Finding close-prime factorizations

This is a proposal for a simple factoring algorithm which actually performs better than regular attacks based on lattice-type algorithms. Although more efficient algorithms exists, this one is very simple to implement, while the more efficient ones are not. Do not quote me on this, though ;-D

Finding a good approximation

The first step is to compute an good approximation of \phi(n). Denote this \phi'. One such good approximation is \phi' = n - 2\sqrt{n} + 1. Let us see why. First, assuming that n = pq, note that

\phi(n) =  (p-1)(q-1) = n - (p+q) + 1

Let us assume there exists a small \gamma such that p = a + \gamma and q = a - \gamma. So, we have that

\delta := \phi' - \phi(n) = 2\sqrt{a^2 - \gamma^2} - 2a = 2a\sqrt{(1 - \gamma^2/a^2)} - 2a.

Now, let \beta = \gamma^2/a^2. Then,

\delta =2a\left(\sqrt{1 - \beta} - 1\right) = 2a \cdot \left[\beta/2 + \mathcal{O}(\beta^2)\right].

Since \beta is assumed to be small (a >> \gamma), we can approximate it as

\delta \approx \gamma^2/a.

Great! So the running time is bounded by \mathcal{O}( \gamma^2/a) if we use pure brute force. Now, we are going to be just a little more clever than that, aren’t we?

Searching efficiently

We will now show how to search for a solution faster than brute force. First, we pick an arbitrary invertible element y and compute all powers of y \bmod n up to a certain parameter b, i.e.,

\mathcal B := \{y^0 \bmod n, y^1 \bmod n, \dots , y^b \bmod n\}

Obviously, this requires \mathcal O(b) time and memory. A reference implementation in Python (with y = 2) would look something like.

# create a look-up table
look_up = {}
z = 1
for i in range(0, b + 1):
   look_up[z] = i
   z = (z * 2) % n

Now, compute

\mu := (y^{\phi'})^{-1} \bmod n.

Note that the exponent of \mu is

-\phi' \bmod \phi(n).

We now do a BSGS-type of algorithm. Generate all powers y^{bi} \bmod n for  0 \leq i \leq b and multiply with \mu forming c:= y^{bi} \cdot \mu. Again, we get an exponent which is

-\phi' + bi \bmod \phi(n).

For each such computed, check against \mathcal{B} is it exists. If so, we have found \phi(n).

# check the table
mu = gmpy.invert(pow(2, phi_approx, n), n)
fac = pow(2, b, n)
j = 0

while True:
    mu = (mu * fac) % n
    j += b
    if mu in look_up:
        phi = phi_approx + (look_up[mu] - j)
        break
    if j > b * b:
        return

This is \mathcal O(b) in time. Once \phi(n) has been found, we can trivially compute the factors in negligible time as

# compute roots
m = n - phi + 1
roots = (m - gmpy.sqrt(m ** 2 - 4 * n)) / 2, \
        (m + gmpy.sqrt(m ** 2 - 4 * n)) / 2

Cool! Since we require that \gamma^2/a \leq b^2 for a solution to be found, the running time is \mathcal O(\gamma/a^{1/2}) and with equal amount of memory required.

A real-world example (sort of)

This little finding inspired me to create a CTF challenge. During the SEC-T CTF 2017, one of the challenges (qproximity) could be solved using this method.

Description

We are given a public key and an encrypted message. The key is

-----BEGIN PUBLIC KEY-----
MIGeMA0GCSqGSIb3DQEBAQUAA4GMADCBiAKBgAOBxiQviVpL4G5d0TmVmjDn51zu
iravDlD4vUlVk9XK79/fwptVzYsjimO42+ZW5VmHF2AUXaPhDC3jBaoNIoa78CXO
ft030bR1S0hGcffcDFMm/tZxwu2/AAXCHoLdjHSwL7gxtXulFxbWoWOdSq+qxtak
zBSZ7R1QlDmbnpwdAgMDEzc=
-----END PUBLIC KEY-----

Extracting the modulus, we find that

\begin{array}{rl} n = &246264974647736414345408205036830544055349190030438864689361084738619 \\ &430136992438500973098730365134506003143847829773369403632725772343146 \\ &864922044439763512753030199250563829152109289871491767838931495698391 \\ &860322173235862868025625353744920431228772475066920885663471105686331 \\ &5465220853759428826555838536733 \end{array}

