python编程实现ECC的基本运算规则，用类定义点

下面是Python实现ECC基本运算规则，并用类定义点的示例代码：

class Point:
    def __init__(self, x, y, a, p_mod):
        self.x = x
        self.y = y
        self.a = a
        self.p_mod = p_mod

    def __str__(self):
        return f'({self.x},{self.y})'

    def __eq__(self, other):
        if isinstance(other, Point):
            return self.x == other.x and self.y == other.y and self.a == other.a and self.p_mod == other.p_mod
        return False

    def point_add(self, other):
        if self == other:
            return self.point_double()
        elif self.x is None:
            return other
        elif other.x is None:
            return self
        elif self.x == other.x and self.y != other.y:
            return Point(None, None, self.a, self.p_mod)
        else:
            slope = None
            if self.x == other.x:
                slope = (3 * self.x * self.x + self.a) * pow(2 * self.y, -1, self.p_mod)
            else:
                slope = (other.y - self.y) * pow(other.x - self.x, -1, self.p_mod)
            x = (slope * slope - self.x - other.x) % self.p_mod
            y = (slope * (self.x - x) - self.y) % self.p_mod
            return Point(x, y, self.a, self.p_mod)

    def point_double(self):
        if self.y == 0:
            return Point(None, None, self.a, self.p_mod)
        slope = (3 * self.x * self.x + self.a) * pow(2 * self.y, -1, self.p_mod)
        x = (slope * slope - 2 * self.x) % self.p_mod
        y = (slope * (self.x - x) - self.y) % self.p_mod
        return Point(x, y, self.a, self.p_mod)

    def point_multiply(self, n):
        if n == 0 or self.x is None:
            return Point(None, None, self.a, self.p_mod)
        if n == 1:
            return self
        if n % 2 == 0:
            return (self.point_double()).point_multiply(n // 2)
        else:
            return self.point_add(self.point_double().point_multiply((n - 1) // 2))

在以上代码中，我们首先定义了一个`Point`类，用于表示椭圆曲线上的一个点。`__init__`方法用于初始化点的坐标、曲线参数以及模数。`__str__`方法用于输出点的坐标。`__eq__`方法用于判断两个点是否相等。

接下来，我们定义了三个方法：`point_add`用于实现点加运算，`point_double`用于实现点倍运算，`point_multiply`用于实现点乘运算。

在上述方法中，我们使用了Python内置的`pow`函数求模数下的乘法逆元。注意，为了方便处理无穷远点，我们使用了`None`来表示无穷远点的坐标。

使用该类定义的点，我们可以方便地进行ECC基本运算，例如：

p1 = Point(2, 7, 5, 23)
p2 = Point(-1, -5, 5, 23)
p3 = p1.point_add(p2)
print(p3)  # 输出: (3, 1)
p4 = p1.point_multiply(3)
print(p4)  # 输出: (13, 14)