SM2算法

简述

SM2是非对称加密算法

它是基于椭圆曲线密码的公钥密码算法标准，其秘钥长度256bit，包含数字签名、密钥交换和公钥加密，用于替换RSA/DH/ECDSA/ECDH等国际算法。可以满足电子认证服务系统等应用需求，由国家密码管理局于2010年12月17号发布。

SM2采用的是ECC 256位的一种，其安全强度比RSA 2048位高，且运算速度快于RSA。随着密码技术和计算技术的发展，目前常用的1024位RSA算法面临严重的安全威胁，我们国家密码管理部门经过研究，决定采用SM2椭圆曲线算法替换RSA算法。SM2算法在安全性、性能上都具有优势。

基础知识：椭圆曲线知识点、SM3算法

获取公私钥：

椭圆曲线方程：$y^2=x^3+ax+b\mathrm{m}\mathrm{o}\mathrm{d}p$

1. $确认a、b、p，确认曲线。$
2. $选择一个点P(x_g,y_g)为基点。$
3. $对曲线做切线、x对称点运行。次数为d,运算倍点为Q$
4. $d为私钥,Q为公钥$

密钥对的生成:

1. $产生随机整数d[1,n-2]$
2. $G为基点，计算点P=(xP,yP)=[d]G;$
3. $密钥对为:(d,P)其中，d为私钥，P为公钥$

一个很典型的例子：

a = 0xFFFFFFFEFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF00000000FFFFFFFFFFFFFFFC
b = 0x28E9FA9E9D9F5E344D5A9E4BCF6509A7F39789F515AB8F92DDBCBD414D940E93
p = 0xFFFFFFFEFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF00000000FFFFFFFFFFFFFFFF
x_g = 0x32c4ae2c1f1981195f9904466a39c9948fe30bbff2660be1715a4589334c74c7
y_g = 0xbc3736a2f4f6779c59bdcee36b692153d0a9877cc62a474002df32e52139f0a0
n = 0xFFFFFFFEFFFFFFFFFFFFFFFFFFFFFFFF7203DF6B21C6052B53BBF40939D54123

SM签名

$M$为待签名消息，数字签名结果为$(r,s)$ ，用户密钥对$(d,P)$。

实现步骤：

1. $e=hash(M)\Rightarrow 获取消息散列值$
2. $产生随机数k\Rightarrow 以便即使是同一个消息，每次签名出来的结果不同。$
3. $使用随机数，计算椭圆曲线点(x_1,y_1)=[k]G$
4. $r=(e+x1)\mathrm{m}\mathrm{o}\mathrm{d}n\Rightarrow 判断：r=0或者r+k=n,继续第2步。$
5. $s=((1+d{)}^{-1}\ast (k-r\ast d))\mathrm{m}\mathrm{o}\mathrm{d}n,若s=0，继续第2步$
6. $r,s为签名信息。$

SM验签

$M$为明文，$(r,s)$为签名结果,用户公钥$P$

实现步骤：

1. $e=hash(M)$
2. $t=(r+s)\mathrm{m}\mathrm{o}\mathrm{d}n$
3. $(x,y)=[s]G+[t]P$
4. $R=(e+x)\mathrm{m}\mathrm{o}\mathrm{d}n$
5. $计算R是否等于r$

$[s]G$ +$[t]P$的结果可以推导出等于$[k]G$。

验证原理

$[s]G+[t]P=sG+(r+s)P$
$=sG+(r+s)dG$
$=sG+sdG+rdG$
$=(1+d)sG+rdG$
$=(1+d)(1+d{)}^{-1}(k-rd)G+rdG$
$=(k-rd)G+rdG$
$=kG-rdG+rdG$
$=kG=(x1,y1)$

SM加密

M为明文字符串

1. $获取随机数k$
2. $(x1,y1)=[k]G$
3. $S=[h]P\Rightarrow h为余因子$
4. $C1=(x2,y2)=[k]P$
5. $t=KDF(x2\parallel y2,klen)\Rightarrow klen为M的长度。KDF是sm2的密钥派生函数$
6. $C2=M+t$
7. $C3=Hash(x2\parallel M\parallel y2)$
8. $C=C1\parallel C2\parallel C3$

