Christoph's last Weblog entries

Entries from April 2017.

Secured OTP Server (ASIS CTF 2017)
9th April 2017

This weekend was ASIS Quals weekend again. And just like last year they have quite a lot of nice crypto-related puzzles which are fun to solve (and not "the same as every ctf").

Actually Secured OTP Server is pretty much the same as the First OTP Server (actually it's a "fixed" version to enforce the intended attack). However the template phrase now starts with enough stars to prevent simple root.:

def gen_otps():
    template_phrase = '*************** Welcome, dear customer, the secret passphrase for today is: '

    OTP_1 = template_phrase + gen_passphrase(18)
    OTP_2 = template_phrase + gen_passphrase(18)

    otp_1 = bytes_to_long(OTP_1)
    otp_2 = bytes_to_long(OTP_2)

    nbit, e = 2048, 3
    privkey = RSA.generate(nbit, e = e)
    pubkey  = privkey.publickey().exportKey()
    n = getattr(privkey.key, 'n')

    r = otp_2 - otp_1
    if r < 0:
        r = -r
    IMP = n - r**(e**2)
    if IMP > 0:
        c_1 = pow(otp_1, e, n)
        c_2 = pow(otp_2, e, n)
    return pubkey, OTP_1[-18:], OTP_2[-18:], c_1, c_2

Now let A = template * 2^(18*8), B = passphrase. This results in OTP = A + B. c therefore is (A+B)^3 mod n == A^3 + 3A^2b + 3AB^2 + B^3. Notice that only B^3 is larger than N and is statically known. Therefore we can calculate A^3 // N and add that to c to "undo" the modulo operation. With that it's only iroot and long_to_bytes to the solution. Note that we're talking about OTP and C here. The code actually produced two OTP and C values but you can use either one just fine.

#!/usr/bin/python3

import sys
from util import bytes_to_long
from gmpy2 import iroot

PREFIX = b'*************** Welcome, dear customer, the secret passphrase for today is: '
OTPbase = bytes_to_long(PREFIX + b'\x00' * 18)

N = 27990886688403106156886965929373472780889297823794580465068327683395428917362065615739951108259750066435069668684573174325731274170995250924795407965212988361462373732974161447634230854196410219114860784487233470335168426228481911440564783725621653286383831270780196463991259147093068328414348781344702123357674899863389442417020336086993549312395661361400479571900883022046732515264355119081391467082453786314312161949246102368333523674765325492285740191982756488086280405915565444751334123879989607088707099191056578977164106743480580290273650405587226976754077483115441525080890390557890622557458363028198676980513

WRAPPINGS = (OTPbase ** 3) // N

C = 13094996712007124344470117620331768168185106904388859938604066108465461324834973803666594501350900379061600358157727804618756203188081640756273094533547432660678049428176040512041763322083599542634138737945137753879630587019478835634179440093707008313841275705670232461560481682247853853414820158909864021171009368832781090330881410994954019971742796971725232022238997115648269445491368963695366241477101714073751712571563044945769609486276590337268791325927670563621008906770405196742606813034486998852494456372962791608053890663313231907163444106882221102735242733933067370757085585830451536661157788688695854436646

x = N * WRAPPINGS + C

val, _ = iroot(x, 3)
bstr = "%x" % int(val)

for i in range(0, len(bstr) // 2):
    sys.stdout.write(chr(int(bstr[2*i:2*i+2], 16)))

print()
Tags: asisctf, crypto, ctf, faust, security, writeup.

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