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# Hastad (crypto 400)
###ENG
[PL](#pl-version)
In the task we get the same unpadded message encrypted via RSA using 3 different public keys, but with the same exponent `e = 3`.
This means we can utilize Hastad Broadcast Attack here to recover the message thanks to Chinese Reminder Theorem.
The public exponent is 3 in all cases so what we have is in fact:
```
m^e = y1 mod n1
m^e = y2 mod n2
m^e = y3 mod n3
```
From Chinese Reminder Theorem we know it's possible to recover `m^e`, and once we have it, we only need to get k-th integer root to get the flag:
```python
from src.crypto_commons.generic import long_to_bytes
from src.crypto_commons.rsa.rsa_commons import hastad_broadcast
def main():
n1 = 95118357989037539883272168746004652872958890562445814301889866663072352421703264985997800660075311645555799745426868343365321502734736006248007902409628540578635925559742217480797487130202747020211452620743021097565113059392504472785227154824117231077844444672393221838192941390309312484066647007469668558141
n2 = 98364165919251246243846667323542318022804234833677924161175733253689581393607346667895298253718184273532268982060905629399628154981918712070241451494491161470827737146176316011843738943427121602324208773653180782732999422869439588198318422451697920640563880777385577064913983202033744281727004289781821019463
n3 = 68827940939353189613090392226898155021742772897822438483545021944215812146809318686510375724064888705296373853398955093076663323001380047857809774866390083434272781362447147441422207967577323769812896038816586757242130224524828935043187315579523412439309138816335569845470021720847405857361000537204746060031
c1 = 64830446708169012766414587327568812421130434817526089146190136796461298592071238930384707543318390292451118980302805512151790248989622269362958718228298427212630272525186478627299999847489018400624400671876697708952447638990802345587381905407236935494271436960764899006430941507608152322588169896193268212007
c2 = 96907490717344346588432491603722312694208660334282964234487687654593984714144825656198180777872327279250667961465169799267405734431675111035362089729249995027326863099262522421206459400405230377631141132882997336829218810171728925087535674907455584557956801831447125486753515868079342148815961792481779375529
c3 = 43683874913011746530056103145445250281307732634045437486524605104639785469050499171640521477036470750903341523336599602288176611160637522568868391237689241446392699321910723235061180826945464649780373301028139049288881578234840739545000338202917678008269794179100732341269448362920924719338148857398181962112
print(long_to_bytes(hastad_broadcast([(c1, n1), (c2, n2), (c3, n3)])))
main()
```
Using some functions from our crypto-commons:
```python
def solve_crt(residue_and_moduli):
"""
Solve CRT for given modular residues and modulus values, eg:
x = 1 mod 3
x = 2 mod 4
x = 3 mod 5
x = 58
residue_and_moduli = [(1,3), (2,4), (3,5)]
:param residue_and_moduli: list of pairs with (modular residue mod n, n)
:return: x
"""
moduli = [x[1] for x in residue_and_moduli]
residues = [x[0] for x in residue_and_moduli]
N = reduce(lambda x, y: x * y, moduli)
Nxs = [N / n for n in moduli]
ds = [modinv(N / n, n) for n in moduli]
mults = [residues[i] * Nxs[i] * ds[i] for i in range(len(moduli))]
return reduce(lambda x, y: x + y, mults) % N
def hastad_broadcast(residue_and_moduli):
"""
Hastad RSA attack for the same message encrypted with the same public exponent e and different modulus.
Requires exactly 'e' pairs as input
:param residue_and_moduli: list of pairs (residue, modulus)
:return: decrypted message
"""
import gmpy2
k = len(residue_and_moduli)
solution, _ = gmpy2.iroot(solve_crt(residue_and_moduli), k)
assert residue_and_moduli[0][0] == pow(int(solution), k, residue_and_moduli[0][1])
return solution
```
And this gives the flag: `theoretical_computer_scientist_johan_torkel_hastad`
###PL version
W zadaniu mamy do czynienia z wiadomością szyfrowaną za pomocą RSA bez paddingu przy użyciu 3 różnych kluczy, ale z tym samym wykładnikiem `e = 3`.
