Broken RSA

Description

My friend had a secret message for me. However, he made a mistake when encrypting the message, and now I am unable to decrypt it! Could you decrypt and recover the message for me?

812B

BrokenRSA.zip

TL;DR

Notice that e = 1616, which makes phi and e not coprime and which is why normal RSA decryption won't work
Perform c%p and c%q and RSA decrypt both with their respective d and phi values
Use tonelli shank's algorithm to perform square root mod of p and q
Perform CRT on the values to get original plaintext

Solution

Initial Analysis

If you want to skip the explanation, you can go to e and phi are not coprime.

from Crypto.Util.number import getPrime, bytes_to_long

p = getPrime(512)
q = getPrime(512)
n = p * q
e = 1616

with open("flag.txt", "rb") as f:
    flag = f.read()

c = pow(bytes_to_long(flag), e, n)

with open("output.txt", "w") as f:
    f.write(f"{p=}\n")
    f.write(f"{q=}\n")
    f.write(f"{c=}\n")
    f.write(f"{e=}\n")

p=12745443776097937486523731981184244411822246670344825081786250598280592311994645818017369055015305576625489048312376829887862697232698960303123727339439477
q=6965691623673764963283701881605906791484292250300568740181083332830802480178930748628555885283810920150672372660644966641894191414330596203094237460731853
c=51246606178817027144048790384294344409078205278660263649138314308484271379271492418276059836372460273067275885546317401508060728580046016291398622205239294202483332638706880399609813252751351178753076424449513521022597436767316972137030942140105737604756013333385824335007108113055833516556345445260667314569
e=1616

Looking at challenge.py, everything looks like the usual RSA encryption scheme, except for the value of e looking like an invalid value. This e is the one that makes the RSA decryption difficult to perform.

In standard RSA encryption scheme, the public exponent e must be an odd number.

If e is even, it will most definitely share a common factor with phi that is != 1, since p-1 will be an even number. This means that e and phi are not coprime.

If e and phi are not coprime, then there may be multiple decryption exponent d that may decrypt the ciphertext.

In this case, normal RSA decryption will not work.

Luckily, we are given the values of p and q, as well as the ciphertext c.

To determine why normal RSA decryption doesn't work and how to decrypt the ciphertext with this e value, we have to first understand the concept behind the RSA cryptosystem: Why and how it works.

The brilliance behind RSA decryption

We know that RSA encryption goes like this:

n = p*q \\ct=m^e \bmod n

When decrypting in normal RSA, these steps are performed:

phi = (p-1)*(q-1) \\d = e^{-1} \bmod phi \\pt = ct^d \bmod n

But why does this decryption method work?

Firstly, we can rewrite $d = e^{-1} \bmod phi$ into $ed = 1+k*phi$ , where $k$ is any finite integer.

We can also rewrite $pt = ct^d \bmod n$ into $pt = m^{ed} \bmod n$ .

By substituting $ed$ into the second $pt$ equation, we have $pt = m^{1+k*phi} \bmod n$ , and thus $pt = m^1m^{k*phi} \bmod n$ .

Using Euler's Theorem, we can then get this equation $pt = m^11^k \bmod n$ , and simplifying it gives us our original plaintext! $pt = m \bmod n$ (that is if our plaintext < n)

What is Euler's Theorem?

https://t5k.org/glossary/page.php?sort=EulersTheorem#:~:text=Euler's%20Theorem%20states%20that,%E2%89%A1%201%20(mod%2012).

But this only works if e and phi are coprime: gcd(e,phi)=1.

e and phi are not coprime

If we check the gcd of the e and phi we have, we note that gcd(e,phi)=16 in this case, so that is a problem.

According to Euler's Theorem, if gcd(e,phi)=1, we will have $ed = 1+k*phi$ .

Since we have gcd(e,phi)=16, so we will have $ed = 16 + k*phi$ . Following the decryption process as stated above, we will end up with $pt = m^{16} \bmod n$ . Since $m^{16}$ is definitely larger than $n$ , we can't take the $16^{th}$ root and call it a day.

But wait, there is the Tonelli-Shanks Algorithm that can compute the modulus square root of any number, given that the modulus is a prime.

