Category: Blockchain

### Bitcoinのsecp256k1
a = 0, b = 7 つまり y^2 = x^3 + 7
p = 2^256 – 2^32 ^ 077
Gx = 0x79be667ef9dcbbac55a06295ce870b07029bfcdb2dce28d959f2815b16f81798
(x座標・16進数)
Gy = 0x483ada7726a3c4655da4fbfc0e1108a8fd17b448a68554199c47d08ffb10d4b8
(y座標)
n = 0xfffffffffffffffffffffffffffffffebaaedce6af48a03bbfd25e8cd0364141
– 256ビットで表す
– ビットコインの秘密鍵は途方もない

secp256k1の計算

gx = 0x79be667ef9dcbbac55a06295ce870b07029bfcdb2dce28d959f2815b16f81798
gy = 0x483ada7726a3c4655da4fbfc0e1108a8fd17b448a68554199c47d08ffb10d4b8
p = 2**256 - 2**32 - 977
print(gy**2 % p == (gx**3 + 7) % p)

gx = 0x79be667ef9dcbbac55a06295ce870b07029bfcdb2dce28d959f2815b16f81798
gy = 0x483ada7726a3c4655da4fbfc0e1108a8fd17b448a68554199c47d08ffb10d4b8
p = 2**256 - 2**32 - 977
n = 0xfffffffffffffffffffffffffffffffebaaedce6af48a03bbfd25e8cd0364141
x = FieldElement(gx, p)
y = FieldElement(gy, p)
seven = FieldElement(7, p)
zero = FieldElement(0, p)
G = Point(x, y, zero, seven)
print(n*G)

class S256Field(FieldElement):

	def __init__(self, num, prime=None):
		super().__init__(num=num, prime=P)

	def __repr__(self):
		return '{:x}'.format(self.num).zfill(64)


class S256Point(Point):

	def __init__(self, x, y, a=None, b=None):
		a, b = S256Field(A), S256Field(B)
		if type(x) == int:
			super().__init__(x=S256Field(x), y=S256Field(y), a=a, b=b)
		else:
			super().__init__(x=x, y=y, a=a, b=b)

	def __rmul__(self, coefficient):
		coef = coefficient % N
		return super().__rmul__(coef)

P = 2**256 - 2**32 - 977
A = 0
B = 7
N = 0xfffffffffffffffffffffffffffffffebaaedce6af48a03bbfd25e8cd0364141
G = S256Point(
	0x79be667ef9dcbbac55a06295ce870b07029bfcdb2dce28d959f2815b16f81798,
	0x483ada7726a3c4655da4fbfc0e1108a8fd17b448a68554199c47d08ffb10d4b8)


print(N*G)

なんか一気に来たな

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

楕円曲線暗号: メッセージの署名と検証

– 実数上の楕円曲線
実数は有理数、無理数を含む数(π, sqrt(2), e +7√19 など)

### 有限体上の楕円曲線
F103 でのy^2 = x^3 + 7
y^2 = 64^2 % 103 = 79
x^3 + 7 = (17^3 + 7) % 103 = 79

有限体に負が存在しないと、y^2の項によって真ん中の線に対して対称になる

a = FieldElement(num=0, prime=223)
b = FieldElement(num=7, prime=223)
x = FieldElement(num=192, prime=223)
y = FieldElement(num=105, prime=223)
p1 = Point(x, y, a, b)
print(p1)

$ python3 app.py
<__main__.Point object at 0x7f1a3ceef2b0>

### 有限体における点の加算

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)
x2 = FieldElement(num=17, prime=prime)
y2 = FieldElement(num=56, prime=prime)
p1 = Point(x1, y1, a, b)
p2 = Point(x2, y2, a, b)
print(p1+p2)

prime = 223
a = FieldElement(num=0, prime=prime)
b = FieldElement(num=7, prime=prime)
p1 = Point(FieldElement(170, prime), FieldElement(142, prime), a, b)
p2 = Point(FieldElement(60, prime), FieldElement(139, prime), a, b)
print(p1+p2)
p1 = Point(FieldElement(47, prime), FieldElement(71, prime), a, b)
p2 = Point(FieldElement(17, prime), FieldElement(56, prime), a, b)
print(p1+p2)
p1 = Point(FieldElement(143, prime), FieldElement(98, prime), a, b)
p2 = Point(FieldElement(76, prime), FieldElement(66, prime), a, b)
print(p1+p2)

