# 基本概念

## **Weierstrass方程**

域$$K$$上的椭圆曲线$$E$$方程为$$E:y^2+a\_1xy+a\_3y=x^3+a\_2x^2+a\_4x+a\_6$$

其中$$a\_1,a\_2,a\_3,a\_4\in K,\Delta\neq0$$

$$\Delta=-d\_2^2d\_8-8d\_4^2-27d\_6^3+9d\_2d\_4d\_6$$

$$d\_2=a\_1^2+4a\_2$$

$$d\_4=2a\_4+a\_1a\_3$$

$$d\_6=a\_3^2+4a\_6$$

$$d\_8=a\_1^2a\_6+4a\_2a\_6-a\_1a\_3a\_4+a\_2a\_3^2-a\_4^2$$

* 无穷远点$$\left{0\left(\infty,\infty\right)\right}=\left{\left(x,y\right)\in L\times L:E:y^2+a\_1xy+a\_3y-x^3-a\_2x^2-a\_4x-a\_6=0\right}$$

## **简化Weierstrass方程**

$$\left(x^\prime,y^\prime\right)\rightarrow\left(\frac{x-3a\_1^2-12a\_2}{36},\frac{y-3a\_1x}{216}-\frac{a\_1^3+4a\_1a\_2-12a\_3}{24}\right)$$

得到$$E:{y^\prime}^2={x^\prime}^3+a\_4x^\prime+a\_6$$

$$\Delta=-16\left(4a\_4^3+27a\_6^2\right)\neq0$$

## **椭圆曲线与群**

* $$K$$为$$\mathbb{R}$$，$$K$$的特征不为$$2,3$$时：

  $$y^2=x^3+a\_4x+a\_6$$

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

  $$y^2=x^3+a\_4x+a\_6\left(\mod p\right)$$

  $$\Delta=-16\left(4a\_4^3+27a\_6^2\right)\neq0\left(\mod p\right)$$
* $$K$$为$$F\_{2^n}$$，$$K$$的特征为$$2$$时：

  $$y^2+xy=x^3+a\_2x^2+a\_6$$

  $$\Delta=a\_6\neq0$$
* 其解为一个二元组$$\<x,y>,x,y\in K$$，将此二元组描画到椭圆曲线上便为一个点，称其为解点
* 解点构成群
  * 单位元：$$0\left(\infty,\infty\right)$$简记为$$0$$
  * 逆元：解点$$R\left(x,y\right)=R^{-1}\left(x,-y\right)$$，$$0\left(\infty,\infty\right)=-0\left(\infty,\infty\right)$$
  * 加法：$$kP=P+\cdots+P$$，有时记为$$P^k$$

    > **椭圆曲线加法**
    >
    > ![](https://upload.wikimedia.org/wikipedia/commons/thumb/c/c1/ECClines.svg/1020px-ECClines.svg.png)


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