# babyRSA

# CustomService

There are 4 options to choose:

- Encrypt demo: The encryption that use a random generated AES key
- Decrypt demo: The decryption of demo version
- Encrypt pro: The encryption that add other information to the cipher encrypted by demo version
- Decrypt pro: The decryption of pro version

Since the ciphertext of pro version using the base 64 of ciphertext of demo version as a substring, we can find the key by xoring the substring to the ciphertext at any position.

Firstly, we obtain the ciphertext of `'Plaintext is : ' + 'A' * 666 + hash('Plaintext is : ' + 'A' * 666)`

from the demo version, let say `s1`

.

Secondly, we obtain the ciphertext of `'A' * 666`

from the pro version, let say `s2`

. Since the base64 has 3 characters as a block of 4 encoded ones, `b64decode(s2)`

has either `b64encode(s1)`

or `b64decode('A' + s1)`

or `b64decode('A' + s1)`

as a substring. By xoring `s2`

with any possible cyclic shifted string of these three, we could find the key which is the subarray that occurs many times in the xored string.

```
def findkey(s1, s2):
KEYLEN = 80
st2, keyc = b64decode(s2), None
for l in range(3):
st1 = b64encode(('A' * l + s1).encode())
for j in range(len(st1)):
k = [st1[(i + j) % len(st1)] ^ st2[i] for i in range(len(st2))]
h = len([i for i in range(len(k) - KEYLEN) if k[i] == k[i + KEYLEN]])
keyc = k if (h > m) else keyc
for i in range(len(keyc) - KEYLEN):
if keyc[i:i+KEYLEN] == keyc[i+KEYLEN:i+2*KEYLEN]:
break
return keyc[i+KEYLEN-i%KEYLEN:i+2*KEYLEN-i%KEYLEN]
```

Now we could find the key in the encryption pro version. This process can be done locally because this key would not change. Next, we obtain the ciphertext of `'Plaintext is : Bae give me flag' + hash('Plaintext is : Bae give me flag')`

from the demo version, let's say `s3`

. Then we obtain the ciphertext of `'something'`

from the pro version, let's say `s4`

. Using the key, we can decrypt `s4`

to take the plaintext. It is a json type data. We just need to change the `enc_data`

field of this plaintext to `s3`

and encrypt it again. Then the plaintext will have the string `'Bae give me flag'`

.

# F3737

First of all, we have a class F3737:

```
class F3737(object):
def __init__(self, components):
self.components = list(map(reduce_modulo_37, components))
assert len(self.components) == 37
def __add__(self, other):
return F3737(
map(lambda x, y: x + y, self.components, other.components))
def __mul__(self, scalar):
return F3737(map(lambda x: x * scalar, self.components))
def __eq__(self, other):
return self.components == other.components
def __str__(self):
return str(self.components)
def __repr__(self):
return repr(self.components)
```

This class is a vector belonging to the Field ${\mathbb F}^{37}_{37}$.

For example: Let $v_1 = (0, 1, 2, ..., 36)$, $v_2 = (1, 1, 1, ..., 1)$

and $c = 3$ ($c$ is a scalar) then we have

The addition is: $v_1 + v_2 = (1, 2, 3, ..., 36, 0)$

The multiplication is: $c * v1 = (0, 3, 6, ..., 34)$

Next, we have a f3737 hash function:

```
def f3737_hash(input):
if input == b'EVALUATE_TO_ZERO':
return F3737([0] * 37)
h = int(sha512(input).hexdigest(), 16)
result = [0] * 37
for i in range(37):
result[i] = h % 37
h = h // 37
return F3737(result)
```

This function takes the sha512 value of the input. Then, it assigns $37$ last digist of the hash value written in base $37$. We can see that :

- (1) It is hard to reverse the input with the given output.
- (2) If the input is
`EVALUATE_TO_ZERO`

, then the output is $0$.

Let's denote this function by $f$ and go to another hash function:

```
def myhash(innput):
i = 0
result = F3737([0] * 37)
while True:
block = innput[BLOCKSIZE * i: BLOCKSIZE * (i + 1)]
if not block:
break
result += f3737_hash(block) * i
i += 1
return result
```

If $g$ is the function belonging to `myhash`

and message $m$ can be divided into $m_0||m_1||...||m_k$ with 16-character string $m_i$, then

$$g(m)=\sum_{i=0}^{k}i * f(m_i)$$

Obviously, we can see that

- (3) The 0-th block (37-th, 74-th, ... also) is evaluated to $0$
- (3) The 1-st block (38-th, 75-th, ... also) is evaluated to $f(m_i)$
- (3) The 2-nd block (39-th, 76-th, ... also) is evaluated to $2*f(m_i)$
- ...

