上篇文章（发布于2023-09-18）给自己挖了个坑，说是要搞定SM9。从国庆前一周开始，到现在一个月时间，这个坑终于填上了。此前信息安全数学基础太差，理解不了SM9双线性对、扩域计算等等，为此还特意选修了现代密码学和近世代数2门专业课，边写代码边上课，带着问题学确实收获不少。说实话，弯路比预想的多，但实现效果却出乎意料的好。
SM9原理就不赘述了。目前，互联网上开源的基于Python原生实现的、确保正确（输出数据与《GBT 38635.2-2020 信息安全技术 SM9标识密码算法 第2部分：算法》附录A列举的数据完全一致）的SM9貌似是没有（我没找到）。我参考了以下代码：
- GitHub – gongxian-ding/gmssl-Python: a Python crypto for sm2/sm3/sm4/sm9
- GitHub – guanzhi/GmSSL: 支持国密SM2/SM3/SM4/SM9/SSL的密码工具箱
- GitHub – funfungho/GmSSL: http://gmssl.org/ v2.5.4
- GitHub – GmSSL/GmSSL-Python: Python binding to the GmSSL library
- GitHub – yaoyuanyylyy/abestudy: Study project for sm9, IBE, ABE implements with JPBC and BC under IDEA
其中，代码①应该是搬运了国外的IBC算法或有限域等运算的Python代码来尝试实现 SM9，虽然其实现并不完全正确，但给了我最初的重要参考。代码②③是C语言实现的GmSSL库，②是新版（还包含js代码实现的版本）包含更多数学上的优化，③是旧版但跟国标描述的更接近。代码④是代码②在Python下的调用，需要先编译②，我还没尝试过，应该是Python环境下最高效的，但没有基于Python原生实现SM9。代码⑤是SM9的Java实现，还有完整的SM9国标PDF。在GitHub上还能搜到不少，没下载的就不列举了。
本次用Python实现SM9国标描述的公钥加密、数字签名、密钥交换算法，除了保证正确性之外，还对实现细节做了若干数学和代码实现上的优化（包括但不限于：用2-NAF优化Miller循环和常数t模幂、最终模幂分解优化、用共轭实现frobenius映射、将Miller循环中的分子分母分别计算以减少模逆、针对Fq4零元较多而设计稀疏乘法、分圆循环子群Gϕ6(Fp2)下平方运算的优化、用Karatsuba 思想方法减少乘法）。不止是把他人的研究成果写进代码，也有自己独创的优化。另有些文章描述的优化方法在理论上是没问题的，但用Python实际实现却效果不佳，这些都要通过大量的代码测试来甄别。最后在不依赖第三方库的情况下，凝练成总计不到700行代码，我相信高效的代码一定不会太冗长。除了SM9国标，还参考了以下论文，在此对作者表示衷心感谢！
[1] 胡芯忆,何德彪,彭聪,等. 一种SM9算法R-ate对的快速实现方法[J]. 密码学报,2022,9(5):936-948. DOI:10.13868/j.cnki.jcr.000559.
[2] 甘植旺,廖方圆. 国密SM9中R-ate双线性对快速计算[J]. 计算机工程,2019,45(6):171-174. DOI:10.19678/j.issn.1000-3428.0054123.
[3] 王明东,何卫国,李军,等. 国密SM9算法R-ate对计算的优化设计[J]. 通信技术,2020,53(9):2241-2244. DOI:10.3969/j.issn.1002-0802.2020.09.025.
[4] 王江涛,樊荣,黄哲. SM9中高次幂运算的快速实现方法[J]. 计算机工程,2023,49(9):118-124,136. DOI:10.19678/j.issn.1000-3428.0065618.
[5] 付柱. R-ate双线性对密码算法的高效实现[D]. 天津:天津大学,2017.
[6] 孙铭玮. SM9标识密码算法关键技术研究[D]. 黑龙江:哈尔滨理工大学,2022.
[7] 李江峰. SM9算法的研究与FPGA实现[D]. 西安:西安电子科技大学,2021.
