NPUCTF2020 共模攻擊

拜讀師傅們的wp頗有收獲，記錄在此，以備日后查閱~

hint

1. hint.py中后半部分代碼給了n,e1,e2,c1,c2可以求出c的值，由c和p可以求得m，由m得到hint

2. c的求解過程就是共模攻擊。共模攻擊代碼^[1]如下（通用）

# py2 sameNAttack.py
import primefac

def same_n_attack(n,e1,e2,c1,c2):
    def egcd(a,b):
        x, lastX = 0, 1
        y, lastY = 1, 0
        while (b != 0):
            q = a // b
            a, b = b, a % b
            x, lastX = lastX - q * x, x
            y, lastY = lastY - q * y, y
        return (lastX, lastY)
    s = egcd(e1,e2)
    s1 = s[0]
    s2 = s[1]
    if s1 < 0:
        s1 = -s1
        c1 = primefac.modinv(c1, n)
        if c1 < 0:
            c1 += n
    elif s2 < 0:
        s2 = -s2
        c2 = primefac.modinv(c2, n)
        if c2 < 0:
            c2 += n
    m = (pow(c1, s1, n) * pow(c2, s2, n)) % n
    return m

3. 得到c后，有這樣的一個表達式：c = m²⁵⁶ mod p，有兩種方法解出m。

　　方法一：借助Python的sympy庫nthroot_mod方法^[2]

# dec.py
from Crypto.Util.number import long_to_bytes
from sameNAttack import same_n_attack
from sympy.ntheory.residue_ntheory import nthroot_mod

# hint
n = 6807492006219935335233722232024809784434293293172317282814978688931711423939629682224374870233587969960713638310068784415474535033780772766171320461281579
e1= 2303413961
e2 = 2622163991
c1 = 1754421169036191391717309256938035960912941109206872374826444526733030696056821731708193270151759843780894750696642659795452787547355043345348714129217723
c2= 1613454015951555289711148366977297613624544025937559371784736059448454437652633847111272619248126613500028992813732842041018588707201458398726700828844249
c = same_n_attack(n, e1, e2, c1, c2)
print c

p = 107316975771284342108362954945096489708900302633734520943905283655283318535709
m = nthroot_mod(c,256,p,all_roots=True)
print m
for i in m:
    print i
    hint = long_to_bytes(i)
    print hint

得到hint：m.bit_length() < 400

task

　　由於hint提示了m有長度限制，所以聯想到Coppersmith定理。Coppersmith定理的內容為：在一個e階的mod n多項式f(x)中，如果有一個根小於n^1/e，就可以運用一個O(log n)的算法求出這些根^[3]。計算可得m是滿足這個情況的。

task中我們可以獲取的信息有：

　　c1 = m^p mod n = m^p mod p*q

　　c2 = m^q mod n = m^q mod p*q

因為p、q為素數，所以由費馬定理可得：

　　m^p ≡ m mod p

　　m^q ≡ m mod q

所以，又有：

　　c1 = m + ip + xpq，可整理成 c1 = m + ip 

　　c2 = m + jq + ypq，可整理成 c2 = m + jq

因此：

　　c1 * c2 = m² + (ip + jq)m + ijn

　　(c1 + c2)m = 2m² + (ip+jq)m 

　　有： m² - (c1 + c2)m + c1 * c2 = ijn ≡ 0 mod n

最終的任務就是求m的值。

sage代碼^[2]如下：

n=128205304743751985889679351195836799434324346996129753896234917982647254577214018524580290192396070591032007818847697193260130051396080104704981594190602854241936777324431673564677900773992273463534717009587530152480725448774018550562603894883079711995434332008363470321069097619786793617099517770260029108149
c1=96860654235275202217368130195089839608037558388884522737500611121271571335123981588807994043800468529002147570655597610639680977780779494880330669466389788497046710319213376228391138021976388925171307760030058456934898771589435836261317283743951614505136840364638706914424433566782044926111639955612412134198
c2=9566853166416448316408476072940703716510748416699965603380497338943730666656667456274146023583837768495637484138572090891246105018219222267465595710692705776272469703739932909158740030049375350999465338363044226512016686534246611049299981674236577960786526527933966681954486377462298197949323271904405241585m

PR.<m> = PolynomialRing(Zmod(n))
f = m^2-(c1+c2)*m+c1*c2
x0 = f.small_roots(X=2^400)
print(x0)

得到x0=4242839043019782000788118887372132807371568279472499477998758466224002905442227156537788110520335652385855

用Python輸出以下就可以：

from Crypto.Util.number import long_to_bytes
x0=4242839043019782000788118887372132807371568279472499477998758466224002905442227156537788110520335652385855
print long_to_bytes(x0)

最終結果為verrrrrrry_345yyyyyyy_rsaaaaaaa_righttttttt?

Reference

[1] FlappyPig. CTF特訓營. 2020

[2] https://shimo.im/docs/6hyIjGkLoRc43JRs

[3] https://www.52pojie.cn/thread-653446-1-8.html