The Tonelli-Shanks Algorithm

We note that our modulus $n=p*q$ and is therefore not a prime, so how do we apply this algorithm?

The only primes we have here are $p$ and $q$ . What if we calculate the ciphertext as a modulus of $p$ or $q$ instead?

So we can calculate our new ciphertext using the given ciphertext: $ctp = c \bmod p$ . This is equivalent to $ctp = m^e \bmod n \bmod p$ which is simplified to $ctp = m^e \bmod p$ (note that $\bmod\ n$ is gone as $p$ is a factor of $n$ )

Now we get the Euler's Totient function for $p$ : $phip = p-1$

Then we calculate dp = inverse(e,phip), then decrypt the ciphertext ptp = pow(ctp,dp,p). Since gcd(e,phip)=4, it means we have $ptp = m^4 \bmod p$ . So we have to use Tonelli twice to get back our plaintext.

Since $p$ is a prime, we can now use Tonelli-Shanks Algorithm (code)! Let's test this with a test flag of our own:

#!/usr/bin/env python3

from Crypto.Util.number import getPrime, bytes_to_long, inverse, long_to_bytes
from egcd import egcd
from gmpy2 import *
from sympy.ntheory.modular import crt

def legendre_symbol(a, p):
    """
    Legendre symbol
    Define if a is a quadratic residue modulo odd prime
    http://en.wikipedia.org/wiki/Legendre_symbol
    """
    ls = pow(a, (p - 1)//2, p)
    if ls == p - 1:
        return -1
    return ls

def prime_mod_sqrt(a, p):
    """
    Square root modulo prime number
    Solve the equation
        x^2 = a mod p
    and return list of x solution
    http://en.wikipedia.org/wiki/Tonelli-Shanks_algorithm
    """
    a %= p

    # Simple case
    if a == 0:
        return [0]
    if p == 2:
        return [a]

    # Check solution existence on odd prime
    if legendre_symbol(a, p) != 1:
        return []

    # Simple case
    if p % 4 == 3:
        x = pow(a, (p + 1)//4, p)
        return [x, p-x]

    # Factor p-1 on the form q * 2^s (with Q odd)
    q, s = p - 1, 0
    while q % 2 == 0:
        s += 1
        q //= 2

    # Select a z which is a quadratic non resudue modulo p
    z = 1
    while legendre_symbol(z, p) != -1:
        z += 1
    c = pow(z, q, p)

    # Search for a solution
    x = pow(a, (q + 1)//2, p)
    t = pow(a, q, p)
    m = s
    while t != 1:
        # Find the lowest i such that t^(2^i) = 1
        i, e = 0, 2
        for i in range(1, m):
            if pow(t, e, p) == 1:
                break
            e *= 2

        # Update next value to iterate
        b = pow(c, 2**(m - i - 1), p)
        x = (x * b) % p
        t = (t * b * b) % p
        c = (b * b) % p
        m = i

    return [x, p-x]

p=12745443776097937486523731981184244411822246670344825081786250598280592311994645818017369055015305576625489048312376829887862697232698960303123727339439477
q=6965691623673764963283701881605906791484292250300568740181083332830802480178930748628555885283810920150672372660644966641894191414330596203094237460731853
c=51246606178817027144048790384294344409078205278660263649138314308484271379271492418276059836372460273067275885546317401508060728580046016291398622205239294202483332638706880399609813252751351178753076424449513521022597436767316972137030942140105737604756013333385824335007108113055833516556345445260667314569
e=1616
n = p*q

m = bytes_to_long(b'LNC2023{this_is_a_flag}')
c = pow(m,e,n)

ctp = c%p
phip = p-1
dp = inverse(e,phip)
ptp = pow(ctp,dp,p)

r1 = prime_mod_sqrt(ptp,p)
for r in r1:
    r2 = prime_mod_sqrt(r,p)
    for rr in r2:
        print(long_to_bytes(rr))

We successfully got our own flag!

But if we try this on the ciphertext given to us, it doesn't work...

We note that p and q are primes each with512 bits. Because the result we get is $\bmod\ p$ ( $result = m \bmod p$ ), it is very likely that the original message is > $p$ , which is > 512 bits.