总之，非常感谢前辈们的工作，没有这些参考，对于一个非密码专业的网安学生，我手上连一本能讲清SM9具体实现的书都没有，想做出自己的实现更是难上加难，事实上这一个月我也的确走了很多弯路。不像SM2、SM3、SM4、ZUC对着国标或教材的算法描述就能实现，SM9涉及的数学基础要厚实得多，把图书馆里椭圆曲线的书都借遍了，它们和国标描述差不多，数学上都是那么几条式子，可这其中暗藏着巨大的知识鸿沟，让我对代码实现一时无从下手。另一个困难，是手上没有已经正确实现的Python代码（写惯了Python再来看Java会感觉代码太散，看C语言就像是当初学了C看汇编语言的感觉。当然，再难也要啃下来）。做其他国密算法实现的时候，更像是在优化，因为前人已经正确实现了，我只是研究如何把性能往上提，只要哪个步骤输出数据跟参考代码不一致，就说明实现错了，方便及时纠正。但做SM9实现，没有Python下的正确参考，双线性对有那么多复杂步骤，任何一步理解错了或一个粗心，都得不到正确结果，还不知道错在哪…………………………………………直到上周五突然输出了正确结果。再经过近一周优化，达到了满意的效果，及时将代码分享出来。
废话多了，上源码：
from random import randrange
from math import ceil
from .SM3 import digest as sm3
# SM9总则(GB_T 38635.1-2020) A.1 系统参数
q = 0XB640000002A3A6F1D603AB4FF58EC74521F2934B1A7AEEDBE56F9B27E351457D # 基域特征
N = 0XB640000002A3A6F1D603AB4FF58EC74449F2934B18EA8BEEE56EE19CD69ECF25 # 群的阶
# 群G1的生成元 P1=(x_p1, y_p1)
x_p1 = 0X93DE051D62BF718FF5ED0704487D01D6E1E4086909DC3280E8C4E4817C66DDDD
y_p1 = 0X21FE8DDA4F21E607631065125C395BBC1C1C00CBFA6024350C464CD70A3EA616
# 群G2的生成元 P2=(x_p2, y_p2)
x_p2 = (0X85AEF3D078640C98597B6027B441A01FF1DD2C190F5E93C454806C11D8806141,
0X3722755292130B08D2AAB97FD34EC120EE265948D19C17ABF9B7213BAF82D65B)
y_p2 = (0X17509B092E845C1266BA0D262CBEE6ED0736A96FA347C8BD856DC76B84EBEB96,
0XA7CF28D519BE3DA65F3170153D278FF247EFBA98A71A08116215BBA5C999A7C7)
HASH_SIZE = 32 # sm3输出256位（32字节）
N_SIZE = 32 # 阶的字节数
KEY_LEN = 128 # 默认密钥位数
K2_len = 256 # MAC函数中密钥K2的位数
def to_byte(x, size=None):
if type(x) is int:
return x.to_bytes(size if size else ceil(x.bit_length() / 8), byteorder='big')
elif type(x) in (str, bytes):
x = x.encode() if type(x) is str else x
return x[:size] if size and len(x) > size else x # 超过指定长度，则截取左侧字符
elif type(x) in (tuple, list):
return b''.join(to_byte(c, size) for c in x)
return bytes(x)[:size] if size else bytes(x)
# 将字节转换为int
def to_int(byte):
return int.from_bytes(byte, byteorder='big')
# 广义的欧几里得除法求模逆（耗时约为slow/SM2代码内get_inverse函数的43%）
def mod_inv(a, mod=q):
if a == 0:
return 0
lm, low, hm, high = 1, a % mod, 0, mod
while low > 1:
r = high // low
lm, low, hm, high = hm - lm * r, high - low * r, lm, low
return lm % mod
class FQ:
def __init__(self, n):
self.n = n
def __add__(self, other):
return FQ(self.n + other.n)
def __sub__(self, other):
return FQ(self.n - other.n)
def __mul__(self, other): # 右操作数可为int
return FQ(self.n * (other.n if type(other) is FQ else other) % q)
def __truediv__(self, other): # 右操作数可为int
return FQ(self.n * mod_inv(other.n if type(other) is FQ else other) % q)
def __pow__(self, other): # 操作数应为int
return FQ(pow(self.n, other, q) if other else 1)
def __eq__(self, other): # 右操作数可为int
return self.n % q == (other.n if type(other) is FQ else other) % q
def __neg__(self):
return FQ(-self.n)
def __repr__(self):
return 'FQ(%064X)' % (self.n % q)
def __bytes__(self):
return to_byte(self.n % q, N_SIZE)
def is_zero(self):
return self.n % q == 0
def inv(self):
return FQ(mod_inv(self.n))
def sqr(self):
return FQ(self.n * self.n % q)
@classmethod
def one(cls):
return cls(1)
@classmethod
def zero(cls):
return cls(0)
class FQ2:
def __init__(self, *coeffs): # 国标中的表示是高位在前，而此处coeffs是低位在前
self.coeffs = coeffs
def __add__(self, other):
(a0, a1), (b0, b1) = self.coeffs, other.coeffs
return FQ2(a0 + b0, a1 + b1)
def __sub__(self, other):
(a0, a1), (b0, b1) = self.coeffs, other.coeffs
return FQ2(a0 - b0, a1 - b1)
def sqr(self):
a0, a1 = self.coeffs
