Cryptopals Challenge 43

Cryptopals Challenge 43 的题解，很简单
DSA key recovery from nonce

题意

已知

m = For those that envy a MC it can be hazardous to your health
So be friendly, a matter of life and death, just like a etch-a-sketch
p = 0x800000000000000089e1855218a0e7dac38136ffafa72eda7859f2171e25e65eac698c1702578b07dc2a1076da241c76c62d374d8389ea5aeffd3226a0530cc565f3bf6b50929139ebeac04f48c3c84afb796d61e5a4f9a8fda812ab59494232c7d2b4deb50aa18ee9e132bfa85ac4374d7f9091abc3d015efc871a584471bb1
q = 0xf4f47f05794b256174bba6e9b396a7707e563c5b
g = 0x5958c9d3898b224b12672c0b98e06c60df923cb8bc999d119458fef538b8fa4046c8db53039db620c094c9fa077ef389b5322a559946a71903f990f1f7e0e025e2d7f7cf494aff1a0470f5b64c36b625a097f1651fe775323556fe00b3608c887892878480e99041be601a62166ca6894bdd41a7054ec89f756ba9fc95302291
r0 = 548099063082341131477253921760299949438196259240
s = 857042759984254168557880549501802188789837994940
k = 0-65536 的一个数

求 x(即 private key)

DSA 复习

算法中应用了下述参数：

p：L bits 长的素数。L 是 64 的倍数，范围是 512 到 1024；

q：p - 1 的 160bits 的素因子；

g： $g =h^{\frac{p-1}{q}}\ mod\ p$，$h$ 满足 $h < p - 1$，$h^{\frac{p-1}{q}}\ mod\ p>1$；

x：$x < q$，x 为私钥；

y：$y = g^{x}\ mod\ p$，$( p, q, g, y )$ 为公钥；

H(x)：One-Way Hash 函数。DSS（FIPS186-4）中选用 SHA-1 或者 SHA-2(Secure Hash Algorithm 系列中的 2 个较新版本，其中 SHA-2 有 4 个，SHA-224，SHA-256，SHA-384，SHA-512，最原始的 SHA 已经不再被使用)。

p, q, g 可由一组用户共享，但在实际应用中，使用公共模数可能会带来一定的威胁。签名及验证协议如下：

P 产生随机数 k，k < q；
P 计算

r = $ g^{k} mod p mod q$

$s = ( k^{-1} (H(m) + xr))\ mod\ q$

签名结果是(r, s)。
验证时计算

$w = s^{-1}mod q$

$u_1 = (H(m) * w)\ mod\ q$

$u_2 = (r * w)\ mod\ q$

$v = (( g^{u_1} * y^{u_2} )\ mod\ p )\ mod\ q$

若 $v = r$，则认为签名有效。

推导 x 的公式

若 $k$ 已知，则

$sk = (H(m)+xr)\ mod\ q$

$xr = (sk - H(m))\ mod\ q$

$x = \frac{sk - H(m)}{r}\ mod\ q$

解题过程

题中给出了 k 的范围，暴力一下就好了。对于每个 k，先求出对应的 r，若 r 等于题中给出的 r0，则证明 k 就是对的。得到了 k，可求 x，即 private key。

方法 1

from Crypto.Hash import SHA
from gmpy2 import *
from time import *
clock()

anSHA1 = '0954edd5e0afe5542a4adf012611a91912a3ec16'
print '[+]Cal SHA-1(M)...'
sha = SHA.new()
sha.update('''For those that envy a MC it can be hazardous to your health
So be friendly, a matter of life and death, just like a etch-a-sketch
''')
H = sha.hexdigest()
print '  [-]SHA-1(M) is:', H
print '  [-]Done!'

p = 0x800000000000000089e1855218a0e7dac38136ffafa72eda7859f2171e25e65eac698c1702578b07dc2a1076da241c76c62d374d8389ea5aeffd3226a0530cc565f3bf6b50929139ebeac04f48c3c84afb796d61e5a4f9a8fda812ab59494232c7d2b4deb50aa18ee9e132bfa85ac4374d7f9091abc3d015efc871a584471bb1
q = 0xf4f47f05794b256174bba6e9b396a7707e563c5b
g = 0x5958c9d3898b224b12672c0b98e06c60df923cb8bc999d119458fef538b8fa4046c8db53039db620c094c9fa077ef389b5322a559946a71903f990f1f7e0e025e2d7f7cf494aff1a0470f5b64c36b625a097f1651fe775323556fe00b3608c887892878480e99041be601a62166ca6894bdd41a7054ec89f756ba9fc95302291
r0 = 548099063082341131477253921760299949438196259240
s = 857042759984254168557880549501802188789837994940

