OMG RNG!

Description

Gifts for everyone!

nc lab-1.ctf.standcon.n0h4ts.com 7903

Overview

Starting an instance and connecting to the server, the server would run chal.py and send you gifts and a secret containing the flag that is encrypted.

We have to figure out a way to decrypt the secret retrieved from the server using the gifts.

TL;DR

  • Use gcd to find value of m

  • Get the previous state by solving linear congruence between m and first gift received.

  • Get previous state until it reaches the first state.

  • Perform XOR of secret with first 2 states.

  • Got flag

Solution

Analysis

chal.py

from Crypto.Util.number import bytes_to_long, getRandomRange

FLAG = bytes_to_long(open("flag.txt", "rb").read())

class MLFG():
    def __init__(self):
        self.m = getRandomRange(2, 2**300)
        self.s0, self.s1 = [getRandomRange(2, self.m-1) for _ in '__']
        self.curr_state = 0
    
    def next(self):
        self.curr_state = (self.s0 * self.s1) % self.m
        self.s0 = self.s1
        self.s1 = self.curr_state
        return self.curr_state


if __name__ == '__main__':
    
    P = MLFG()
    secret = FLAG ^ P.s0 ^ P.s1
    
    [P.next() for _ in range(1337)] # discard some states :p

    for _ in range(5):
        ch = input("\n[1] Get a random gift \n[2] Get Secret! \n\n[?]> ").strip()
        if ch == '1':
            print("\nHere's your gift:", hex(P.next()))
        elif ch == '2':
            print("\nHere's your secret:", hex(secret))
        else:
            exit("\nOops! Invalid input!")
    else:
        exit("\nEnough Gifts!")

It's a pretty short code, so let's quickly break it down.

Analyzing chal.py

The MLFG class gets initialized first.

When the MLFG class gets initialized, it grabs a large random number and stores it as m in the class. Then it randomly generates 2 other numbers smaller than m, and stores it in s0 and s1 in the class. curr_state is then assigned with the value 0.

class MLFG():
    def __init__(self):
        self.m = getRandomRange(2, 2**300)
        self.s0, self.s1 = [getRandomRange(2, self.m-1) for _ in '__']
        self.curr_state = 0

The next() function is pretty self explanatory.

def next(self):
        self.curr_state = (self.s0 * self.s1) % self.m
        self.s0 = self.s1
        self.s1 = self.curr_state
        return self.curr_state

Searching up MLFG, it stands for Multiplicative Lagged Fibonacci Generator. I tried to search for any writeups on such challenges but couldn't find any. The key takeaway from looking it up is that this algorithm is a Pseudo-Random Number Generator (PRNG).

According to the MLFG class, we can see how it performs as a PRNG:

  1. It first generates a random number as a modulus (m)

  2. It then generates 2 random numbers to start with (s0 and s1)

  3. To get a random value, it multiplies both s0 and s1 and mod m and returns it (this is what makes it Multiplicative as the name of the class suggest). So curr_state would be your random value.

  4. It then stores the value of s1 into s0, and the current value (curr_state) generated into s1 (this is what makes it Lagged Fibonacci).

Here are some images to help you visualize it. (assume f() is a function (self.s0 * self.s1) % self.m. Note that the numbers in the picture are just an example.)

See why it is related to Fibonacci?

Now, the last few lines of code is to produce the secret where the flag is encrypted through xor. We can easily find the flag if we find the values it got xored with (goal is to find the original value of s0 and s1).

You then get to retrieve the value of secret and the next few random values generated.

P = MLFG()
    secret = FLAG ^ P.s0 ^ P.s1
    
    [P.next() for _ in range(1337)] # discard some states :p

    for _ in range(5):
        ch = input("\n[1] Get a random gift \n[2] Get Secret! \n\n[?]> ").strip()
        if ch == '1':
            print("\nHere's your gift:", hex(P.next()))
        elif ch == '2':
            print("\nHere's your secret:", hex(secret))
        else:
            exit("\nOops! Invalid input!")
    else:
        exit("\nEnough Gifts!")

Finding value of 'm'

We will first have to grab the next few states of MLFG. I went to grab 4 states and then the secret.