return FQ2((a0 * a0 - (a1 * a1 << 1)) % q, (a0 * a1 << 1) % q) # (a0^2 - 2 * a1^2, 2 * a0 * a1)
def sqr_u(self):
a0, a1 = self.coeffs
return FQ2(-(a0 * a1 << 2) % q, (a0 * a0 - (a1 * a1 << 1)) % q) # (-4 * a0 * a1, a0^2 - 2 * a1^2)
def mul_b_u(self, b): # 带参数乘法
(a0, a1), (b0, b1) = self.coeffs, b.coeffs
return FQ2(-(a0 * b1 + a1 * b0 << 1) % q, (a0 * b0 - (a1 * b1 << 1)) % q) # (-2*(a0*b1+a1*b0), a0*b0-2*a1*b1)
def __mul__(self, other):
if type(other) is int:
a0, a1 = self.coeffs
return FQ2(a0 << 1, a1 << 1) if other == 2 else FQ2(a0 * other % q, a1 * other % q)
(a0, a1), (b0, b1) = self.coeffs, other.coeffs
a0b0, a1b1 = a0 * b0, a1 * b1 # Karatsuba 思想方法（节约一次乘法），实测此处约有5%提升，用在其他地方未见性能提升
return FQ2((a0b0 - (a1b1 << 1)) % q, ((a0 + a1) * (b0 + b1) - (a0b0 + a1b1)) % q) # (a0*b0-2*a1*b1,a0*b1+a1*b0)
def __rmul__(self, other):
return self.__mul__(other)
def __truediv__(self, other):
if type(other) is int:
other_inv = mod_inv(other)
return FQ2([c * other_inv % q for c in self.coeffs])
return self * other.inv()
def __pow__(self, other): # 实际运行此函数的对象都是分圆循环子群Gϕ6(Fp2)中的元素
if other == 0:
return self.one()
t = self
for ri in bin(other)[3:]:
t = t.sqr2() * self if ri == '1' else t.sqr2()
return t
def inv(self):
a0, a1 = self.coeffs
if a0 == 0:
return FQ2(0, -mod_inv(a1 << 1)) # (0, -(2 * a1)^-1)
if a1 == 0:
return FQ2(mod_inv(a0), 0) # (a0^-1, 0)
k = mod_inv(a0 * a0 + (a1 * a1 << 1)) # k = (a0^2 + 2 * a1^2)^-1
return FQ2(a0 * k % q, -a1 * k % q) # (a0 * k, -a1 * k)
def conjugate(self): # 共轭
a0, a1 = self.coeffs
return self.__class__(a0, -a1)
def get_fp_list(self): # 返回所有基域元素（高位在前）
if type(self) is FQ2:
return [i % q for i in self[::-1]]
return [y for x in self[::-1] for y in x.get_fp_list()] if self.coeffs else [0] * 4 # 注意FQ4对象零值的处理
def __repr__(self):
return '%s(%s)' % (self.__class__.__name__, ', '.join('%064X' % i for i in self.get_fp_list()))
def __bytes__(self): # 字节串高位在前
return to_byte(self.get_fp_list(), N_SIZE)
def __eq__(self, other):
return self.get_fp_list() == other.get_fp_list()
def __neg__(self):
return self.__class__(*[-c for c in self.coeffs])
def __getitem__(self, item):
return self.coeffs[item]
def is_zero(self):
return all(c % q == 0 for c in self.coeffs) if type(self) is FQ2 else all(c.is_zero() for c in self.coeffs)
@classmethod
def one(cls):
return FQ2_one if cls is FQ2 else (FQ12_one if cls is FQ12 else FQ4_one)
@classmethod
def zero(cls):
return FQ2_zero if cls is FQ2 else ()
class FQ4(FQ2): # 零元的coeffs为空，可优化FQ12稀疏乘法运算
def __add__(self, other):
if not self.coeffs:
return other
if not other.coeffs:
return self
(a0, a1), (b0, b1) = self.coeffs, other.coeffs
return FQ4(a0 + b0, a1 + b1)
def __sub__(self, other):
if not self.coeffs:
return -other
if not other.coeffs:
return self
(a0, a1), (b0, b1) = self.coeffs, other.coeffs
return FQ4(a0 - b0, a1 - b1)
def sqr(self):
if not self.coeffs:
return FQ4_zero
a0, a1 = self.coeffs
return FQ4(a0.sqr() + a1.sqr_u(), a0 * a1 * 2) # (a0^2 + a1^2 * u, 2 * a0 * a1)
def sqr_v(self):
if not self.coeffs:
return FQ4_zero
a0, a1 = self.coeffs
return FQ4(a0.mul_b_u(a1) * 2, a0.sqr() + a1.sqr_u()) # (2 * a0 * a1 * u, a0^2 + a1^2 * u)
def mul_b_v(self, b): # 带参数乘法
if not self.coeffs or not b.coeffs:
return FQ4_zero
(a0, a1), (b0, b1) = self.coeffs, b.coeffs
return FQ4(a0.mul_b_u(b1) + a1.mul_b_u(b0), a0 * b0 + a1.mul_b_u(b1)) # (a0*b1*u+a1*b0*u, a0*b0+a1*b1*u)
