=Q\left(x\_2,y\_2\right)=2P\left(x\_1,y\_1\right)$$
>
>   $$\left{\begin{array}{**lr**} x\_3=\lambda^2+\lambda+a\_2\ y\_3=x\_1^2+\left(\lambda+1\right)x\_3\ \lambda=\left(x\_1^2+y\_1\right)/\left(x\_1\right) \end{array} \right.$$