Coming to the challenge, we first see the generation and verification:

```
ALLOWED_LETTERS = string.ascii_letters + string.digits + '_'
def generate_password(size):
rand = random.SystemRandom()
password = ''.join(rand.choice(ALLOWED_LETTERS) for _ in range(size))
password_hash_value = myhash(password.encode())
return password, password_hash_value
def verify_password(password, hash_value):
if all(c in ALLOWED_LETTERS for c in password):
if myhash(password.encode()) == hash_value:
return True
return False
```

What we could see from these two functions is:

- (4) The generation function takes a random string
`password`

(containing ascii letters, digits and underscore) with specified length`size`

as input and returns it together with its hash value`password_hash_value`

computed from`myhash`

function. - (5) The verification function takes the string
`password`

and a hash value`hash_value`

, then returns`True`

if the`hash_value`

is the value computed from`myhash`

function with`password`

as input and`False`

otherwise.

Okay, now it's time to see what we need to get the flag:

```
def main():
import signal
signal.alarm(10)
for level in [1, 2, 3]:
_, hash_value = generate_password(32 * level)
print("Hash value: {}".format(hash_value))
try:
user_input_password = input("Your guess: ")
if not verify_password(user_input_password, hash_value):
raise Exception()
except:
print("Wrong password!")
return
print("Congratulations! Here's the FLAG:")
from secret import flag
print(flag)
```

There are three challenges and $10$ seconds for each. In i-th level, a `hash_value`

is pre-computed for a $32*i$-character string. Then we have to find a string `user_input_password`

that have the hash value equals to `hash_value`

.

Let's come from the naive approach. Obviously, we needs to pass the first level. This level contains only two 16-character strings, let's call $str_1$ and $str_2$. From (3), we can see that the `hash_value`

of this level is $f(str_2)$. It is only a block!! Then what we think next is to find $str_2$ from $z=f(str_2)$. From the beginning, may be you can not notice the characteristic (1) of the function $f$. But going by this way, you have to reverse it to obtain $str_2$. Assuming that if we could find $str_2$ by this way, then an input of SHA512 function could be found from the output which has $37$ known last digits in base $37$. This would make SHA512 not secure anymore so the characteristic (1) said that we should go another way to find flag.

It is also the time to ask whether we need to find the original input. If so then SHA512 would be not secure anymore as we discuss before. Since the input need not to be the original one, its length also could be different. This leads to the question that what if the input has really long length. There would be $37$ types of block (like (3)) and the sum of these block is $\sum_{36}^{i=0}i.a_{i}$. We want to have this value equals to the hash value:

$$\sum_{i=0}^{36}i.a_{i}=f(str_2)$$

Where we could have $a_i$? We can use the `f3737_hash`

function to compute it. You can explore this way more, but with me this seems to be a hard problem the output of `f3737_hash`

is considered as unpredictable. The only thing we got here is to have as many pre-computed $f(x)$ as we want.

Thinking a little deeper, what is the role of (2) in this problem? It could make any block immediately evaluated to zero. Using this way, we can put any pre-computed $f(x)$ at any block and `EVALUATE_TO_ZERO`

(let's say null) to others. In other words, we can control the scalar multiplying the vector in the output. So what we need is to find the suitable scalar for pre-computed $f(x)$.

For example, if $x_i = f('ii...ii')$, then we have to find $c_i$ such that:

$$\sum_{i=0}^{k}c_i.x_i = z$$.

This can be written by:

$$

\begin{pmatrix}

x_{0,0} & x_{1,0} & ... & x_{k,0}\\

x_{0,1} & x_{1,1} & ... & x_{k,1}\\

... & ... & ... & ...\\

x_{0,36} & x_{1,36} & ... & x_{k,36}

\end{pmatrix}

\begin{pmatrix}

c_0\\

c_1\\

...\\

c_k

\end{pmatrix}

~=~

\begin{pmatrix}

z_0\\

z_1\\

...\\

z_{36}

\end{pmatrix}

$$

Let's say $M.c=z$. Well, it is more clear now. We would choose $k=36$ and find $37$ pairwise independent pre-computed vector to solve these equations.

Taking $37$ first letter a..z, A..K, computing $f(a...a), ..., f(K...K)$ and arranging into a $37\times 37$ matrix by row, we will have $M$. $z$ is the `hash_value`

taken from server. Using sage we can compute $c=M^{-1}.z$.

Finally, we need to map $c$ to the string.

This can be done by:

```
def findkey(v):
str = ["EVALUATE_TO_ZERO"] * 37
for i in range(len(v)):
if v[i] == 0:
continue
k = v[i]
while k < len(str) and str[k] != "EVALUATE_TO_ZERO":
k += 37
if k > len(str):
str += ["EVALUATE_TO_ZERO"] * 37
str[k] = s[i]
return str
```

For example:

- $c = (1, 2, 3, 0, ..., 0) \rightarrow m = 'a...a' + 'b...b' + 'c...c' + null * 34$
- $c = (1, 1, 0, 0, ..., 0) \rightarrow m = 'a...a' + null * 36 + 'b...b' + null * 36$
- $c = (1, 3, 1, 0, ..., 0) \rightarrow m = 'a...a' + null + 'b...b' + null * 34 + 'c...c' + null * 36$

Then the challenge is solved.

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