def __mul__(self, other):
if not self.coeffs:
return FQ4_zero
if type(other) is int:
a0, a1 = self.coeffs
return FQ4(a0 * other, a1 * other)
if not other.coeffs:
return FQ4_zero
(a0, a1), (b0, b1) = self.coeffs, other.coeffs
return FQ4(a0 * b0 + a1.mul_b_u(b1), a0 * b1 + a1 * b0) # (a0*b0+a1*b1*u, a0*b1+a1*b0)
def inv(self):
if not self.coeffs:
return FQ4_zero
a0, a1 = self.coeffs
k = (a1.sqr_u() - a0.sqr()).inv()
return FQ4((-a0 * k), a1 * k)
class FQ12(FQ2):
def __add__(self, other):
(a0, a1, a2), (b0, b1, b2) = self.coeffs, other.coeffs
return FQ12(a0 + b0, a1 + b1, a2 + b2)
def __sub__(self, other):
(a0, a1, a2), (b0, b1, b2) = self.coeffs, other.coeffs
return FQ12(a0 - b0, a1 - b1, a2 - b2)
def sqr(self):
a0, a1, a2 = self.coeffs
return FQ12(a0.sqr() + a1.mul_b_v(a2) * 2, a0 * a1 * 2 + a2.sqr_v(), a0 * a2 * 2 + a1.sqr())
def __mul__(self, other):
(a0, a1, a2), (b0, b1, b2) = self.coeffs, other.coeffs
return FQ12(a0 * b0 + a1.mul_b_v(b2) + a2.mul_b_v(b1), a0 * b1 + a1 * b0 + a2.mul_b_v(b2),
a0 * b2 + a1 * b1 + a2 * b0)
def sqr2(self): # 分圆循环子群Gϕ6(Fp2)中的元素平方
a, b, c = self.coeffs
a2, b2, c2v = a.sqr(), b.sqr(), c.sqr_v()
return FQ12(a2 + (a2 - a.conjugate()) * 2, c2v + (c2v + b.conjugate()) * 2, b2 + (b2 - c.conjugate()) * 2)
def pow_t(self): # 只可用于分圆循环子群Gϕ6(Fp2)中的元素，求逆为共轭
c, _inv = self, self.frobenius6()
for ti in t_NAF:
c = c.sqr2()
if ti == '1':
c = c * self
elif ti == '2': # 用2代替-1
c = c * _inv
return c
def inv(self):
a0, a1, a2 = self.coeffs
a0_2, a1_2 = a0.sqr(), a1.sqr()
if a2.is_zero():
k = (a0 * a0_2 + a1.mul_b_v(a1_2)).inv()
return FQ12(a0_2 * k, (-a0 * a1 * k), a1_2 * k)
t0, t1, t2 = a1_2 - a0 * a2, a0 * a1 - a2.sqr_v(), a0_2 - a1.mul_b_v(a2)
t3 = a2 * (t1.sqr() - t0 * t2).inv()
return FQ12(t2 * t3, (-t1 * t3), t0 * t3)
def frobenius(self):
(a0, a1), (b0, b1), (c0, c1) = self.coeffs
a = FQ4(a0.conjugate(), a1.conjugate() * alpha3)
b = FQ4(b0.conjugate() * alpha1, b1.conjugate() * alpha4)
c = FQ4(c0.conjugate() * alpha2, c1.conjugate() * alpha5)
return FQ12(a, b, c)
def frobenius2(self):
a, b, c = self.coeffs
return FQ12(a.conjugate(), b.conjugate() * alpha2, c.conjugate() * alpha4)
def frobenius3(self):
(a0, a1), (b0, b1), (c0, c1) = self.coeffs
a = FQ4(a0.conjugate(), -a1.conjugate() * alpha3)
b = FQ4(b0.conjugate() * alpha3, b1.conjugate())
c = FQ4(-c0.conjugate(), c1.conjugate() * alpha3)
return FQ12(a, b, c)
def frobenius6(self):
a, b, c = self.coeffs
return FQ12(a.conjugate(), -b.conjugate(), c.conjugate())
class ECC_Point:
def __init__(self, *pt): # 采用Jacobian射影坐标计算，输入仿射坐标后会转换为Jacobian射影坐标
self.pt = pt if len(pt) == 3 else (*pt, pt[0].one())
@classmethod
def from_byte(cls, byte): # 输入bytes类型仿射坐标，构建点对象
fp_num = len(byte) // (N_SIZE << 1) # 单个坐标包含的域元素个数
if fp_num in (1, 2) and len(byte) % N_SIZE == 0:
fp_list = [to_int(byte[i:i + N_SIZE]) for i in range(0, len(byte), N_SIZE)] # 将bytes转换为域元素列表
if fp_num == 1:
return cls(FQ(fp_list[0]), FQ(fp_list[1]))
x_list, y_list = fp_list[fp_num - 1::-1], fp_list[:fp_num - 1:-1] # 从bytes到FQ2对象保存的域元素，需翻转高低位顺序
return cls(FQ2(*x_list), FQ2(*y_list))
return False
def is_inf(self):
return self[2].is_zero()
def is_on_curve(self): # 检查点是否满足曲线方程 y^2 == x^3 + b
x, y, z = self.pt
return y ** 2 == x ** 3 + (_b1 if type(x) is FQ else _b2) * z ** 6
def double(self):
x, y, z = self.pt
T1, _y2 = x.sqr() * 3, y.sqr()