The factors are found as follows

def close_factor(n, b):

    # approximate phi
    phi_approx = n - 2 * gmpy.sqrt(n) + 1

    # create a look-up table
    look_up = {}
    z = 1
    for i in range(0, b + 1):
        look_up[z] = i
        z = (z * 2) % n

    # check the table
    mu = gmpy.invert(pow(2, phi_approx, n), n)
    fac = pow(2, b, n)
    j = 0

    while True:
        mu = (mu * fac) % n
        j += b
        if mu in look_up:
            phi = phi_approx + (look_up[mu] - j)
            break
        if j > b * b:
            return

    m = n - phi + 1
    roots = (m - gmpy.sqrt(m ** 2 - 4 * n)) / 2, \
            (m + gmpy.sqrt(m ** 2 - 4 * n)) / 2

    return roots

n = 2462649746477364143454082050368305440553491900304388646893610847386194301369924385009730987303651345060031438478297733694036327257723431468649220444397635127530301992505638291521092898714917678389314956983918603221732358628680256253537449204312287724750669208856634711056863315465220853759428826555838536733
b = 10000000

close_factor(n, b)

The code runs in matter of seconds and gives the factors

\begin{array}{rl} p = &156928319511723700336966188419841178779822856669047426731288203806365 \\ &781043486977181195772183437407585036370841822568787188893070958887580 \\ &0968904667752571 \end{array}

and

\begin{array}{rl} q = &156928319511723700336966188419841178779822856669047426731288203806365 \\ &838576177312234882203162849666809217683250766146317773098495921211843 \\ &5829588059735623 \end{array}

Using these, we construct the flag

SECT{w3ll_th4t_wasnt_2_h4rd?}
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Polictf 2017 – Lucky Consecutive Guessing

This is a short snippet to solve the LCG on Polictf’17. The basic idea is to reduce the problem aX_i + b = X_{i+1} \bmod M to aX_i = X_{i+1} \bmod M. This problem is quite easy to solve.

The basic idea is to view the problem as the latter and correct the discrepancies (\delta = (b, ab, a^2b,\dots)). So, for instance, if we sample

V = \texttt{1310585136, 1634517111, 2548394614, 745784911},

then these values contain the implicit discrepancies. By performing correction of the values (subtracting \sum_k^i \delta_k from the corresponding V_i we obtain a corrected value).

So, now we can use LLL to solve the simpler problem. Once a solution is found, we use the discrepancies to correct it to the case aX_i + b = X_{i+1} \bmod M.

import random

class LinearCongruentialGenerator:

    def __init__(self, a, b, nbits):
        self.a = a
        self.b = b
        self.nbits = nbits
        self.state = random.randint(0, 1 << nbits)

    def nextint(self):
        self.state = ((self.a * self.state) + self.b) % (1 << self.nbits)
        return self.state >> (self.nbits - 32)

    def break_lcg(a,b,p,i,j,outputs):
        deltas = [b % p, (a*b + b) % p, (a^2*b + a*b + b) % p, (a^3*b + a^2*b + a*b + b) % p]
        Y = [((val << (i - j)) ) for val, delta in zip(outputs, deltas)]
        L = matrix([
                   [p,    0,  0, 0 ],
                   [a^1, -1,  0, 0 ],
                   [a^2,  0, -1, 0 ],
                   [a^3,  0,  0, -1]
        ])
        B = L.LLL()
        Y = vector([x - y for x, y in zip(Y, deltas)])
        target = vector([ round(RR(w) / p) * p - w for w in B * vector(Y) ])
        states = list(B.solve_right(target))
        return [x + y + z for x, y, z in zip(Y, states, deltas)]

# parameters
p = 2^85
a = 0x66e158441b6995
b = 0xb
n = 85
r = 32
sequence = [1310585136, 1634517111, 2548394614, 745784911]

# break the LCG
start_seed = break_lcg(a,b,p,n,r,sequence)

# run it to sample required values
lcg = LinearCongruentialGenerator(a, b, n)
lcg.state = start_seed[0]
for i in range(0, 104):
    print lcg.nextint()