ほう、なるほどー

class Point:

	def __init__(self, x, y, a, b):
		self.a = a
		self.b = b
		self.x = x
		self.y = y
		if self.x is None and self.y is None:
			return
		if self.y**2 != self.x**3 + a * x + b:
			raise ValueError('({},{}) is not on the curve'.format(x, y))

	def __add__(self, other):
		if self.a != other.a or self.b != other.b:
			raise TypeError('Points {}, {} are not on the same curve'.format(self, other))

		if self.x == other.x and self.y != other.y:
			return self.__class__(None, None, self.a, self.b)

		if self.x is None:
			return other
		if other.x is None:
			return self

	def __eq__(self, other):
		return self.x == other.x and self.y == other.y \
			and self.a == other.a and self.b == other.b

	def __ne__(self, other):
		return not (self == other)

p1 = Point(-1, -1, 5, 7)
p2 = Point(-1, 1, 5, 7)
inf = Point(None, None, 5, 7)
print(p1 + inf)
print(inf + p2)
print(p1 + p2)

$ python3 app.py
<__main__.Point object at 0x7f014fc2fbb0>
<__main__.Point object at 0x7f014fbf07f0>
None

### x1 != x2 の点の加算
P1 = (x1,y1), P2 = (x2, y2), P3 = (x3, y3)
s = (y2 – y1)/(x2 – x1)
x3 = s^2 – x1 – x2
y3 = s(x1 – x3) – y1

x1, y1 = 2, 5
x2, y2 = -1, -1
s = (y2 - y1) / (x2 - x1)
x3 = s**2 - x1 - x2
y3 = s * (x1 - x3) - y1
print(x3, y3)

		if self.x != other.x:
			s = (other.y - self.y) / (other.x - self.x)
			x = s**2 - self.x - other.x
			y = s * (self.x - x) - self.y
			return self.__class__(x, y, self.a, self.b)

### P1 = P2 の時の点の加算
x3 = s^2 – 2×1
y3 = s(x1 – x3) – y1

a, x1, y1 = 5, -1, -1
s = (3 * x1**2 + a) / (2 * y1)
x3 = s**2 - 2*x1
y3 = s * (x1 - x3) - y1
print(x3, y3)

		if self == other:
			s = (3 * self.x**2 + self.a) / (2 * self.y)
			x = s**2 - 2 * self.x
			y = s * (self.x - x) - self.y
			return self.__class__(x, y, self.a, self.b)

		if self == other and self.y == 0 * self.x:
			return self.__class__(None, None, self.a, self.b)

なるほどー

### 楕円曲線 定義
y^2 = x^3 + ax + b

class Point:

	def __init__(self, x, y, a, b):
		self.a = a
		self.b = b
		self.x = x
		self.y = y
		if self.y**2 != self.x**3 + a * x + b:
			raise ValueError('({},{}) is not on the curve'.format(x, y))

	def __eq__(self, other):
		return self.x == other.x and self.y == other.y \
			and self.a == other.a and self.b == other.b

	def __ne__(self, other):
		return not (self == other)

p1 = Point(-1, -1, 5, 7)
p1 = Point(-1, -2, 5, 7)

一次方程式
y = ax^2 + bx + c

二次方程式
y = ax^3 + bx^2 + cx + d

楕円曲線は左辺がy^2となっている　
x軸に対して対称のグラフ

ビットコインに用いられる楕円曲線はsecp256k1と呼ばれる
y^2 = x^3 + 7

def on_curve(x, y):
	return y**2 == x**3 + 5*x + 7

print(on_curve(2,4))
print(on_curve(-1,-1))
print(on_curve(18,77))
print(on_curve(5,7))