T2, T3 = x * _y2 * 4, _y2.sqr() * 8
x3 = T1.sqr() - T2 * 2
y3 = T1 * (T2 - x3) - T3
z3 = y * z * 2
return ECC_Point(x3, y3, z3)
def zero(self):
cls = self[0].__class__
return ECC_Point(cls.one(), cls.one(), cls.zero())
def __add__(self, p2):
if self.is_inf():
return p2
if p2.is_inf():
return self
(x1, y1, z1), (x2, y2, z2) = self.pt, p2.pt
z1_2, z2_2 = z1.sqr(), z2.sqr()
T1, T2 = x1 * z2_2, x2 * z1_2
T3, T4, T5 = T1 - T2, y1 * z2_2 * z2, y2 * z1_2 * z1
T6, T7, T3_2 = T4 - T5, T1 + T2, T3.sqr()
T8, T9 = T4 + T5, T7 * T3_2
x3 = T6.sqr() - T9
T10 = T9 - x3 * 2
y3 = (T10 * T6 - T8 * T3_2 * T3) * _2_inv
z3 = z1 * z2 * T3
return ECC_Point(x3, y3, z3)
def multiply(self, n): # 算法一：二进制展开法
if n in (0, 1):
return self if n else self.zero()
Q = self
for i in bin(n)[3:]:
Q = Q.double() + self if i == '1' else Q.double()
return Q
def __mul__(self, n): # 算法三：滑动窗法
k = bin(n)[2:]
l, r = len(k), 5 # 滑动窗口为5效果较好
if r >= l: # 如果窗口大于k的二进制位数，则本算法无意义
return self.multiply(n)
P_ = {1: self, 2: self.double()} # 保存P[j]值的字典
for i in range(1, 1 << (r - 1)):
P_[(i << 1) + 1] = P_[(i << 1) - 1] + P_[2]
t = r
while k[t - 1] != '1':
t -= 1
hj = int(k[:t], 2)
Q, j = P_[hj], t
while j < l:
if k[j] == '0':
Q = Q.double()
j += 1
else:
t = min(r, l - j)
while k[j + t - 1] != '1':
t -= 1
hj = int(k[j:j + t], 2)
Q = Q.multiply(1 << t) + P_[hj]
j += t
return Q
def __rmul__(self, n):
return self.__mul__(n)
def __eq__(self, p2):
(x1, y1, z1), (x2, y2, z2) = self.pt, p2.pt
z1_2, z2_2 = z1.sqr(), z2.sqr()
return x1 * z2_2 == x2 * z1_2 and y1 * z2_2 * z2 == y2 * z1_2 * z1
def __neg__(self):
x, y, z = self.pt
return ECC_Point(x, -y, z)
def __getitem__(self, item):
return self.pt[item]
def __repr__(self):
return '%s%s' % (self.__class__.__name__, self.normalize())
def __bytes__(self):
return to_byte(self.normalize())
def normalize(self):
x, y, z = self.pt
if not hasattr(self, 'normalize_tuple'):
if z != z.one():
z_inv = z.inv()
z_inv_2 = z_inv.sqr()
x, y = x * z_inv_2, y * z_inv_2 * z_inv
self.normalize_tuple = (x.n, y.n) if type(x) is FQ else (x, y)
return self.normalize_tuple
def frobenius(self):
x, y, z = self.pt
return ECC_Point(x.conjugate(), y.conjugate(), z.conjugate() * alpha1)
def frobenius2_neg(self):
x, y, z = self.pt
return ECC_Point(x, -y, z * alpha2)
FQ2_one, FQ2_zero = FQ2(1, 0), FQ2(0, 0) # FQ2单位元、零元
FQ4_one, FQ4_zero = FQ4(FQ2_one, FQ2_zero), FQ4() # FQ4单位元、零元
FQ12_one = FQ12(FQ4_one, FQ4_zero, FQ4_zero) # FQ12单位元
P1 = ECC_Point(FQ(x_p1), FQ(y_p1)) # 群G1的生成元
P2 = ECC_Point(FQ2(*x_p2[::-1]), FQ2(*y_p2[::-1])) # 群G2的生成元
_b1, _b2 = FQ(5), FQ2(0, 5) # b2=βb=(1,0)*5
alpha1 = 0X3F23EA58E5720BDB843C6CFA9C08674947C5C86E0DDD04EDA91D8354377B698B # -2^((q - 1)/12)
alpha2 = 0XF300000002A3A6F2780272354F8B78F4D5FC11967BE65334 # -2^((q - 1)/6)
alpha3 = 0X6C648DE5DC0A3F2CF55ACC93EE0BAF159F9D411806DC5177F5B21FD3DA24D011 # -2^((q - 1)/4)
alpha4 = 0XF300000002A3A6F2780272354F8B78F4D5FC11967BE65333 # -2^((q - 1)/3)
alpha5 = 0X2D40A38CF6983351711E5F99520347CC57D778A9F8FF4C8A4C949C7FA2A96686
_2_inv = 0X5B2000000151D378EB01D5A7FAC763A290F949A58D3D776DF2B7CD93F1A8A2BF # 1/2
_3div2 = 0X5B2000000151D378EB01D5A7FAC763A290F949A58D3D776DF2B7CD93F1A8A2C0 # 3/2
R_ate_a_NAF = '00100000000000000000000000000000000000010001020200020200101000020' # a=6t+2的二进制非相邻表示(2-NAF)（去首1）
t_NAF = '10000000000000000000000000000000000000102020010000201020001010' # t的二进制非相邻表示(2-NAF)（去首1）