Generating wallpaper-consistent terminal colors with K-means++

K-means algorithm

Assume that we have to following image:

Vg3ve2E.jpg

We can represent the image in 3D-space as follows:

scatter0

Suppose that we want to find eight dominant colors in the image. We could create a histogram and take the eight most common values. This would be fine, but often, very similar colors would repeat. Instead, we can try to find eight centroids in the image and find clusters of points beloning to each centroid (say, a point p belongs to a centroid p_{\text{centroid}} if \|p - p_{\text{centroid}}\|_2 < R for some radius R). All points not belonging to a centroid will be penalized using some heuristic. This is basically the K-means clustering algorithm.

Usage

The algorithm can be used to generate set of dominant colors to be used for instance in the terminal. Running the above algorithm on the image, we get

colors.png

or as list

['#321331', '#e1a070', '#621d39', '#e05a40', '#9ed8aa', '#8b3860', '#f7d188', '#ab3031']

with some minor adjustment

{
"color": [
"#ab3031","#621d39","#e05a40","#e1a070","#9ed8aa","#521f50","#8b3860","#f7d188","#ab5a5b","#623b4b","#e08c7b","#e1b99c","#c4d8c8","#715171","#8b5770","#f7e4c0"
],
"foreground": "#c5c8c6",
"background": "#282a2e"
}

In iTerm, it might look like this

scrot

This can be achieved using the following code:

#!/usr/bin/env python
import sys, os
from sklearn.cluster import KMeans
from PIL import Image

nbrcentroids = 10
# Constant increase in colors
beta = 10
# Multplicative factor in background
gamma = 0.4
rgb2hex = lambda rgb: '#%s' % ''.join(('%02x' % min(p + beta, 255) for p in rgb))
darken = lambda rgb : (p * gamma for p in rgb)

def getcentroids(filename, n=8):
    img = Image.open(filename)
    img.thumbnail((100, 100))
    # Run K-means algorithm on image
    kmeans = KMeans(init='k-means++', n_clusters=n)
    kmeans.fit(list(img.getdata()))
    # Get centroids from solution
    rgbs = [map(int, c) for c in kmeans.cluster_centers_]
    return rgbs

def set_colors_gnome(centroids):
    centroids = sorted(centroids, key=lambda rgb: sum(c**2 for c in rgb))
    prefix = 'gsettings set org.pantheon.terminal.settings '
    # Set background and foreground
    os.system(prefix + 'background \"%s\"' % rgb2hex(darken(centroids[0])))
    os.system(prefix + 'foreground \"%s\"' % rgb2hex(centroids[-1]))
    # Set ANSI colors
    colors = ':'.join(rgb2hex(centroid) for centroid in centroids[1:-1])
    os.system(prefix + 'palette \"' + colors + ':' + colors + '\"')

def bar(mode):
    write = sys.stdout.write
    for i in range(0, nbrcentroids):
        write('\033[0;3%dmBAR ' % i)
    write('\n')

centroids = getcentroids(sys.argv[1], n=nbrcentroids)
set_colors_gnome(centroids)
bar(0);

It is on Github too!

PlaidCTF’17 – Multicast

We are given a challenge which contains a Sage script, which holds the following lines of code

nbits = 1024
e = 5
flag = open("flag.txt").read().strip()
assert len(flag) <= 64
m = Integer(int(flag.encode('hex'),16))
out = open("data.txt","w")

for i in range(e):
    while True:
        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)
        ni = p*q
        phi = (p-1)*(q-1)
        if gcd(phi, e) == 1:
            break

    while True:
        ai = randint(1,ni-1)
        if gcd(ai, ni) == 1:
            break

    bi = randint(1,ni-1)
    mi = ai*m + bi
    ci = pow(mi, e, ni)
    out.write(str(ai)+'\n')
    out.write(str(bi)+'\n')
    out.write(str(ci)+'\n')
    out.write(str(ni)+'\n')