SM解密

C为密文字符串,klen为密文中C2的长度

1. $C1=C里面获取，验证C1是否满足椭圆曲线。\Rightarrow C2长度确定，可以获取C1内容。$
2. $S=[h]C1，S为无穷点，退出。$
3. $(x2,y2)=[d]C1$
4. $t=KDF(m2\parallel y2,klen)$
5. $M=C2+t$
6. $u=Hash(x2\parallel M\parallel y2),u?==C3$
7. $M为明文$

代码实现

首先，需要导入包gmssl

pip install gmssl

生成公私钥算法
sm2utils.py

from random import SystemRandom

class CurveFp:
	def __init__(self, A, B, P, N, Gx, Gy, name):
		self.A = A
		self.B = B
		self.P = P
		self.N = N
		self.Gx = Gx
		self.Gy = Gy
		self.name = name

sm2p256v1 = CurveFp(
	name="sm2p256v1",
	A=0xFFFFFFFEFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF00000000FFFFFFFFFFFFFFFC,
	B=0x28E9FA9E9D9F5E344D5A9E4BCF6509A7F39789F515AB8F92DDBCBD414D940E93,
	P=0xFFFFFFFEFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF00000000FFFFFFFFFFFFFFFF,
	N=0xFFFFFFFEFFFFFFFFFFFFFFFFFFFFFFFF7203DF6B21C6052B53BBF40939D54123,
	Gx=0x32C4AE2C1F1981195F9904466A39C9948FE30BBFF2660BE1715A4589334C74C7,
	Gy=0xBC3736A2F4F6779C59BDCEE36B692153D0A9877CC62A474002DF32E52139F0A0
)

def multiply(a, n, N, A, P):
	return fromJacobian(jacobianMultiply(toJacobian(a), n, N, A, P), P)

def add(a, b, A, P):
	return fromJacobian(jacobianAdd(toJacobian(a), toJacobian(b), A, P), P)

def inv(a, n):
	if a == 0:
		return 0
	lm, hm = 1, 0
	low, high = a % n, n
	while low > 1:
		r = high//low
		nm, new = hm-lm*r, high-low*r
		lm, low, hm, high = nm, new, lm, low
	return lm % n

def toJacobian(Xp_Yp):
	Xp, Yp = Xp_Yp
	return (Xp, Yp, 1)

def fromJacobian(Xp_Yp_Zp, P):
	Xp, Yp, Zp = Xp_Yp_Zp
	z = inv(Zp, P)
	return ((Xp * z**2) % P, (Yp * z**3) % P)

def jacobianDouble(Xp_Yp_Zp, A, P):
	Xp, Yp, Zp = Xp_Yp_Zp
	if not Yp:
		return (0, 0, 0)
	ysq = (Yp ** 2) % P
	S = (4 * Xp * ysq) % P
	M = (3 * Xp ** 2 + A * Zp ** 4) % P
	nx = (M**2 - 2 * S) % P
	ny = (M * (S - nx) - 8 * ysq ** 2) % P
	nz = (2 * Yp * Zp) % P
	return (nx, ny, nz)

def jacobianAdd(Xp_Yp_Zp, Xq_Yq_Zq, A, P):
	Xp, Yp, Zp = Xp_Yp_Zp
	Xq, Yq, Zq = Xq_Yq_Zq
	if not Yp:
		return (Xq, Yq, Zq)
	if not Yq:
		return (Xp, Yp, Zp)
	U1 = (Xp * Zq ** 2) % P
	U2 = (Xq * Zp ** 2) % P
	S1 = (Yp * Zq ** 3) % P
	S2 = (Yq * Zp ** 3) % P
	if U1 == U2:
		if S1 != S2:
			return (0, 0, 1)
		return jacobianDouble((Xp, Yp, Zp), A, P)
	H = U2 - U1
	R = S2 - S1
	H2 = (H * H) % P
	H3 = (H * H2) % P
	U1H2 = (U1 * H2) % P
	nx = (R ** 2 - H3 - 2 * U1H2) % P
	ny = (R * (U1H2 - nx) - S1 * H3) % P
	nz = (H * Zp * Zq) % P
	return (nx, ny, nz)