hlen = 320 # 8 * ceil(5 * log(N, 2) / 32)
# 线函数g T,Q(P)，求过点T和Q的直线在P上的值（分母在最终模幂时值为1，可消去）
def g(T, Q, P=None):
if P:
(xT, yT, zT), (xQ, yQ, zQ), (xP, yP) = T, Q, P
zT_2, zQ_2 = zT.sqr(), zQ.sqr()
zQ_3, t1 = zQ * zQ_2, (xT * zQ_2 - xQ * zT_2) * zT * zQ
b1, t2 = t1 * zQ_3, (yT * zQ_3 - yQ * zT * zT_2) * zQ
a0, a4 = t1 * yQ - t2 * xQ, t2 * zQ_2 * xP
else: # 当P为空时，g T,T(P)，求过点T的切线在P上的值
(xT, yT, zT), (xP, yP) = T, Q
zT_2, t1 = zT.sqr(), xT.sqr() * _3div2
b1, a0, a4 = zT * zT_2 * yT, yT.sqr() - t1 * xT, t1 * zT_2 * xP
return FQ12(FQ4(a0, -b1 * yP), FQ4_zero, FQ4(a4, FQ2_zero))
# BN曲线上R_ate对的计算
def e(P, Q):
T, nQ, f, P_xy = Q, -Q, FQ12_one, P.normalize()
for ai in R_ate_a_NAF:
f, T = f.sqr() * g(T, P_xy), T.double()
if ai == '1':
f, T = f * g(T, Q, P_xy), T + Q
elif ai == '2': # 用2代替-1
f, T = f * g(T, nQ, P_xy), T + nQ
Q1, nQ2 = Q.frobenius(), Q.frobenius2_neg()
return final_exp(f * g(T, Q1, P_xy) * g(T + Q1, nQ2, P_xy))
# 最终模幂
def final_exp(f):
m = f.frobenius6() * f.inv() # f^(p^6 - 1)
s = m.frobenius2() * m # m^(p^2 + 1)
# 困难部分 s^(p^3 + (6t^2+1)p^2 + (-36t^3-18t^2-12t+1)p + (-36t^3-30t^2-18t-2))
s_6t = s.pow_t() ** 6
s_6t2, s_12t = s_6t.pow_t(), s_6t.sqr2()
s_6t3, s_12t2, s_18t = s_6t2.pow_t(), s_6t2.sqr2(), s_6t * s_12t
s_36t3, s_18t2, a2 = s_6t3 ** 6, s_6t2 * s_12t2, s_6t2 * s
s_30t2, a1 = s_12t2 * s_18t2, (s_36t3 * s_18t2 * s_12t).frobenius6() * s
a0 = (s_36t3 * s_30t2 * s_18t * s.sqr2()).frobenius6()
return s.frobenius3() * a2.frobenius2() * a1.frobenius() * a0
# SM9算法(GB_T 38635.2-2020) 5.3.6定义的密钥派生函数
# Z为bytes类型，klen表示输出密钥比特长度（8的倍数）；输出为bytes类型
def KDF(Z, klen=KEY_LEN):
ksize, K = klen >> 3, bytearray()
for ct in range(1, ceil(ksize / HASH_SIZE) + 1):
K.extend(sm3(Z + to_byte(ct, 4)))
return K[:ksize]
# SM9算法(GB_T 38635.2-2020) 5.3.2.2和5.3.2.3定义的密码函数
def H(i, Z):
Ha = to_int(KDF(to_byte(i, 1) + Z, hlen))
return Ha % (N - 1) + 1
# SM9算法(GB_T 38635.2-2020) 5.3.5定义的消息认证码函数
def MAC(K2, Z):
return sm3(Z + K2)
class SM9: # SM9算法(GB_T 38635.2-2020)
def __init__(self, ID='', ks=None, Ppub_s=None, ke=None, Ppub_e=None, hid_s=1, hid_e=3, is_KGC=False):
self.ID = ID
if is_KGC: # 作为密钥生成中心
self.ks = ks if ks and 0 < ks % N < N else randrange(1, N) # 未提供签名主密钥则随机生成
self.Ppub_s = Ppub_s if Ppub_s and Ppub_s == P2 * self.ks else P2 * self.ks # 确保签名主公钥与签名主私钥匹配
self.ke = ke if ke and 0 < ke % N < N else randrange(1, N) # 未提供加密主密钥则随机生成
self.Ppub_e = Ppub_e if Ppub_e and Ppub_e == P1 * self.ke else P1 * self.ke # 确保加密主公钥与加密主私钥匹配
else: # 作为用户
self.ds, self.Ppub_s, self.de, self.Ppub_e = ks, Ppub_s, ke, Ppub_e
self.gs, self.ge = e(P1, Ppub_s), e(Ppub_e, P2)
self.hid_s_byte, self.hid_e_byte = [hid if type(hid) is bytes else to_byte(hid, 1) for hid in [hid_s, hid_e]]
def KGC_gen_user(self, ID):
ID_byte = to_byte(ID)
t1 = (H(1, ID_byte + self.hid_s_byte) + self.ks) % N
if t1 == 0: # 需重新产生签名主密钥，并更新所有用户的签名密钥
return False
t2 = self.ks * mod_inv(t1, N) % N
ds = P1 * t2 # 用户签名私钥
t1 = (H(1, ID_byte + self.hid_e_byte) + self.ke) % N
if t1 == 0: # 需重新产生加密主密钥，并更新所有用户的加密密钥
return False
t2 = self.ke * mod_inv(t1, N) % N
de = P2 * t2 # 用户加密私钥
return SM9(ID, ds, self.Ppub_s, de, self.Ppub_e, self.hid_s_byte, self.hid_e_byte)
# 6.2 数字签名生成算法
def sign(self, M, r=None, outbytes=True):
l = 0
while l == 0:
r = r if r else randrange(1, N) # A2
w = bytes(self.gs ** r) # A3
h = H(2, to_byte(M) + w) # A4
