[Bitcoin] 楕円曲線暗号 スカラー倍算

同じ点を加算することができるため、次のような表記になる
(170,142)+(170,142) = 2(170,142)

この加算は何度も繰り返し行うことができ、スカラー倍算と呼ぶ
計算をせずに予測することが非常に難しい

prime = 223
a = FieldElement(num=0, prime=prime)
b = FieldElement(num=7, prime=prime)
x1 = FieldElement(num=192, prime=prime)
y1 = FieldElement(num=105, prime=prime)
p = Point(x1, y1, a, b)
print(p+p)
x1 = FieldElement(num=143, prime=prime)
y1 = FieldElement(num=98, prime=prime)
p = Point(x1, y1, a, b)
print(p+p)
x1 = FieldElement(num=47, prime=prime)
y1 = FieldElement(num=71, prime=prime)
p = Point(x1, y1, a, b)
print(p+p)
print(p+p+p+p)
print(p+p+p+p+p+p+p+p)
print(p+p+p+p+p+p+p+p+p+p+p+p+p+p+p+p+p+p+p+p+p)

for s in range(1, 21):
	result = s*p
	print('{}*(47,71)=({},{})'.format(s,result.x.num, result.y.num))

スカラー倍数ははっきりとしたパターンがないため、逆の離散対数は難しい

## 数学の群

prime = 223
a = FieldElement(num=0, prime=prime)
b = FieldElement(num=7, prime=prime)
x = FieldElement(num=15, prime=prime)
y = FieldElement(num=86, prime=prime)
p = Point(x, y, a, b)
inf = Point(None, None, a, b)
product = p
count = 1
while product != inf:
	product += p
	count += 1
print(count)

    def __rmul__(self, coefficient):
        coef = coefficient
        current = self
        result = self.__class__(None, None, self.a, self.b)
        while coef:
            if coef & 1:
                result += current
            current += current
            coef >>= 1
        return result