…along with a text file containing the encrypted flag. Let us first parse the file and put in a JSON structure

data = {'n' : [n0, n1, ...], 'a' : [a0, a1, ...], ...}

We see that the flag is encrypted several times, up to affine transformation. If we define a polynomial g_i(x) = (a_ix + b_i)^5 - c_i \mod N_i, this has a root g_i(m) = 0 \mod N_i. We could try to find this root using Coppersmith, but it turns out it not possible since the size of (am + b)^5 is larger than N_i. However, due to a publication by Håstad, we can construct the following:

g(x) = \sum \phi_i(N_i) g_i(x)

where

\phi_i \mod N_j = \begin{cases}0 & \quad \text{if } i \neq j\\1 & \quad \text{when } i = j\\\end{cases}

It is easy to construct \phi_i as

\phi_i = \frac{1}{N_i}\cdot \prod_k N_k \cdot [\prod_k N_k / N_i]^{-1}_{N_i}

using the Chinese Remainder Theorem. Clearly, g(x) also has a root g(m) = 0 \mod (\prod N_i) (this is easy to check!). Since m^5 is strictly smaller than \prod_k N_k, the polynomial roots of g(x) can be found using Coppersmith!

p = data['n'][0] * data['n'][1] * data['n'][2] * data['n'][3] * data['n'][4]
PR. = PolynomialRing(Zmod(p))
f = 0

# compute the target polynomial
for i in range(0, 5):
    q = p / data['n'][i]
    qinv = inverse_mod(q, data['n'][i])
    f = f + q * qinv * ((data['a'][i]*x + data['b'][i])^5 - data['c'][i])

# make f monic
f = f * inverse_mod(f[5], p)

print f.small_roots(X=2^512, beta=1)[0]

The code outputs the long integer representation of

PCTF{L1ne4r_P4dd1ng_w0nt_s4ve_Y0u_fr0m_H4s7ad!}

ASIS CTF’17

F4 Phantom

With F-4 Phantom II, we want to break the encryption! Please help us!!

‍‍‍nc 66.172.27.77 54979

We get a key-generation function, as seen below.

def gen_pubkey(e, p):
    assert gmpy.is_prime(p) != 0
    B = bin(p).strip('0b')
    k = random.randrange(len(B))
    k, l = min(k, len(B) - k), max(k, len(B) - k)
    assert k != l
    BB = B[:k] + str(int(B[k]) ^ 1) + B[k+1:l] + str(int(B[l]) ^ 1) + B[l+1:]
    q = gmpy.next_prime(int(BB, 2))
    assert p != q
    n = p*q
    key = RSA.construct((long(n), long(e)))
    pubkey = key.publickey().exportKey("PEM")
    return n, p, q, pubkey

So, pq = p \cdot (p \pm 2^k \pm 2^l + \Delta), where \Delta = \texttt{next\_prime}(p \pm 2^k \pm 2^l) - (p \pm 2^k \pm 2^l). The density of primes (well, asymptotically) is \pi(n) \sim \frac{x}{log x}, so we expect \Delta to be quite small.

We define a := p \pm 2^k \pm 2^l. Now, we solve the following equation

x \cdot (x \pm 2^k \pm 2^l + \Delta) = N \iff x \cdot (x + a) - N = 0 \iff x = \frac{a}{2} \pm \sqrt{\frac{a^2}{4} + N}

Now, we know that the roots should be numerically close to p and q. So, we may simply call \texttt{next\_prime} on them and check if they divide N. Note that we need to do this for different hypotheses (different k).

import gmpy, base64
from Crypto.PublicKey import RSA

def solve(a):
   a2 = a ** 2/4
   L = gmpy.sqrt(a2 + N) # L > a/2
   C_1 = gmpy.next_prime(-a/2 + L)
   if N % C_1 == 0: return C_1
   C_2 = gmpy.next_prime(a/2 + L)
   if N % C_2 == 0: return C_2
   return False

# just some public key
pem_data1 = '''
MIGmMA0GCSqGSIb3DQEBAQUAA4GUADCBkAKBhyW0uQDhy7/eShVn6VQWQisC8sda
zl8xJmCe8eEPGNfDVMFgN7Ll4WdRLYE4T8dOvoMb99Cmjhym7Orb/qfP5q/BpTQy
5hr1w5RZVtYMqQ5I+M4qg7E8kkVhGJh/13bNwqEYvV6VNSCK+bgFWVGMelTQam31
Fiq6wsTLbIBCrLie52Cil1584QIEC8rPFw==
'''

pub =  RSA.importKey(base64.b64decode(pem_data1))
N = pub.n
m = (len(bin(N))+1)/2-2
print '[ ] Solving equations...'