We can test this by testing with a flag shorter than $p$ , and then testing with a flag longer than $p$ .

And our theory is correct, the original message is most likely > 512 bits...

Plaintext > 512 bits

So how do we get the plaintext if it is more than 512 bits?

Since we can decrypt using $p$ as a modulus, it means we can also decrypt using $q$ as a modulus. This means we can end up with 2 different equations with 2 different mods:

resultp = m \bmod p \\resultq = m \bmod q

Since $p$ and $q$ are coprime, the original message is the same but the modulus are different, we can use Chinese Remainder Theorem (CRT) to solve for the original message.

Chinese Remainder Theorem

We can use the crt function in the sympy python library. We will need to supply a list of mods [p,q] and a list of results [resp,resq]

#!/usr/bin/env python3

from Crypto.Util.number import getPrime, bytes_to_long, inverse, long_to_bytes
from egcd import egcd
from gmpy2 import *
from sympy.ntheory.modular import crt

def legendre_symbol(a, p):
    """
    Legendre symbol
    Define if a is a quadratic residue modulo odd prime
    http://en.wikipedia.org/wiki/Legendre_symbol
    """
    ls = pow(a, (p - 1)//2, p)
    if ls == p - 1:
        return -1
    return ls

def prime_mod_sqrt(a, p):
    """
    Square root modulo prime number
    Solve the equation
        x^2 = a mod p
    and return list of x solution
    http://en.wikipedia.org/wiki/Tonelli-Shanks_algorithm
    """
    a %= p

    # Simple case
    if a == 0:
        return [0]
    if p == 2:
        return [a]

    # Check solution existence on odd prime
    if legendre_symbol(a, p) != 1:
        return []

    # Simple case
    if p % 4 == 3:
        x = pow(a, (p + 1)//4, p)
        return [x, p-x]

    # Factor p-1 on the form q * 2^s (with Q odd)
    q, s = p - 1, 0
    while q % 2 == 0:
        s += 1
        q //= 2

    # Select a z which is a quadratic non resudue modulo p
    z = 1
    while legendre_symbol(z, p) != -1:
        z += 1
    c = pow(z, q, p)

    # Search for a solution
    x = pow(a, (q + 1)//2, p)
    t = pow(a, q, p)
    m = s
    while t != 1:
        # Find the lowest i such that t^(2^i) = 1
        i, e = 0, 2
        for i in range(1, m):
            if pow(t, e, p) == 1:
                break
            e *= 2

        # Update next value to iterate
        b = pow(c, 2**(m - i - 1), p)
        x = (x * b) % p
        t = (t * b * b) % p
        c = (b * b) % p
        m = i

    return [x, p-x]

p=12745443776097937486523731981184244411822246670344825081786250598280592311994645818017369055015305576625489048312376829887862697232698960303123727339439477
q=6965691623673764963283701881605906791484292250300568740181083332830802480178930748628555885283810920150672372660644966641894191414330596203094237460731853
c=51246606178817027144048790384294344409078205278660263649138314308484271379271492418276059836372460273067275885546317401508060728580046016291398622205239294202483332638706880399609813252751351178753076424449513521022597436767316972137030942140105737604756013333385824335007108113055833516556345445260667314569
e=1616
n = p*q

phi = (p-1)*(q-1)
d = inverse(e,phi//4)

# m = bytes_to_long(b'LNC2023{this_is_a_very_long_test_flag_tuwinuiwneugnfuhe2hf923hu91fw}')
# print(m)
# ct = pow(m,e,n)
ct = c
ctq = ct % q
ctp = ct % p
phiq = q-1
phip = p-1
dq = inverse(e,phiq)
dp = inverse(e,phip)
ptq = pow(ctq,dq,q)
ptp = pow(ctp,dp,p)

#tonellishank
resq = prime_mod_sqrt(ptq,q)
resq = prime_mod_sqrt(resq[0],q) # guess which num between 2

resp = prime_mod_sqrt(ptp,p)
resp = prime_mod_sqrt(resp[1],p) # guess which num between 2

mod = [p,q]
res = [resp[0],resq[0]]
x = crt(mod,res)[0]
print(long_to_bytes(x))

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