To oznacza że możemy użyć Hastad Broadcast Attack aby odzyskać wiadomość za pomocą Chińskiego Twierdzenia o Resztach.
Wykładnik dla wszystkich 3 szyfrogramów wynosi 3 więc mamy:
```
m^e = y1 mod n1
m^e = y2 mod n2
m^e = y3 mod n3
```
Z Chińskiego Twierdzenia o Resztach wiemy że można odzyskać z takiego układu `m^e`, a wtedy potrzebujemy już tylko wyliczyć k-ty pierwiastek całkowity aby dostać flagę:
```python
from src.crypto_commons.generic import long_to_bytes
from src.crypto_commons.rsa.rsa_commons import hastad_broadcast
def main():
n1 = 95118357989037539883272168746004652872958890562445814301889866663072352421703264985997800660075311645555799745426868343365321502734736006248007902409628540578635925559742217480797487130202747020211452620743021097565113059392504472785227154824117231077844444672393221838192941390309312484066647007469668558141
n2 = 98364165919251246243846667323542318022804234833677924161175733253689581393607346667895298253718184273532268982060905629399628154981918712070241451494491161470827737146176316011843738943427121602324208773653180782732999422869439588198318422451697920640563880777385577064913983202033744281727004289781821019463
n3 = 68827940939353189613090392226898155021742772897822438483545021944215812146809318686510375724064888705296373853398955093076663323001380047857809774866390083434272781362447147441422207967577323769812896038816586757242130224524828935043187315579523412439309138816335569845470021720847405857361000537204746060031
c1 = 64830446708169012766414587327568812421130434817526089146190136796461298592071238930384707543318390292451118980302805512151790248989622269362958718228298427212630272525186478627299999847489018400624400671876697708952447638990802345587381905407236935494271436960764899006430941507608152322588169896193268212007
c2 = 96907490717344346588432491603722312694208660334282964234487687654593984714144825656198180777872327279250667961465169799267405734431675111035362089729249995027326863099262522421206459400405230377631141132882997336829218810171728925087535674907455584557956801831447125486753515868079342148815961792481779375529
c3 = 43683874913011746530056103145445250281307732634045437486524605104639785469050499171640521477036470750903341523336599602288176611160637522568868391237689241446392699321910723235061180826945464649780373301028139049288881578234840739545000338202917678008269794179100732341269448362920924719338148857398181962112
print(long_to_bytes(hastad_broadcast([(c1, n1), (c2, n2), (c3, n3)])))
main()
```
Używając kilku funkcji z naszych crypto-commons:
```python
def solve_crt(residue_and_moduli):
"""
Solve CRT for given modular residues and modulus values, eg:
x = 1 mod 3
x = 2 mod 4
x = 3 mod 5
x = 58
residue_and_moduli = [(1,3), (2,4), (3,5)]
:param residue_and_moduli: list of pairs with (modular residue mod n, n)
:return: x
"""
moduli = [x[1] for x in residue_and_moduli]
residues = [x[0] for x in residue_and_moduli]
N = reduce(lambda x, y: x * y, moduli)
Nxs = [N / n for n in moduli]
ds = [modinv(N / n, n) for n in moduli]
mults = [residues[i] * Nxs[i] * ds[i] for i in range(len(moduli))]
return reduce(lambda x, y: x + y, mults) % N
def hastad_broadcast(residue_and_moduli):
"""
Hastad RSA attack for the same message encrypted with the same public exponent e and different modulus.
Requires exactly 'e' pairs as input
:param residue_and_moduli: list of pairs (residue, modulus)
:return: decrypted message
"""
import gmpy2
k = len(residue_and_moduli)
solution, _ = gmpy2.iroot(solve_crt(residue_and_moduli), k)
assert residue_and_moduli[0][0] == pow(int(solution), k, residue_and_moduli[0][1])
return solution
```
Co daje flagę: `theoretical_computer_scientist_johan_torkel_hastad`
