# 验证基点G在曲线上
## 1. 复合域、曲线和基点G
复合域为  。
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。
曲线方程为：
基点  为：
## 2. 转换基点G
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：
```python
def to_field_repr(val, bits=15, count=17):
mask = (1 << bits) - 1
result = []
for _ in range(count):
result.append(val & mask)
val >>= bits
return result
def print_field(label, field_data):
print(f"{label} = to_field([")
for i in range(len(field_data)):
print(f" 0x{field_data[i]:04X},", end="\n" if (i + 1) % 4 == 0 else " ")
print("])")
Gx_int = 0x56fdcbc6a27acee0cc2996e0096ae74feb1acf220a2341b898b549440297b8cc
Gy_int = 0x20da32e8afc90b7cf0e76bde44496b4d0794054e6ea60f388682463132f931a7
Gx_field = to_field_repr(Gx_int)
Gy_field = to_field_repr(Gy_int)
print_field("Gx", Gx_field)
print()
print_field("Gy", Gy_field)
```
输出结果可以看到正是 WinRarConfig.hpp 文件中 435~456 行的内容：
```
Gx = to_field([
0x38CC, 0x052F, 0x2510, 0x45AA,
0x1B89, 0x4468, 0x4882, 0x0D67,
0x4FEB, 0x55CE, 0x0025, 0x4CB7,
0x0CC2, 0x59DC, 0x289E, 0x65E3,
0x56FD
])
Gy = to_field([
0x31A7, 0x65F2, 0x18C4, 0x3412,
0x7388, 0x54C1, 0x539B, 0x4A02,
0x4D07, 0x12D6, 0x7911, 0x3B5E,
0x4F0E, 0x216F, 0x2BF2, 0x1974,
0x20DA
])
```
## 3. 验证基点G和PK是否在曲线上
一个验证基点  和PK是否在曲线上的 Python 脚本如下：
```python
# GF(((2^15)^17))
def gf2_15_add(a, b):
return a ^ b
def gf2_15_mul(a, b):
# 模 x^15 + x + 1
res = 0
for i in range(15):
if (b >> i) & 1:
res ^= a << i
# 模多项式 x^15 + x + 1
for i in range(29, 14, -1):
if (res >> i) & 1:
res ^= 0b1000000000000011 << (i - 15)
return res & 0x7FFF # 15-bit mask
def gf2_15_poly_add(a, b):
return [gf2_15_add(x, y) for x, y in zip(a, b)]
def gf2_15_poly_mul(a, b):
res = [0] * 33
for i in range(17):
for j in range(17):
res[i + j] ^= gf2_15_mul(a[i], b[j])
return res
def gf2_15_17_mod(poly):
# 模 y^17 + y^3 + 1
for i in range(len(poly) - 1, 16, -1):
if poly[i]:
poly[i - 17] ^= poly[i]
poly[i - 14] ^= poly[i]
poly[i] = 0
return poly[:17]
def gf2_15_17_mul(a, b):
return gf2_15_17_mod(gf2_15_poly_mul(a, b))
def gf2_15_17_square(a):
return gf2_15_17_mul(a, a)
def gf2_15_17_add(a, b):
return gf2_15_poly_add(a, b)
def gf2_15_17_eq(a, b):
return all(x == y for x, y in zip(a, b))
def is_on_curve(x, y, b):
y2 = gf2_15_17_square(y)
xy = gf2_15_17_mul(x, y)
lhs = gf2_15_17_add(y2, xy)
x2 = gf2_15_17_square(x)
x3 = gf2_15_17_mul(x2, x)
rhs = gf2_15_17_add(x3, b)
return gf2_15_17_eq(lhs, rhs)
def to_field(arr):
assert len(arr) == 17
return arr[:]
# 参数定义
b = [0x00] * 17
b = [161] + [0]*16 # GF((2^15)^17) 中的常数元素，等价于 b[0] = 0xA1
Gx = to_field([0x38CC, 0x052F, 0x2510, 0x45AA, 0x1B89, 0x4468, 0x4882, 0x0D67,
0x4FEB, 0x55CE, 0x0025, 0x4CB7, 0x0CC2, 0x59DC, 0x289E, 0x65E3, 0x56FD])
Gy = to_field([0x31A7, 0x65F2, 0x18C4, 0x3412, 0x7388, 0x54C1, 0x539B, 0x4A02,
0x4D07, 0x12D6, 0x7911, 0x3B5E, 0x4F0E, 0x216F, 0x2BF2, 0x1974, 0x20DA])
PKx = to_field([0x3A1A, 0x1109, 0x268A, 0x12F7, 0x3734, 0x75F0, 0x576C, 0x2EA4,
0x4813, 0x3F62, 0x0567, 0x784D, 0x753D, 0x6D92, 0x366C, 0x1107, 0x3861])
PKy = to_field([0x6C20, 0x6027, 0x1B22, 0x7A87, 0x43C4, 0x1908, 0x2449, 0x4675,
0x7933, 0x2E66, 0x32F5, 0x2A58, 0x1145, 0x74AC, 0x36D0, 0x2731, 0x12B6])
# 验证
print("验证基点 G 是否在曲线上：", is_on_curve(Gx, Gy, b))
print("验证 PK 是否在曲线上：", is_on_curve(PKx, PKy, b))
```