l = (r - h) % N # A5
S = self.ds * l # A6
return to_byte([h, S]) if outbytes else (h, S)
# 6.4 数字签名验证算法
def verify(self, ID, M_, sig):
h_, S_ = (to_int(sig[:N_SIZE]), ECC_Point.from_byte(sig[N_SIZE:])) if type(sig) is bytes else sig
if not 0 < h_ < N or not S_ or not S_.is_on_curve(): # B1、B2
return False
t = self.gs ** h_ # B4
h1 = H(1, to_byte(ID) + self.hid_s_byte) # B5
P = P2 * h1 + self.Ppub_s # B6
u = e(S_, P) # B7
w_ = bytes(u * t) # B8
h2 = H(2, to_byte(M_) + w_) # B9
return h_ == h2
# A 发起协商（也可用作B生成rB、RB；outbytes=True时输出bytes）
# 7.2 密钥交换协议 A1-A3
def agreement_initiate(self, IDB, r=None, outbytes=True):
QB = P1 * H(1, to_byte(IDB) + self.hid_e_byte) + self.Ppub_e # A1
rA = r if r else randrange(1, N) # A2
RA = QB * rA # A3
return rA, bytes(RA) if outbytes else RA
# B 响应协商（option=True时计算选项部分）
# 7.2 密钥交换协议 B1-B6
def agreement_response(self, RA, IDA, option=False, rB=None, klen=KEY_LEN, outbytes=True):
RA = ECC_Point.from_byte(RA) if type(RA) is bytes else RA
if not RA or not RA.is_on_curve(): # B4
return False, 'RA不属于椭圆曲线群G1'
rB, RB = self.agreement_initiate(IDA, rB, outbytes) # B1-B3
g1, g2 = e(RA, self.de), bytes(self.ge ** rB) # B4
g1, g3 = bytes(g1), bytes(g1 ** rB) # B4
tmp_byte = to_byte([IDA, self.ID, RA, RB])
SKB = KDF(tmp_byte + g1 + g2 + g3, klen) # B5
if not option:
return True, (RB, SKB)
self.tmp_byte2 = g1 + sm3(g2 + g3 + tmp_byte)
SB = sm3(to_byte(0x82, 1) + self.tmp_byte2) # B6（可选部分）
return True, (RB, SKB, SB)
# A 协商确认
# 7.2 密钥交换协议 A5-A8
def agreement_confirm(self, rA, RA, RB, IDB, SB=None, option=False, klen=KEY_LEN):
RB = ECC_Point.from_byte(RB) if type(RB) is bytes else RB
if not RB or not RB.is_on_curve(): # A5
return False, 'RB不属于椭圆曲线群G1'
g1_, g2_ = bytes(self.ge ** rA), e(RB, self.de) # A5
g2_, g3_ = bytes(g2_), bytes(g2_ ** rA) # A5
tmp_byte = to_byte([self.ID, IDB, RA, RB])
if option and SB: # A6（可选部分）
tmp_byte2 = g1_ + sm3(g2_ + g3_ + tmp_byte)
S1 = sm3(to_byte(0x82, 1) + tmp_byte2)
if S1 != SB:
return False, 'S1 != SB'
SKA = KDF(tmp_byte + g1_ + g2_ + g3_, klen) # A7
if not option or not SB:
return True, SKA
SA = sm3(to_byte(0x83, 1) + tmp_byte2) # A8
return True, (SKA, SA)
# B 协商确认（可选部分）
# 7.2 密钥交换协议 B8
def agreement_confirm2(self, SA):
if not hasattr(self, 'tmp_byte2'):
return False, 'step error'
S2 = sm3(to_byte(0x83, 1) + self.tmp_byte2)
if S2 == SA:
del self.tmp_byte2
return True, ''
return False, 'S2 != SA'
# 8.2 密钥封装算法
def encaps(self, IDB, klen, r=None, outbytes=True):
K = bytes()
while K == bytes(len(K)):
r, C = self.agreement_initiate(IDB, r, outbytes) # A1-A3
w = bytes(self.ge ** r) # A5
K = KDF(to_byte([C, w, IDB]), klen)
return K, C
# 8.4 密钥封装算法
def decaps(self, C, klen):
C = ECC_Point.from_byte(C) if type(C) is bytes else C
if not C or not C.is_on_curve(): # B1
return False, 'C不属于椭圆曲线群G1'
w_ = bytes(e(C, self.de)) # B2
K_ = KDF(to_byte([C, w_, self.ID]), klen) # B3
return (True, K_) if K_ != bytes(len(K_)) else (False, 'K为全0比特串')
# 9.2 加密算法
def encrypt(self, IDB, M, r=None, outbytes=True):
M = to_byte(M)
K, C1 = self.encaps(IDB, (len(M) << 3) + K2_len, r, outbytes) # A1-A6.a.1
K1, K2 = K[:len(M)], K[len(M):] # A6.a.1
C2 = bytes(M[i] ^ K1[i] for i in range(len(M))) # A6.a.2
C3 = MAC(K2, C2) # A7
return to_byte([C1, C3, C2]) if outbytes else (C1, C3, C2)
# 9.4 解密算法
def decrypt(self, C):
C3_start, C3_end = N_SIZE << 1, (N_SIZE << 1) + HASH_SIZE
