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155 lines
5.4 KiB
Markdown
155 lines
5.4 KiB
Markdown
# 验证基点G在曲线上
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## 1. 复合域、曲线和基点G
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复合域为  。
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基域为  ，即域中每个元素是一个 `15-bit` 的数字。扩展次数是17，所以整个复合域元素可以看作是一个17项的多项式，每项都是  中的数。
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曲线方程为：
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<p align="center">
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<img src="assets/formula/8-light.svg#gh-light-mode-only">
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<img src="assets/formula/8-dark.svg#gh-dark-mode-only">
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</p>
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基点  为：
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<p align="center">
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<img src="assets/formula/9-light.svg#gh-light-mode-only">
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<img src="assets/formula/9-dark.svg#gh-dark-mode-only">
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</p>
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## 2. 转换基点G
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将基点 （以大整数形式）转换为  所需的 `15-bit` 小端表示形式。一个转换基点  的 Python 脚本如下：
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```python
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def to_field_repr(val, bits=15, count=17):
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mask = (1 << bits) - 1
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result = []
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for _ in range(count):
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result.append(val & mask)
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val >>= bits
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return result
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def print_field(label, field_data):
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print(f"{label} = to_field([")
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for i in range(len(field_data)):
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print(f" 0x{field_data[i]:04X},", end="\n" if (i + 1) % 4 == 0 else " ")
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print("])")
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Gx_int = 0x56fdcbc6a27acee0cc2996e0096ae74feb1acf220a2341b898b549440297b8cc
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Gy_int = 0x20da32e8afc90b7cf0e76bde44496b4d0794054e6ea60f388682463132f931a7
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Gx_field = to_field_repr(Gx_int)
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Gy_field = to_field_repr(Gy_int)
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print_field("Gx", Gx_field)
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print()
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print_field("Gy", Gy_field)
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```
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输出结果可以看到正是 WinRarConfig.hpp 文件中 435~456 行的内容：
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```
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Gx = to_field([
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0x38CC, 0x052F, 0x2510, 0x45AA,
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0x1B89, 0x4468, 0x4882, 0x0D67,
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0x4FEB, 0x55CE, 0x0025, 0x4CB7,
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0x0CC2, 0x59DC, 0x289E, 0x65E3,
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0x56FD
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])
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Gy = to_field([
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0x31A7, 0x65F2, 0x18C4, 0x3412,
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0x7388, 0x54C1, 0x539B, 0x4A02,
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0x4D07, 0x12D6, 0x7911, 0x3B5E,
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0x4F0E, 0x216F, 0x2BF2, 0x1974,
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0x20DA
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])
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```
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## 3. 验证基点G和PK是否在曲线上
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一个验证基点  和PK是否在曲线上的 Python 脚本如下：
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```python
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# GF(((2^15)^17))
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def gf2_15_add(a, b):
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return a ^ b
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def gf2_15_mul(a, b):
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# 模 x^15 + x + 1
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res = 0
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for i in range(15):
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if (b >> i) & 1:
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res ^= a << i
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# 模多项式 x^15 + x + 1
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for i in range(29, 14, -1):
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if (res >> i) & 1:
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res ^= 0b1000000000000011 << (i - 15)
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return res & 0x7FFF # 15-bit mask
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def gf2_15_poly_add(a, b):
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return [gf2_15_add(x, y) for x, y in zip(a, b)]
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def gf2_15_poly_mul(a, b):
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res = [0] * 33
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for i in range(17):
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for j in range(17):
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res[i + j] ^= gf2_15_mul(a[i], b[j])
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return res
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def gf2_15_17_mod(poly):
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# 模 y^17 + y^3 + 1
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for i in range(len(poly) - 1, 16, -1):
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if poly[i]:
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poly[i - 17] ^= poly[i]
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poly[i - 14] ^= poly[i]
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poly[i] = 0
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return poly[:17]
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def gf2_15_17_mul(a, b):
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return gf2_15_17_mod(gf2_15_poly_mul(a, b))
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def gf2_15_17_square(a):
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return gf2_15_17_mul(a, a)
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def gf2_15_17_add(a, b):
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return gf2_15_poly_add(a, b)
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def gf2_15_17_eq(a, b):
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return all(x == y for x, y in zip(a, b))
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def is_on_curve(x, y, b):
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y2 = gf2_15_17_square(y)
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xy = gf2_15_17_mul(x, y)
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lhs = gf2_15_17_add(y2, xy)
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x2 = gf2_15_17_square(x)
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x3 = gf2_15_17_mul(x2, x)
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rhs = gf2_15_17_add(x3, b)
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return gf2_15_17_eq(lhs, rhs)
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def to_field(arr):
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assert len(arr) == 17
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return arr[:]
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# 参数定义
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b = [0x00] * 17
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b = [161] + [0]*16 # GF((2^15)^17) 中的常数元素，等价于 b[0] = 0xA1
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Gx = to_field([0x38CC, 0x052F, 0x2510, 0x45AA, 0x1B89, 0x4468, 0x4882, 0x0D67,
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0x4FEB, 0x55CE, 0x0025, 0x4CB7, 0x0CC2, 0x59DC, 0x289E, 0x65E3, 0x56FD])
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Gy = to_field([0x31A7, 0x65F2, 0x18C4, 0x3412, 0x7388, 0x54C1, 0x539B, 0x4A02,
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0x4D07, 0x12D6, 0x7911, 0x3B5E, 0x4F0E, 0x216F, 0x2BF2, 0x1974, 0x20DA])
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PKx = to_field([0x3A1A, 0x1109, 0x268A, 0x12F7, 0x3734, 0x75F0, 0x576C, 0x2EA4,
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0x4813, 0x3F62, 0x0567, 0x784D, 0x753D, 0x6D92, 0x366C, 0x1107, 0x3861])
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PKy = to_field([0x6C20, 0x6027, 0x1B22, 0x7A87, 0x43C4, 0x1908, 0x2449, 0x4675,
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0x7933, 0x2E66, 0x32F5, 0x2A58, 0x1145, 0x74AC, 0x36D0, 0x2731, 0x12B6])
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# 验证
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print("验证基点 G 是否在曲线上：", is_on_curve(Gx, Gy, b))
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print("验证 PK 是否在曲线上：", is_on_curve(PKx, PKy, b))
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```
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