Crypto - 100 Points

Common Modulus 1: Simple Common Modulus Attack
Common Modulus 2: Common Modulus Attack with common exponent divisor
Common Modulus 3: Common Modulus Attack with common exponent divisor + message padding

### Writeup

The challenge title was pretty self explanatory.

textbook RSA is vulnerable to Common Modulus Attack.

RSA works like the following $c = m^e \mod N$

If you encrypt the same message with the same $N$ like:
$C_1 = M^{e_1} \mod N$
$C_2 = M^{e_2} \mod N$

Then $\gcd(e_1, e_2)=d$, this means that $a$ and $b$ exists such that $e_1a + e_2b=d$.

This is usefull since: %

In the case where $e_1$ and $e_2$ don’t share any factor, $\gcd(e_1, e_2)=1$ so $M^d = M^1 = M$.
In the case where $e_1$ and $e_2$ share some factors, we end up with $M^d$.

Common Modulus 1 was the first easy case.

In Common Modulus 2 both the exponents were multiplied by $3$, so $d=3$ then $M^d=M^3$
Luckily our $M^3$ is smaller than our $N$ so we can retrive the flag by applying the cube-root.

In Common Modulus 3, the greatest common divisor is $17$ and unfortunately $M^{17}>N$.
We need to remove the padding from $M$ until $M^{17}<N$ then take the 17th-root as before.

The message/flag is padded with the following code:

flag = key.FLAG.encode('hex')

while len(flag) * 4 < 8192:
flag += '00'

FLAG = long(flag[:-2], 16)


len(FLAG) should be 2046, the previous flags were CBCTF{<32_char_here>} so converted in hex we have len(flag) = 78.

$2046-78=1968$

Adding 00 to an hex number it’s the same as multiplying by $2^4$,
so we need to multiply the padded $M$ by $2^{-4}$ to remove all the 00

Let $d=17$ and $i=1968$, retrive $M$ with

The complete python script is available here

CBCTF{6ac2afd2fc108894db8ab21d1e30d3f3}
CBCTF{d65718235c137a94264f16d3a51fefa1}
CBCTF{b5c96e00cb90d11ec6eccdc58ef0272d}