for mm in range(m, m+3):
  for j in range(2, m/2+1):
    k = mm - j
    l = j
    retval = solve(2**(k-1) + 2**(l-1))
    if retval: 
      q = retval
      break
    retval = solve(2**(k-1) - 2**(l-1))
    if retval: 
      q = retval
      break

p = N / q
print '[+] Found p = {}... and q = {}...'.format(str(p)[:40], str(q)[:40])
λ python solve.py
[ ] Solving equations...
[+] Found p = 1381282812140256921334162381757305071089... and q = 1381282811302268925712750063033928508701...

Now, it turns out that \text{gcd}(e, (p-1)(q-1)) \neq 1. We cannot decrypt the message! If we look closely, we see that \text{gcd}(e, (p-1)) = 1. Hmm… so, we can decrypt the message c^d = m \mod p. And the message is the flag, which is constant. What if we sample two ciphertexts encrypting the same message but under different keys? Yes… so, basically, we have m \mod p_1 and m \mod p_2. Then, we can combine it using CRT! This is easy!

import gmpy
from Crypto.Util.number import long_to_bytes


enc1 = 63188108518214820361083256140053967663112132356420859206347143811869148973386950682507343981284848159232100220605963292020722612854139075311063776086523677522160515093598202087755508512714885329251980275085813640578839753523295579661559881983237388178475574713319340090857313275483787001349560782781513895950569634984688753665
q1 = 33452043035349425454164058954054458228134102234436666511159820871022348004023966976017694615018111901825497223286811050478588079083387098695346723120647425399335947
p1 = 33452043035349425454164058954054813129854949698738692548175391185736387867969615080539316436404430573352896343366560167202569408949342999951142479436993639588965567
e1 = 3562731839

enc2 = 162331112890791758781057932826106636167735138703054666826574266304486608255768782351247144197937186145526374008317633308191215438756014724244242639042178681790803086016105986729819920057318114565244866632716401408212076926367974085730803964312385570202673351687846963322223962280999264618753873289802828995706526584379102468
q2 = 1381282812140256921334162381757305071089685391539884775199049048751523002134390007428038278188586333291047282664366863789424963612326970767160301759006158962675449
p2 = 1381282811302268925712750063033928508701820008572424411412024462643800411901779755548441592138469189655615818433739872652769585433967353091413641137354055362924329
e2 = 197840663

d1 = gmpy.invert(e1, p1-1)
d2 = gmpy.invert(e2, p2-1)

assert(gmpy.gcd(e1, p1-1) == 1)
assert(gmpy.gcd(e2, p2-1) == 1)

m = (pow(enc1, d1, p1) * p2 * gmpy.invert(p2, p1) + pow(enc2, d2, p2) * p1 * gmpy.invert(p1, p2)) % (p1*p2)
print long_to_bytes(m)

…prints:

********** great job! the flag is: ASIS{Still____We_Can_Solve_Bad_F4!}

Alright!

A fine OTP server

Connect to OTP generator server, and try to find one OTP.

nc 66.172.27.77 35156

We get the code for generating OTPs:

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

We note that \texttt{otp\_1}^3 < N. Because of this, the task is really trivial. We can simply compute the cubic root without taking any modulus into consideration. \texttt{gmpy2.iroot(arg, 3)} can solve this efficiently! This gives us:

ASIS{0f4ae19fefbb44b37f9012b561698d19}

Secured OTP server

Connect to OTP generator server, and try to find one OTP.
This is secure than first server 🙂

nc 66.172.33.77 12431

This is almost identical to the previous, but the OTP is chosen such that it will overflow the modulus when cubed.

    template_phrase = '*************** Welcome, dear customer, the secret passphrase for today is: '
    A = bytes_to_long(template_phrase + '00' * 18)

