Cipher RSA Wiener P-Q-E

• htb-brainfuck

• https://crypto.stackexchange.com/questions/19444/rsa-given-q-p-and-e

Definitions

• p - prime random 1

• q - prime random 2

• n

• e

• ct - cipher text

• pt - plain text

Encryption:

> cat encrypt.sage 


nbits = 1024
password = open("/root/root.txt").read().strip()
enc_pass = open("output.txt","w")
debug = open("debug.txt","w")
m = Integer(int(password.encode('hex'),16))

p = random_prime(2^floor(nbits/2)-1, lbound=2^floor(nbits/2-1), proof=False)
q = random_prime(2^floor(nbits/2)-1, lbound=2^floor(nbits/2-1), proof=False)
n = p*q
phi = (p-1)*(q-1)
e = ZZ.random_element(phi)
while gcd(e, phi) != 1:
    e = ZZ.random_element(phi)

c = pow(m, e, n)
enc_pass.write('Encrypted Password: '+str(c)+'\n')
debug.write(str(p)+'\n')
debug.write(str(q)+'\n')
debug.write(str(e)+'\n')

Debug Outputs

> cat debug.txt 

7493025776465062819629921475535241674460826792785520881387158343265274170009282504884941039852933109163193651830303308312565580445669284847225535166520307
7020854527787566735458858381555452648322845008266612906844847937070333480373963284146649074252278753696897245898433245929775591091774274652021374143174079
30802007917952508422792869021689193927485016332713622527025219105154254472344627284947779726280995431947454292782426313255523137610532323813714483639434257536830062768286377920010841850346837238015571464755074669373110411870331706974573498912126641409821855678581804467608824177508976254759319210955977053997


> cat output.txt 

Encrypted Password: 44641914821074071930297814589851746700593470770417111804648920018396305246956127337150936081144106405284134845851392541080862652386840869768622438038690803472550278042463029816028777378141217023336710545449512973950591755053735796799773369044083673911035030605581144977552865771395578778515514288930832915182

Decode Script

def egcd(a, b):
    x,y, u,v = 0,1, 1,0
    while a != 0:
        q, r = b//a, b%a
        m, n = x-u*q, y-v*q
        b,a, x,y, u,v = a,r, u,v, m,n
        gcd = b
    return gcd, x, y

def main():

    p = 7493025776465062819629921475535241674460826792785520881387158343265274170009282504884941039852933109163193651830303308312565580445669284847225535166520307
    q = 7020854527787566735458858381555452648322845008266612906844847937070333480373963284146649074252278753696897245898433245929775591091774274652021374143174079
    e = 30802007917952508422792869021689193927485016332713622527025219105154254472344627284947779726280995431947454292782426313255523137610532323813714483639434257536830062768286377920010841850346837238015571464755074669373110411870331706974573498912126641409821855678581804467608824177508976254759319210955977053997
    ct = 44641914821074071930297814589851746700593470770417111804648920018396305246956127337150936081144106405284134845851392541080862652386840869768622438038690803472550278042463029816028777378141217023336710545449512973950591755053735796799773369044083673911035030605581144977552865771395578778515514288930832915182

    # compute n
    n = p * q

    # Compute phi(n)
    phi = (p - 1) * (q - 1)

    # Compute modular inverse of e
    gcd, a, b = egcd(e, phi)
    d = a

    print( "n:  " + str(d) );

    # Decrypt ciphertext
    pt = pow(ct, d, n)
    print( "pt: " + str(pt) )

if __name__ == "__main__":
    main()

Execution

> python3 crack.py 
n:  8730619434505424202695243393110875299824837916005183495711605871599704226978295096241357277709197601637267370957300267235576794588910779384003565449171336685547398771618018696647404657266705536859125227436228202269747809884438885837599321762997276849457397006548009824608365446626232570922018165610149151977
pt: 24604052029401386049980296953784287079059245867880966944246662849341507003750

Human Readable

Python2:

>>> pt = 24604052029401386049980296953784287079059245867880966944246662849341507003750
>>> str(hex(pt))
'0x3665666331613564626238393034373531636536353636613330356262386566L'
>>> str(hex(pt)[2:-1])
'3665666331613564626238393034373531636536353636613330356262386566'
>>> str(hex(pt)[2:-1]).decode('hex')
'6efc1a5dbb8904751ce6566a305bb8ef'


Python3:

>>> pt = 24604052029401386049980296953784287079059245867880966944246662849341507003750
>>> pt = pt.to_bytes(((pt.bit_length() + 7) // 8), "big").decode()
>>> print(pt)
6efc1a5dbb8904751ce6566a305bb8ef  ..flag!!