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2022 | corCTF | crypto

exchanged

#algebra

题目

you could make an exchange out of this

exchanged.py 
from Crypto.Util.number import *
from Crypto.Cipher import AES
from Crypto.Util.Padding import pad
from hashlib import sha256
from secrets import randbelow

p = 142031099029600410074857132245225995042133907174773113428619183542435280521982827908693709967174895346639746117298434598064909317599742674575275028013832939859778024440938714958561951083471842387497181706195805000375824824688304388119038321175358608957437054475286727321806430701729130544065757189542110211847
a = randbelow(p)
b = randbelow(p)
s = randbelow(p)

print("p =", p)
print("a =", a)
print("b =", b)
print("s =", s)

a_priv = randbelow(p)
b_priv = randbelow(p)

def f(s):
    return (a * s + b) % p

def mult(s, n):
    for _ in range(n):
        s = f(s)
    return s

A = mult(s, a_priv)
B = mult(s, b_priv)

print("A =", A)
print("B =", B)

shared = mult(A, b_priv)
assert mult(B, a_priv) == shared

flag = open("flag.txt", "rb").read()
key = sha256(long_to_bytes(shared)).digest()[:16]
iv = long_to_bytes(randint(0, 2**128))
cipher = AES.new(key, AES.MODE_CBC, iv=iv)
print(iv.hex() + cipher.encrypt(pad(flag, 16)).hex())
output.txt 
p = 142031099029600410074857132245225995042133907174773113428619183542435280521982827908693709967174895346639746117298434598064909317599742674575275028013832939859778024440938714958561951083471842387497181706195805000375824824688304388119038321175358608957437054475286727321806430701729130544065757189542110211847
a = 118090659823726532118457015460393501353551257181901234830868805299366725758012165845638977878322282762929021570278435511082796994178870962500440332899721398426189888618654464380851733007647761349698218193871563040337609238025971961729401986114391957513108804134147523112841191971447906617102015540889276702905
b = 57950149871006152434673020146375196555892205626959676251724410016184935825712508121123309360222777559827093965468965268147720027647842492655071706063669328135127202250040935414836416360350924218462798003878266563205893267635176851677889275076622582116735064397099811275094311855310291134721254402338711815917
s = 35701581351111604654913348867007078339402691770410368133625030427202791057766853103510974089592411344065769957370802617378495161837442670157827768677411871042401500071366317439681461271483880858007469502453361706001973441902698612564888892738986839322028935932565866492285930239231621460094395437739108335763
A = 27055699502555282613679205402426727304359886337822675232856463708560598772666004663660052528328692282077165590259495090388216629240053397041429587052611133163886938471164829537589711598253115270161090086180001501227164925199272064309777701514693535680247097233110602308486009083412543129797852747444605837628
B = 132178320037112737009726468367471898242195923568158234871773607005424001152694338993978703689030147215843125095282272730052868843423659165019475476788785426513627877574198334376818205173785102362137159225281640301442638067549414775820844039938433118586793458501467811405967773962568614238426424346683176754273
e0364f9f55fc27fc46f3ab1dc9db48fa482eae28750eaba12f4f76091b099b01fdb64212f66caa6f366934c3b9929bad37997b3f9d071ce3c74d3e36acb26d6efc9caa2508ed023828583a236400d64e

解题思路

  • \(f(x)=ax+b\ (mod\ p)\)，其中 \(p,a,b\) 均已知

  • \(A, B\) 分别为 \(s\) 经过 \(a_{priv}, b_{priv}\) 次 \(f\) 运算后的结果，需要根据 \(A,B\) 求 \(s\) 经过 \(a_{priv}+b_{priv}\) 次 \(f\) 运算后的结果

    \(mult(s, n)=a^{n}s+b(a^{n-1} + a^{n-2} + \dotsb + 1)\)

  • \(ax+b\) 的形式不能直接 \(A\cdot B\) 来求 shared，可进行转换，设存在 \(t\) 使得 \(a(x+t) = f(x) + t\)，解得 \(t = \frac{b}{a-1}\)

\[\begin{equation} \begin{split} f(mult(s,n-1))+t &= a(mult(s,n-1)+t) \\ &=a(a^{n-1}s+b\cdot(a^{n-2}+a^{n-3}+\dotsb+1)+t) \\ &=a(a^{n-1}s+t\cdot(a-1)\cdot(a^{n-2}+a^{n-3}+\dotsb+1)+t) \\ &=a(a^{n-1}s+t\cdot(a^{n-1}-1)+t) \\ &=a^n(s+t) \end{split} \end{equation}\]
  • 那么，\(A+t=a^{a_{priv}}(s+t),B+t=a^{b_{priv}}(s+t)\)，shared 即 \(mult(s,a_{priv}+b_{priv})=a^{a_{priv}+b_{priv}}(s+t)-t=\frac{(A+t)(B+t)}{s+t}-t\)
from Crypto.Util.number import long_to_bytes, inverse
from Crypto.Cipher import AES
from hashlib import sha256

*vars, enc = open("output.txt").readlines()
for v in vars: exec(v)
iv, ct = bytes.fromhex(enc[:32]), bytes.fromhex(enc[32:])

t = b * inverse(a - 1, p) % p
shared = ((A + t) * (B + t) * inverse(s + t, p) - t) % p

key = sha256(long_to_bytes(shared)).digest()[:16]
cipher = AES.new(key, AES.MODE_CBC, iv=iv)
print(cipher.decrypt(ct))

Flag

corctf{th1s_lcg_3xch4ng3_1s_4_l1ttl3_1ns3cur3_f0r_n0w}

最后更新: 2022年9月18日 16:04:25
Contributors: YanhuiJessica

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