Note that we can write an OTP as m = A \cdot 2^k + B, where B < 2^k. So, we have m^3 = A^3 + 3A^2D + 3AD^2 + D^3. The observant reader have noticed that if we remove A^3, it will not wrap around N. Now, we can mod out the terms containing A, i.e., find m - A^3 \mod A. Now we are back in the scenario of previous challenge… so, we can use the method from before! Hence, (m - A^3 \bmod A)^{1/3}. Finally, we get

ASIS{gj____Finally_y0u_have_found_This_is_Franklin-Reiter’s_attack_CongratZ_ZZzZ!_!!!}

DLP

You should solve a DLP challenge, but how? Of course , you don’t expect us to give you a regular and boring DLP problem!

nc 146.185.143.84 28416

The ciphertext is generated using the following function:

def encrypt(nbit, msg):
    msg = bytes_to_long(msg)
    p = getPrime(nbit)
    q = getPrime(nbit)
    n = p*q
    s = getPrime(4)
    enc = pow(n+1, msg, n**(s+1))
    return n, enc

We have that e = (n+1)^m \mod n^s = n^m + {m \choose {m-1}}n^{m-1} + \cdots + + {m \choose 2}n^2 + {m \choose 1}n + 1. We can write this as {m \choose 1}n + 1 + \mathcal{O}(n^2). Now, we can eliminate the ordo term by taking e \mod n^2 = {m \choose 1}n + 1 \iff e - 1 \mod n^2 = mn. Assuming that m < n, we can determine m by simple division. So, \frac{e - 1 \mod n^2}{n} = m. In Python, we have that

Turns out this is the case, and, hence, we obtain

ASIS{Congratz_You_are_Discrete_Logarithm_Problem_Solver!!!}

unsecure ASIS sub-d

ASIS has many insecure sub-domains, but we think they are over HTTPS and attackers can’t leak the private data, what do you think?

So, we get a PCAP. The first thing we do is to extract data binwalk -e a.pcap. This generates a folder with all the certificates used.

Then, we look at the moduli.

#!/bin/bash
FILES=*.crt
for f in $FILES
do
  openssl x509 -inform der -in $f -noout -text -modulus 
done

This generates a file with all moduli. Let us try something simple! Common moduli! For each pair, we check if \text{gcd}(N_i, N_j) \neq 1. If so, we have found a factor. Turns out two moduli have a common factor, so we can factor each of them and decrypt their traffic:

p1 = 146249784329547545035308340930254364245288876297216562424333141770088412298746469906286182066615379476873056564980833858661100965014105001127214232254190717336849507023311015581633824409415804327604469563409224081177802788427063672849867055266789932844073948974256061777120104371422363305077674127139401263621 
q1 = 136417036410264428599995771571898945930186573023163480671956484856375945728848790966971207515506078266840020356163911542099310863126768355608704677724047001480085295885211298435966986319962418547256435839380570361886915753122740558506761054514911316828252552919954185397609637064869903969124281568548845615791

p2 = 159072931658024851342797833315280546154939430450467231353206540935062751955081790412036356161220775514065486129401808837362613958280183385111112210138741783544387138997362535026057272682680165251507521692992632284864412443528183142162915484975972665950649788745756668511286191684172614506875951907023988325767 
q2 = 136417036410264428599995771571898945930186573023163480671956484856375945728848790966971207515506078266840020356163911542099310863126768355608704677724047001480085295885211298435966986319962418547256435839380570361886915753122740558506761054514911316828252552919954185397609637064869903969124281568548845615791

We can now generate two PEM-keys

d1 = gmpy.invert(e, (p1 - 1)*(q1 - 1))
key = RSA.construct((long(p1*q1), long(e), long(d1)))
f = open('privkey.pem','w')
f.write(key.exportKey('PEM'))
f.close()

Putting it into Wireshark, we obtain two images:

flag
flag2

I totally agree.

Alice, Bob and Rob

We have developed a miniature of a crypto-system. Can you break it?
We only want to break it, don’t get so hard on our system!