We will be focusing on this line of code:

self.curr_state = (self.s0 * self.s1) % self.m
self.s0 = self.s1
self.s1 = self.curr_state

Since we have grabbed 4 consecutive states of MLFG, we have something like this:

  • gift0 = (unknown s0 * unknown s1) % m

  • gift1 = (unknown s1 * gift0) % m

  • gift2 = (gift0 * gift1) % m

  • gift3 = (gift1 * gift2) % m

One thing to note, modulus equations can also be expressed in another way that is true if you think about it:

(xy mod m = z) === (xy - n1m = z), where n1 is a random integer

Similarly,

(gift2 = (gift0 * gift1) % m) === (gift2 = (gift0 * gift1) - n1m)

By rearranging the equation...

(gift0 * gift1) - gift2 = n1m

We then have the following equations...

(gift0 * gift1) - gift2 = n1m

(gift1 * gift2) - gift3 = n2m

So now we have n1m and n2m, we can find the value of m by finding the gcd of both of them since m would be larger than n1 and n2!

m acquired!

Finding previous state

Let's focus on this equation as we want to find the previous state:

  • gift1 = (unknown s1 * gift0) % m

We basically have the value of everything except for the previous state (unknown s1)

I had a little trouble solving this due to my lack of knowledge on modular arithmetic, but searching it up gives us something called Linear Congruence. The linear congruence problem is to find the value of X given the equation aX (mod m) = b, which is exactly what we were looking for!

I managed to find and use a linear congruence function implemented in Python https://stackoverflow.com/questions/48252234/how-to-solve-a-congruence-system-in-python.

Now we are able to find the previous state! To find previous state of the previous state, we will just have to use the value of the previous state we got.

Example:

gift1 = (unknown s1 * gift0) % m
# We have m, gift0, and gift1. Use linear congruence to find 'unknown s1'
gift0 = (unknown s0 * unknown s1) % m
# We have m, gift0, and 'unknown s1'. Use linear congruence to find 'unknown s0'
unknown s1 = (prev prev state * unknown s0) % m
# We have m, 'unknown s1', and 'unknown s0'. Use linear congruence to find 'prev prev state'

We can then just keep finding the previous states until we find the flag!

Note that the linear congruence function may return errors at times because some of the values provided may not have any solutions (this happens as the the gcd of the modulus and the starting number does not equally divide the result, so 2x mod 8 = 51 have no solution because gcd(2,8)=2, and 51%2 != 0)

Solve.py

#!/usr/bin/env python3

from pwn import *
from Crypto.Util.number import long_to_bytes

# You dont have to use extended euclidean algorithm to find gcd
def extended_gcd(a: int, b: int) -> int:
    if a == 0:
        return b, 0, 1
    else:
        gcd, x, y = extended_gcd(b % a, a)
        return gcd, y - (b // a) * x, x

# Note that the `linear congruence` function may return errors at times, please run again to get new numbers if this happens
# aX = b (mod m)
def linear_congruence(a,b,m):
	if b == 0:
		return 0

	if a < 0:
		a = -a
		b = -b

	b %= m
	while a > m:
		a -= m
	return (m * linear_congruence(m, -b, a) + b) // a

server = "lab-1.ctf.standcon.n0h4ts.com"
port = 10172
p = remote(server,port)

gifts = []
for _ in range(4):
	p.clean()
	p.sendline(b'1')
	p.recvuntil(b"Here's your gift: ")
	gifts.append(int(p.recvline().strip().decode(),16))
print("Gifts:",gifts)

p.clean()
p.sendline(b'2')
p.recvuntil(b"Here's your secret: ")
secret = int(p.recvline().strip().decode(),16)

print("Secret:",secret)
p.close()

phi1 = ((gifts[0] * gifts[1]) - gifts[2])
phi2 = ((gifts[1] * gifts[2]) - gifts[3])

gcd,x,y = extended_gcd(phi1,phi2)
m = gcd
print("m:",m)
r0 = gifts[0]
r1 = gifts[1]

l = []

for _ in range(1345):
	tmp = r0
	r0 = linear_congruence(r0,r1,m)
	l.append(r0)
	r1 = tmp
	test = long_to_bytes(secret^r0^r1)
	if b"STANDCON" in test:
		print("\n\nFlag:",test.decode())
		exit()
	print('.',end='')

# Check:
# l = l[::-1]
# for _ in range(len(l)):
# 	curr_state = (r0 * r1) % m
# 	r0 = r1
# 	r1 = curr_state
# print(curr_state)
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