ビットコインの曲線とsecp256k1の実装

secp256k1は
a=0, b=7 すなわち　y**2 = x **3 + 7
p(※有限体の素数prime) = 2**256 – 2**32 – 977
Gx = 0x79be667ef9dcbbac55a06295ce870b07029bfcdb2dce28d959f2815b16f81798
Gy = 0x483ada7726a3c4655da4fbfc0e1108a8fd17b448a68554199c47d08ffb10d4b8
n(※群位) = 0xfffffffffffffffffffffffffffffffebaaedce6af48a03bbfd25e8cd0364141
すなわち2*256は32バイトで記憶することができ、秘密鍵の記憶が比較的簡単にできるという特徴を持つ

gx = 0x79be667ef9dcbbac55a06295ce870b07029bfcdb2dce28d959f2815b16f81798
gy = 0x483ada7726a3c4655da4fbfc0e1108a8fd17b448a68554199c47d08ffb10d4b8
p = 2**256 – 2**32 – 977
print(gy**2 % p == (gx**3 + 7) % p)
$ python3 test.py
True

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)

256ビットの数を常に64文字で表示する。

P = 2**256 - 2**32 - 977

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)

a, bの初期化

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)

群の位数

N = 0xfffffffffffffffffffffffffffffffebaaedce6af48a03bbfd25e8cd0364141

class S256Point(Point):
    def __rmul__(self, coefficient):
        coef = coefficient % N
        return super().__rmul__(coef)

x, yも定数として定義Gしておく

G = S256Point(
    0x79be667ef9dcbbac55a06295ce870b07029bfcdb2dce28d959f2815b16f81798,
    0x483ada7726a3c4655da4fbfc0e1108a8fd17b448a68554199c47d08ffb10d4b8
)

from ecc import G, N
print(vars(N*G))
$ python3 main.py
{‘a’: 0000000000000000000000000000000000000000000000000000000000000000, ‘b’: 0000000000000000000000000000000000000000000000000000000000000007, ‘x’: None, ‘y’: None}