C1, C3, C2 = (C[:C3_start], C[C3_start:C3_end], C[C3_end:]) if type(C) is bytes else C
res, K_ = self.decaps(C1, (len(C2) << 3) + K2_len) # B1-B3.a.1
if not res:
return False, K_.replace('C', 'C1')
K1_, K2_ = K_[:len(C2)], K_[len(C2):] # B3.a.1
if K1_ == bytes(len(K_)):
return False, 'K1\'为全0比特串'
u = MAC(K2_, C2) # B4
if u != C3:
return False, 'u != C3'
return True, bytes(C2[i] ^ K1_[i] for i in range(len(C2))) # B3.a.2
使用时先创建KGC对象，由它生成用户对象，再由用户对象完成公钥加密、数字签名或密钥交换，例子如下：
IDA, IDB, message = 'Alice', 'Bob', 'Chinese IBS standard'
kgc = SM9(ks=0x130E78459D78545CB54C587E02CF480CE0B66340F319F348A1D5B1F2DC5F4,
ke=0x2E65B0762D042F51F0D23542B13ED8CFA2E9A0E7206361E013A283905E31F, is_KGC=True)
sm9_A, sm9_B = kgc.KGC_gen_user(IDA), kgc.KGC_gen_user(IDB)
assert bytes(sm9_A.gs).hex().swapcase().endswith('F0F071D7D284FCFB')
print("-----------------test sign and verify---------------")
r = 0x033C8616B06704813203DFD00965022ED15975C662337AED648835DC4B1CBE
signature = sm9_A.sign(message, r)
assert signature.hex().swapcase().endswith('827CC2ACED9BAA05')
assert sm9_B.verify(IDA, message, signature)
print("success")
print("-----------------test key agreement---------------")
rA = 0x5879DD1D51E175946F23B1B41E93BA31C584AE59A426EC1046A4D03B06C8
rA, RA = sm9_A.agreement_initiate(IDB, rA) # A发起协商
# A将RA发送给B
rB = 0x018B98C44BEF9F8537FB7D071B2C928B3BC65BD3D69E1EEE213564905634FE
res, content = sm9_B.agreement_response(RA, IDA, True, rB) # B响应协商
if not res:
print('B报告协商错误：', content)
return
RB, SKB, SB = content
# B将RB、SB发送给A
res, content = sm9_A.agreement_confirm(rA, RA, RB, IDB, SB, True) # A协商确认
if not res:
print('A报告协商错误：', content)
return
SKA, SA = content
assert SKA.hex().swapcase() == '68B20D3077EA6E2B825315836FDBC633'
# A将SA发送给B
res, content = sm9_B.agreement_confirm2(SA) # B协商确认
if not res:
print('B报告协商错误：', content)
return
assert SKA == SKB
print("success")
print("-----------------test encrypt and decrypt---------------")
message = 'Chinese IBE standard'
kgc = SM9(ks=kgc.ks, Ppub_s=kgc.Ppub_s,
ke=0x01EDEE3778F441F8DEA3D9FA0ACC4E07EE36C93F9A08618AF4AD85CEDE1C22, is_KGC=True)
sm9_A, sm9_B = kgc.KGC_gen_user(IDA), kgc.KGC_gen_user(IDB)
C = sm9_A.encrypt(IDB, message, 0xAAC0541779C8FC45E3E2CB25C12B5D2576B2129AE8BB5EE2CBE5EC9E785C)
assert C.hex().swapcase().endswith('378CDD5DA9513B1C')
res, content = sm9_B.decrypt(C)
if not res:
print('解密错误：', content)
return
assert message == content.decode()
print("success")虽然和代码①实现的结果不一样，但手头也只有这一个Python的原生实现，但还是比较一下。
此前介绍国密算法的系列文章如下：
三篇SM2：
国密算法 SM2 公钥加密 数字签名 密钥交换 更高效、依赖更少的开源python代码_国密算法 开源-CSDN博客
国密算法 SM2 公钥加密 数字签名 密钥交换 全网最高效的开源python代码_-CSDN博客
国密算法 SM2 公钥加密 非对称加密 数字签名 密钥协商 python实现完整代码_qq_43339242的博客-CSDN博客_python sm2
SM3：国密算法 SM3 消息摘要 杂凑算法 哈希函数 散列函数 python实现完整代码_qq_43339242的博客-CSDN博客_国密sm3
SM4：国密算法 SM4 对称加密 分组密码 python实现完整代码_qq_43339242的博客-CSDN博客_python国密算法库
ZUC：国密算法 ZUC流密码 祖冲之密码 python代码完整实现_qq_43339242的博客-CSDN博客_国密算法代码
对上述几个算法和实现不了解的，建议点进去看看。下面这篇文章是对上述的汇总：
国密算法 SM2公钥密码 SM3杂凑算法 SM4分组密码 python代码完整实现_qq_43339242的博客-CSDN博客_python sm2
上述所有国密算法的完整Python实现代码和测试代码，库名叫hggm，托管在码云：hggm – 国密算法 SM2 SM3 SM4 python实现完整代码: 国密算法 SM2公钥密码 SM3杂凑算法 SM4分组密码 python代码完整实现 效率高于所有公开的python国密算法库 (gitee.com)
至此，公开国密算法的一家子（SM2、SM3、SM4、SM9和ZUC）已经团聚了，在国密算法Python原生实现领域，目前来看，性能应该都做到了网上开源最优。
咱们所处的世界还很不太平，建设网络安全强国的脚步一刻也不会停，任重道远。
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