This is McElice PKC. The ciphertexts are generated by splitting each byte in blocks of four bits. The following matrix is used as public key:

G = \begin{pmatrix}1& 1& 0& 0& 0& 1& 1& 0\\ 1& 1& 1& 1& 1& 1& 1& 1\\ 0& 1& 1& 0& 1& 1& 0& 0 \\ 0& 1& 1& 1& 0& 0& 1& 0 \end{pmatrix}

The ciphertext is generated as \mathbf{m}G + \mathbf{e}, which is a function from 4 bits to a byte. \mathbf{e} is an error (or pertubation) vector with only one bit set. This defines a map f: \mathbb{F}_4 \rightarrow \mathbb{F}_8. So, each plaintext byte is two ciphertext bytes.

We can first create a set of codewords

P = numpy.matrix([[1, 1, 0, 0, 0, 1, 1, 0], [1, 1, 1, 1, 1, 1, 1, 1], [0, 1, 1, 0, 1, 1, 0, 0],[0, 1, 1, 1, 0, 0, 1, 0]])
image = {} # set of codewords
for i in range(0, 2**4):
    C = (numpy.array([int(b) for b in (bin(i))[2:].zfill(4)]) * P % 2).tolist()[0]
    image[int(''.join([str(c) for c in C]), 2)] = i

Then, go through each symbol in the ciphertext, flip all possible bits (corresponding to zeroing out \mathbf{e}) and perform lookup in the set of codewords P (compute the intersection between the Hamming ball of the ciperhext block and P).

f = open('flag.enc','r')
out = ''
for i in xrange(18730/2):
    blocks = f.read(2)
    j = ord(blocks[0])
    C1 = 0
    C2 = 0
    for i in range(0, 8):
        if j ^ 2**i in image:
            C1 = image[j ^ 2**i] << 4
    j = ord(blocks[1])
    for i in range(0, 8):
        if j ^ 2**i in image:
            C2 = image[j ^ 2**i]
    out += chr(C1+C2)
f=open('decrypted','w')
f.write(out)

Turns out it is a PNG:

test.png

Confidence DS CTF ’17 – Public Key Infrastructure

Log in as admin to get the flag.

nc pki.hackable.software 1337

This was a fun challenge, because it required many steps to complete. The first thing we notice is that MD5 is being used:

def h(x):
    return int(hashlib.md5(x).hexdigest(), 16)

def makeMsg(name, n):
    return 'MSG = {n: ' + n + ', name: ' + name + '}'

def makeK(name, n):
    return 'K = {n: ' + n + ', name: ' + name + ', secret: ' + SECRET + '}'

def sign(name, n):
    k = h(makeK(name, n))
    r = pow(G, k, P) % Q
    s = (modinv(k, Q) * (h(makeMsg(name, n)) + PRIVATE * r)) % Q
    return (r*Q + s)

Naturally, we cannot register as \texttt{admin}. An immediate conclusion is that we need to some kind of collision attack. But how? We cannot exploit any collision in the \texttt{admin} account. What if we can create a collision in r while keeping s distinct?

from coll import Collider, md5pad, filter_disallow_binstrings
import os

def nullpad(payload):
    return payload + b'\x00' * (64 - len(payload) % 64)

def left_randpad(payload):
    random_string = os.urandom(64)
    return random_string[:len(payload) % 64] + payload

# nullpad the prefix so that it does not interfere
# with the two collision blocks
prefix = nullpad(b'K = {n: ')
suffix = b', name: groci, secret:'

collider = Collider()
collider.bincat(prefix)
collider.safe_diverge()
c1, c2 = collider.get_last_coll()
collider.bincat(suffix)
cols = collider.get_collisions()

# here are our collisions!
A = next(cols)
B = next(cols)

The above code creates a collision such that k = h(makeK(name, n)) generates the same value. The problem is that the server does not return the signature, but str(pow(sign(name, n), 65537, int(n.encode('hex'), 16))).

To get the signature, which is computed as Z = (r \cdot Q + s)^{65537} \mod M, we need to perform some additional computations. Obviously, we could factor the modulus M and compute \phi(M) and then obtain the signature as easy as Z^{65537^{-1} \mod \phi(M)}… but factoring…

factoring

The probability that it is smooth enough is not very high. Also, the number is around 300 digits so not a good candidate for msieve or other factoring software… so what then?

