> For the complete documentation index, see [llms.txt](https://kos0ng.gitbook.io/ctfs/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://kos0ng.gitbook.io/ctfs/write-up/2021/mdt4.0-quals/cryptography.md).
# Cryptography
## Baby Key (592 pts)
### Description
\-
### Solution
Diberikan source code sebagai berikut
```python
#!/usr/bin/env python3
from Crypto.PublicKey import RSA
from gmpy2 import digits
key = RSA.import_key(open('key.pem').read())
msg = int(digits(int.from_bytes(open('flag.txt', 'rb').read(), 'little'), 8))
enc = int.to_bytes(pow(msg, key.e, key.n), 128, 'little')
open('flag.enc', 'wb').write(enc)
```
Sepertinya tidak ada yang aneh dari source code tersebut , jadi selanjutnya kami coba cek keynya dan ternyata nilai e yang diberikan besar.
Dari sini kami coba lakukan common attack pada rsa dengan public exponent besar yaitu menggunakan wiener attack dan ternyata berhasil. Berikut script yang kami gunakan
{% code title="continued\_fractions.py" %}
```python
def rational_to_contfrac(x,y):
# Converts a rational x/y fraction into a list of partial quotients [a0, ..., an]
a = x // y
pquotients = [a]
while a * y != x:
x, y = y, x - a * y
a = x // y
pquotients.append(a)
return pquotients
def convergents_from_contfrac(frac):
# computes the list of convergents using the list of partial quotients
convs = [];
for i in range(len(frac)): convs.append(contfrac_to_rational(frac[0 : i]))
return convs
def contfrac_to_rational (frac):
# Converts a finite continued fraction [a0, ..., an] to an x/y rational.
if len(frac) == 0: return (0,1)
num = frac[-1]
denom = 1
for _ in range(-2, -len(frac) - 1, -1): num, denom = frac[_] * num + denom, num
return (num, denom)
```
{% endcode %}
{% code title="test\_rsa.py" %}
```python
from continued_fractions import *
from Crypto.Util.number import *
n = 136375679787927244696289795885005648815108289212464497859431615538673042375946778524473848398525037451190261199521819004739278031645357897677734698610951953541404381824146329328624288281641905271183010840974530269547547895439947481142131892305619548317029068789911069885321453469718238390495448090075373321029
e = 9934026283243447335784142672216496968427437150405829331521710602854826988915623834828401428975156770075612396739682567963382915587369749409382765177454083732327254135875299330600089331382217653825675232007843740755445444239391851451767573941155945865534339871956773517088821953019398585021686663749551991183
tmp = open('flag.enc', 'r').read()
c = bytes_to_long(tmp[::-1])
def egcd(a, b):
if a == 0: return (b, 0, 1)
g, x, y = egcd(b % a, a)
return (g, y - (b // a) * x, x)
def mod_inv(a, m):
g, x, _ = egcd(a, m)
return (x + m) % m
def isqrt(n):
x = n
y = (x + 1) // 2
while y < x:
x = y
y = (x + n // x) // 2
return x
def crack_rsa(e, n):
frac = rational_to_contfrac(e, n)
convergents = convergents_from_contfrac(frac)
for (k, d) in convergents:
if k != 0 and (e * d - 1) % k == 0:
phi = (e * d - 1) // k
s = n - phi + 1
# check if x*x - s*x + n = 0 has integer roots
D = s * s - 4 * n
if D >= 0:
sq = isqrt(D)
if sq * sq == D and (s + sq) % 2 == 0: return d
d = crack_rsa(e, n)
print(long_to_bytes(int(str(pow(c,d,n)),8))[::-1])
```
{% endcode %}
Flag : MDT4.0{small\_d\_\_i\_mean\_d\_as\_RSA\_private\_key}
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