def jacobianMultiply(Xp_Yp_Zp, n, N, A, P):
	Xp, Yp, Zp = Xp_Yp_Zp
	if Yp == 0 or n == 0:
		return (0, 0, 1)
	if n == 1:
		return (Xp, Yp, Zp)
	if n < 0 or n >= N:
		return jacobianMultiply((Xp, Yp, Zp), n % N, N, A, P)
	if (n % 2) == 0:
		return jacobianDouble(jacobianMultiply((Xp, Yp, Zp), n // 2, N, A, P), A, P)
	if (n % 2) == 1:
		return jacobianAdd(jacobianDouble(jacobianMultiply((Xp, Yp, Zp), n // 2, N, A, P), A, P), (Xp, Yp, Zp), A, P)

class PrivateKey:
	def __init__(self, curve=sm2p256v1, secret=None):
		self.curve = curve
		self.secret = secret or SystemRandom().randrange(1, curve.N)

	def publicKey(self):
		curve = self.curve
		xPublicKey, yPublicKey = multiply((curve.Gx, curve.Gy), self.secret, A=curve.A, P=curve.P, N=curve.N)
		return PublicKey(xPublicKey, yPublicKey, curve)

	def toString(self):
		return "{}".format(str(hex(self.secret))[2:].zfill(64))

class PublicKey:
	def __init__(self, x, y, curve):
		self.x = x
		self.y = y
		self.curve = curve

	def toString(self, compressed=True):
		return {
			True:  str(hex(self.x))[2:],
			False: "{}{}".format(str(hex(self.x))[2:].zfill(64), str(hex(self.y))[2:].zfill(64))
		}.get(compressed)


if __name__ == "__main__":
    priKey = PrivateKey()
    pubKey = priKey.publicKey()
    print(priKey.toString())
    print(pubKey.toString(compressed = False))

加解密算法
封装类 sm2encryp.py

from gmssl import sm2
from base64 import b64encode, b64decode
# sm2的公私钥
SM2_PRIVATE_KEY = '00B9AB0B828FF68872F21A837FC303668428DEA11DCD1B24429D0C99E24EED83D5'
SM2_PUBLIC_KEY = 'B9C9A6E04E9C91F7BA880429273747D7EF5DDEB0BB2FF6317EB00BEF331A83081A6994B8993F3F5D6EADDDB81872266C87C018FB4162F5AF347B483E24620207'

sm2_crypt = sm2.CryptSM2(public_key=SM2_PUBLIC_KEY, private_key=SM2_PRIVATE_KEY)


class sm2Encrypt:
    # 加密
    def encrypt(self, info):
        encode_info = sm2_crypt.encrypt(info.encode(encoding="utf-8"))
        encode_info = b64encode(encode_info).decode()  # 将二进制bytes通过base64编码
        return encode_info

    # 解密
    def decrypt(self, info):
        decode_info = b64decode(info.encode())  # 通过base64解码成二进制bytes
        decode_info = sm2_crypt.decrypt(decode_info).decode(encoding="utf-8")
        return decode_info


if __name__ == "__main__":
    origin_pwd = '123456'
    sm2 = sm2Encrypt()
    # 加密的密码
    encrypy_pwd = sm2.encrypt(origin_pwd)
    print(encrypy_pwd)
    # 解密的密码
    decrypt_pwd = sm2.decrypt(encrypy_pwd)
    print(decrypt_pwd)

当跟sm2encryp.py在一个文件夹是可以直接引用它

from sm2encryp import sm2Encrypt
pass_encrypt = sm2Encrypt()
pwd = pass_encrypt.decrypt("H24OlVZgSTtevCW138O+C5PlZp8OiD920JnpVr7r9ndkGBWFZUVDD48iIVrZRnamgosV5910m9k0438WpIyi0guEt8F5inG7Y5A51whRfdPZ+qdvWVQxI857CBEzkb3h1bMp1ETQ")
print(pwd)

参考

https://www.jianshu.com/p/efc43060e0aa
https://blog.csdn.net/u013137970/article/details/84573200
https://blog.csdn.net/u014651560/article/details/113744296