print '[+]Searching key...'
for k in xrange(65537):
    r = pow(g, k, p) % q
    if r == r0:
        print '  [-]key Found!'
        print '  [-]key is:', k

        print '    [-]Cal x...'
        x = (s*k - int(H,16)) * invert(r,q) % q
        print '    [-]x is:', x
        
        print '    [+]Cal SHA-1(x)...'
        sha = SHA.new() 
        sha.update('%x' %(x))
        H = sha.hexdigest()
        print '      [-]SHA-1(x) is:', H
        print '      [-]Done!' 
        
        print '[+]SHA-1(x) == %s:' %anSHA1, H == anSHA1
        assert H == anSHA1
        break

print '[!]All Done!'
print '[!]Timer:', round(clock()), 's'

结果

[+]Cal SHA-1(M)...
  [-]SHA-1(M) is: d2d0714f014a9784047eaeccf956520045c45265
  [-]Done!
[+]Searching key...
  [-]key Found!
  [-]key is: 16575
    [-]Cal x...
    [-]x is: 125489817134406768603130881762531825565433175625
    [+]Cal SHA-1(x)...
      [-]SHA-1(x) is: 0954edd5e0afe5542a4adf012611a91912a3ec16
      [-]Done!
[+]SHA-1(x) == 0954edd5e0afe5542a4adf012611a91912a3ec16: True
[!]All Done!
[!]Timer: 5.0 s

方法 2

from Crypto.Hash import SHA
from gmpy2 import *
from time import *
clock()

ansSHA1 = '0954edd5e0afe5542a4adf012611a91912a3ec16'
print '[+]Cal SHA-1(M)...'
sha = SHA.new()
sha.update('''For those that envy a MC it can be hazardous to your health
So be friendly, a matter of life and death, just like a etch-a-sketch
''')
ansH = int(sha.hexdigest(),16)
print '  [-]SHA-1(M) is:', ansH
print '  [-]Done!'

p = 0x800000000000000089e1855218a0e7dac38136ffafa72eda7859f2171e25e65eac698c1702578b07dc2a1076da241c76c62d374d8389ea5aeffd3226a0530cc565f3bf6b50929139ebeac04f48c3c84afb796d61e5a4f9a8fda812ab59494232c7d2b4deb50aa18ee9e132bfa85ac4374d7f9091abc3d015efc871a584471bb1
q = 0xf4f47f05794b256174bba6e9b396a7707e563c5b
g = 0x5958c9d3898b224b12672c0b98e06c60df923cb8bc999d119458fef538b8fa4046c8db53039db620c094c9fa077ef389b5322a559946a71903f990f1f7e0e025e2d7f7cf494aff1a0470f5b64c36b625a097f1651fe775323556fe00b3608c887892878480e99041be601a62166ca6894bdd41a7054ec89f756ba9fc95302291
r = 548099063082341131477253921760299949438196259240
s = 857042759984254168557880549501802188789837994940

print '[+]Searching key...'

for k in xrange(65537):
    x = (s*k - ansH) * invert(r,q) % q
    sha = SHA.new() 
    sha.update('%x' %(x))
    H = sha.hexdigest()

    if H == ansSHA1:
        r = pow(g, k, p) % q
        print '  [-]key Found!'
        print '  [-]key is:', k
        print '    [-]x is:', x
        print '    [+]Cal SHA-1(x)...'
        print '      [-]SHA-1(x) is:', H
        print '      [-]Done!'     
        print '[+]SHA-1(x) == %s:' %ansSHA1, H == ansSHA1
        assert H == ansSHA1        
        break

print '[!]All Done!'
print '[!]Timer:', round(clock()), 's'

结果

[+]Cal SHA-1(M)...
  [-]SHA-1(M) is: 1203536487446634476431084512413493461057142018661
  [-]Done!
[+]Searching key...
  [-]key Found!
  [-]key is: 16575
    [-]x is: 125489817134406768603130881762531825565433175625
    [+]Cal SHA-1(x)...
      [-]SHA-1(x) is: 0954edd5e0afe5542a4adf012611a91912a3ec16
      [-]Done!
[+]SHA-1(x) == 0954edd5e0afe5542a4adf012611a91912a3ec16: True
[!]All Done!
[!]Timer: 0.0 s