ECC椭圆曲线加密算法—加解密(SageMath实现)

简介

ECC椭圆曲线加密,它的安全性基于椭圆曲线上的离散对数问题。

比特币和目前的二代居民身份证都采用了ECC作为加密算法。

ECC椭圆曲线函数为:

\[y^{2}=x^{3}+ax+b\ (mod\ p) \]

ECC算法如下:

椭圆曲线Ep(a,b)(p为模数),基点(生成元)G(x,y),G点的阶数n,私钥k,公钥K(x,y),随机整数r,明文为一点m(x,y),密文为两点c1(x,y)和c2(x,y)
(其中基点G,明文m,公钥K,密文c1、c2都是椭圆曲线E上的点)
选择私钥k(k<n)
得到公钥K = k*G
选择随机整数r(r<n)
加密:
c1 = m+r*K
c2 = r*G
解密:
m = c1-k*c2(= c1-r*K)

关于椭圆曲线的更多知识,可以参考Kalafinaian师傅的文章:https://www.cnblogs.com/Kalafinaian/p/7392505.html

SageMath可以直接计算椭圆曲线加法和椭圆曲线乘法。

椭圆曲线运算(SageMath):

点u(x,y),整数a,点v(x,y),点w(x,y)
a_inv = inverse_mod(a,p) #a_inv是a关于模p的乘法逆元a_inv
v = a*u
u = v*a_inv
w = u+v

加解密脚本

SageMath加密脚本:

'''
加密
椭圆曲线选取时,模数p应是一个大质数
常用的有几个公开的椭圆曲线,如Secp256k1、Secp256r1等
'''
p = 115792089210356248762697446949407573530086143415290314195533631308867097853951
a = 115792089210356248762697446949407573530086143415290314195533631308867097853948
b = 41058363725152142129326129780047268409114441015993725554835256314039467401291
E = EllipticCurve(GF(p),[a,b]) #建立椭圆曲线E
G = E(101981543389703054444906888236965100227902100585263327233246492901054535785571,105947302391877180514060433855403037184838385483621546199124860815209826713886) #选择一点作为生成元
n = G.order() #G的阶数
k = 78772200542717449282831156601030024198219944170436309154595818823706214492400
K = k*G
r = 3546765
m = E(80764032034929976879602863302323059647882062252124869895215418422992624743795,4964654783828069942602279691168356721024126126864424301508238062949726916347) #取E上一点m作为明文
c1 = m+r*K
c2 = r*G
print(c1)
print(c2)

SageMath解密脚本:

'''解密'''
p = 115792089210356248762697446949407573530086143415290314195533631308867097853951
a = 115792089210356248762697446949407573530086143415290314195533631308867097853948
b = 41058363725152142129326129780047268409114441015993725554835256314039467401291
k = 78772200542717449282831156601030024198219944170436309154595818823706214492400
E = EllipticCurve(GF(p),[a,b]) #建立椭圆曲线E
c1 = E(55527726590533087179712343802771216661752045890626636388680526348340802301667,99976146729305231192119179111453136971828647307627310904093286590128902629941)
c2 = E(85460365972589567444123006081329559170090723413178386022601904195400422637884,58249081362527056631776731740177334121295518073095154119886890634279528757192)
m = c1-k*c2
print(m)

其他

使用Crypto.PublicKey.ECC生成ECC密钥:

from Crypto.PublicKey import ECC

#生成ECC密钥
key = ECC.generate(curve='NIST P-256') #使用椭圆曲线NIST P-256

#输出密钥(包括私钥k,基点G)
print(key)

#公钥(point_x,point_y是基点G的坐标)
print(key.public_key())

#椭圆曲线
print(key.curve)

#私钥k
print(key.d)

#导出为pem密钥文件
print(key.export_key(format='PEM'))

#导入密钥文件
key = ECC.import_key(f.read())

通过fastecdsa.Curve可以查到公开椭圆曲线的参数

import fastecdsa.curve as curve

#P-384的a
curve.P384.a

#P-384的b
curve.P384.b

#P-384的p
curve.P384.p

几种公开椭圆曲线参数:

#NIST P-256(Secp256r1)
#p = 2^224(2^32 − 1) + 2^192 + 2^96 − 1
p = 115792089210356248762697446949407573530086143415290314195533631308867097853951
a = 115792089210356248762697446949407573530086143415290314195533631308867097853948
b = 41058363725152142129326129780047268409114441015993725554835256314039467401291

#Secp256k1(比特币使用)
#p = 2^256 − 2^32 − 2^9 − 2^8 − 2^7 − 2^6 − 2^4 − 1 = 2^256 – 2^32 – 977 
p = 115792089237316195423570985008687907853269984665640564039457584007908834671663
a = 0
b = 7

#NIST P-384
#p = 2^384 – 2^128 – 2^96 + 2^32 – 1
p = 39402006196394479212279040100143613805079739270465446667948293404245721771496870329047266088258938001861606973112319
a = -3
b = 27580193559959705877849011840389048093056905856361568521428707301988689241309860865136260764883745107765439761230575

#NIST P-521
p = 6864797660130609714981900799081393217269435300143305409394463459185543183397656052122559640661454554977296311391480858037121987999716643812574028291115057151
a = -3
b = 1093849038073734274511112390766805569936207598951683748994586394495953116150735016013708737573759623248592132296706313309438452531591012912142327488478985984

SageMath取椭圆曲线上随机一点:

E = EllipticCurve(GF(p),[a,b])
E.random_point() #取椭圆曲线E上随机一点

sagemath计算椭圆曲线上的离散对数问题(数据量不能太大)

a = 1234577
b = 3213242
p = 7654319
E = EllipticCurve(GF(p),[a,b])
G = E(5234568, 2287747) #生成元
#k = 1584718
K = E(2366653, 1424308) #公钥

#求解私钥,自动选择bsgs或Pohlig Hellman算法
discrete_log(K,G,operation='+')

#求解私钥,Pollard rho算法
discrete_log_rho(K,G,operation='+')

#求解私钥,Pollard Lambda算法,能够确定所求值在某一小范围时效率较高
discrete_log_lambda(K,G,(1500000,2000000),operation='+')

使用openssl查看ECC的pem密钥文件信息

#查看ECC私钥信息
openssl ec -in p384-key.pem -text -noout

#查看ECC公钥信息
openssl ec -pubin -in public.pem -text -noout
posted @ 2020-10-12 22:30  lnjoy  阅读(6572)  评论(0编辑  收藏  举报