### 点の加算
曲線上の二つの点を演算して、三つ目の点を得る
第三の点をx軸に対して対称にした点が点の加算の解になる
　L 点の加算は容易に予測できない(同一性、可換性、結合性、可逆性)

なるほど、こういうことをやっとるんか

四則演算(加算、減算、乗算、除算)について閉じている有限体を作るために利用できる道具がモジュロ演算
モジュロ演算：1つの数を別の数で割った余りの数を求めること
(3 – 16) % 12 = 11
(12 + 843) % 60 = 15
(23 + 97) % 60 = 0

pythonのモジュロ演算

print(7 % 3)
print(-27 % 13) 

$ python3 app.py
1
12

有限体上の加算が閉じていることを確認する必要がある
F19 = {0, 1, 2…18}

F57

prime = 57
print((44+33)%prime)
print((9-29)%prime)
print((17+42+49)%prime)
print((52-30-38)%prime)

### 加算と減算

// 省略
	def __add__(self, other):
		if self.prime != other.prime:
			raise TypeError('Cannot add two numbers in different Fields')
		num = (self.num + other.num) % self.prime
		return self.__class__(num, self.prime)

	def __sub__(self, other):
		if self.prime != other.prime:
			raise TypeError('Cannot subtract two numbers in different Fields')
		num = (self.num - other.num) % self.prime
		return self.__class__(num, self.prime)		

a = FieldElement(7, 13)
b = FieldElement(12, 13)
c = FieldElement(6, 13)
print(a+b==c)

### 乗算とべき乗
べき乗は乗算を繰り返し

prime = 97
print(95*45*31 % prime)
print(17*13*19*44 % prime)
print(17**7 * 77**49 % prime)

乗算

	def __mul__(self, other):
		if self.prime != other.prime:
			raise TypeError('Cannot multiply two numbers in different Fields')
		num = (self.num * other.num) % self.prime
		return self.__class__(num, self.prime)

a = FieldElement(3, 13)
b = FieldElement(12, 13)
c = FieldElement(10, 13)
print(a*b==c)

べき乗

	def __pow__(self, exponent):
		num = (self.num ** exponent) % self.prime
		return self.__class__(num, self.prime)

a = FieldElement(3, 13)
b = FieldElement(1, 13)
print(a**3==b)

for prime in (7, 11, 17, 31):
	print([pow(i, prime-1, prime) for i in range(1, prime)])

### 有限体上の除算

prime = 31
print(3*pow(24, prime-2, prime) % prime)
print(pow(17, prime-4, prime))
print(pow(4, prime-5, prime)*11 % prime)

	def __truediv__(self, other):
		if self.prime != other.prime:
			raise TypeError('Cannot divide two numbers in different Fields')
		num = self.num * pow(other.num, self.prime - 2, self.prime) % self.prime
		return self.__class__(num, self.prime)

いきなり難しいな

$ sudo apt install python3-virtualenv
$ virtualenv -p python3 .venv
$ .venv/vin/activate
$ . .venv/bin/activate
$ pip3 install -r requirements.txt

// vagrant上でjupyter notebookを起動する
$ jupyter notebook –no-browser –ip=0.0.0.0

ビットコインは相互に依存するコンポーネントが多数存在する
楕円曲線暗号(ECC)が基礎となる
L 署名アルゴリズム、検証アルゴリズム　… トランザクションの仕組みの中心
　　L シェノア署名、秘匿トランザクションなどの技術

### 有限体
有限の個数からなる集合と演算
集合が閉じている、加法単位元、乗法単位元、加法逆元、乗法逆元

### 有限集合
Fp = {0,1,2,…p-1}
体の位数は素数のべき乗となる

class FieldElement:

	def __init__(self, num, prime):
		if num >= prime or num < 0:
			error = 'Num {} not in field range 0 to {}'.format(
				num, prime - 1)
			raise ValueError(error)
		self.num = num
		self.prime = prime

	def __repr__(self):
		return 'FieldElement_{}({})'.format(self.prime, self.num)

	def __eq__(self, other):
		if other is None:
			return False
		return self.num == other.num and self.prime == other.prime

a = FieldElement(7, 13)
b = FieldElement(6, 13)
print(a==b)
print(a==a)

$ python3 app.py
False
True