Well… what if 2^{96} \cdot A + c and 2^{96} \cdot B + c both are prime? Then, we can easily recover the signature as Z^{65537^{-1} \mod (M-1)}! No need for time-consuming factoring! Embodied in Python, it could look like this:

import hashlib
from Crypto.Util.number import isPrime

def num2b(i):
    c = hex(i).strip('L')
    c = (len(c) % 2) * '0' + c[2:]
    return c.decode('hex')

# get the payloads
A = int(A[64:64*3].encode('hex'), 16)
B = int(B[64:64*3].encode('hex'), 16)

# PURE BRUTEFORCE, NOTHING FANCY HERE!
# append the same data to both payload until
# they resulting numbers are BOTH prime
mul = 2 ** 96
for i in range(1, 2**20, 2):
    if isPrime(AA + i) and isPrime(BB + i):
        print AA+i
        print BB+i
        break

As an example, we obtain

\begin{array}{rcl} n_1 &= &  105830311539247723296176540884596952305269885491629657210626981367998\\ &&347855559972583535368860650411949121405830046936667132754993342040282\\ &&396579458814796410615711380733886667185822027819749278255976029130752\\ &&166929087394904457033345839480235723175883533744993499706397792250975\\ &&3538392909994754306314762383722990106537161406818023\\ \end{array}

and

\begin{array}{rcl} n_2 &=&  105830311539247723296176540884596952305269885520672956204333681813273\\ &&006022550217898599044674310459977554450027911751595002340337873829980\\ &&015343362188266640170082456168713800027076364038815247707214102650115\\ &&472228269311796278082939271876115926467296299772272093195558205398667\\ &&4873073960726016011965433749481512129478713571026663\\ \end{array}

Putting this into action, we might do something like:

def connect(name, n):
   name_encoded = base64.b64encode(name)
   n_encoded = base64.b64encode(n)
   server = remote('pki.hackable.software', 1337)
   payload = 'register:' + name_encoded + ',' + n_encoded
   server.send(payload + '\n')
   return pow(int(server.recvline()), modinv(65537, n1-1), n1)

payload1 = nullpad('K = {n: ') + num2b(n1)
payload2 = nullpad('K = {n: ') + num2b(n2)

name = 'groci'
PORT = 1331

assert(h(makeK(name, payload1)) == h(makeK(name, payload2)))
assert(h(makeMsg(name, payload1)) != h(makeMsg(name, payload2)))

# Get first signature
sig = connect(name, payload1)
r1, s1 = sig / Q, sig % Q
z1 = h(makeMsg(name, payload1))

# Get second signature
connect(name, payload2)
sig = pow(int(server.recvline()), modinv(65537, n2-1), n2)
r2, s2 = sig / Q, sig % Q
z2 = h(makeMsg(name, payload2))

# Make sure we got a nonce re-use
assert(r1 == r2)

# OK, use standard nonce re-use technique...
delta = modinv(((s1 - s2) % Q), Q)
k = ( ((z1 - z2) % Q) * delta) % Q
r_inv = modinv(r1, Q)
PRIVATE = (((((s1 * k) % Q) - z1) % Q) * r_inv) % Q

# Now that we know the private key, 
# lets forge/sign the admin account!
name = 'admin'
n = 'snelhest'
k = h(makeK(name, n))

r = pow(G, k, P) % Q
s = (modinv(k, Q) * (h(makeMsg(name, n)) + PRIVATE * r)) % Q
sig = r*Q + s

# connect and login
name_encoded = base64.b64encode(name)
n_encoded = base64.b64encode(n)
server = remote('pki.hackable.software', 1337)
payload = 'login:' + name_encoded + ',' + n_encoded + ',' + base64.b64encode(num2b(sig))
server.send(payload + '\n')
print server.recvline()

We obtain the following signatures:

r_1 = 455070882754437660339639715779025537190883529105 s_1 = 719850042802710537609987071764516609303635832220
r_2 = 455070882754437660339639715779025537190883529105 s_2 = 584220801769294572739392594733340817152152143208

Now, with our forced nonce re-use, we can use standard techniques to obtain the private key and sign our \texttt{admin} account, which gives the flag
DrgnS{ThisFlagIsNotInterestingJustPasteItIntoTheScoreboard}!

211092_242669842430795_4056741_n.jpg

Note to reader: I apologize for the excessive use of memes.