stars.txt

```
n = 3704315853248399520931286149673707861739862840174474318850719536726710629666865320437415173536083044248674656366508763974563867624182606586607673025903533434817835086465783354244510555600264634130273148019420890567027535547227590914410411615928952211983453962602151306110276831526240787368162679547412294786291064966782230443213405903174988451410995303610907533223263661657022610213941595505402486344210348467569618138176118521585956414022200085910233519821468163410050367827176390736810757738681372850691835129931503547843319099082181577780559763017889775357876998458807632809093618617485403908641927128567353193360668090395039602867558019689535059386138345918370693234076176881713132466541413701890144073499822831857803210562795039757846366860142062528042547185814880309533531009787005636747054029265024429693545769057677993257539297043796211840831462670392733400569034198413996091467275311168017215341595275384890539948027166145573735873646315369542533040843746818921047007048215557270397023005318175507358018082722469914231819553975967320629679189163059501755587530922829237056878993018295219879385447112790139581486354382509344797829578897288431682124840984604266323302273393857383811762984514580342125613255178652414822963564869351218173787050628467442637469971056029077563975073591774597269174531187818103532433894598280645766078102633738987646941950999789200401692007650522695229851082972906186018526013063835926631066293946238364751626692658502230496793712404951869930251498399478950443234400311619244641598277508121436749410796181780508379156355104722456111706582162269020945596481112763783522150990226114867335614536816138186566447583639848730080522260064509836147032605761376505234474572072334454737916929928957071355381652814660308963367515589390876610261412033928965154188453244893992884372548886256587154030720268721740548470139041848679945758053674161556432495785493981806766135281532135845427813009741588593121034452658170351001464780761847912542795779189019845240208570375066538167569500377796309764656262247487848676594103588076166586891212714090941313565555203618668077425758661000673064635747737323512221035673339963530741105199538223119
e = 65537
c = 1890542018923620935112404571390267463044771404340539852654454163421322784131374626938125628909369500484672852457699752836794851668177554752694569480833589506281060764802478167809475843864863010224742636449171460711562400590580946293278022177973604217197729862697501014955273338012893824413982401698433351247436693175518346363430547563575237429026208733403944381002182612963632178995085161195304895642449121609426847712983417558339924558444470082314710007938285805497203551318491773326164557602529684001253659352725160735497149899998922033507206044442056198143593433947379784128199737258656118578219421919085097879160406490718541327506724660681519907530750871792173102790503851758679070200134015226988860086382708659309820815068856900840522602932358921124524038033117760324955208431843589240233038158065364666936016479598474989719135380119445900393078743516628362378875759274530774051128410372267608865052931691829589285501692827537962441398960003650438584754477318978505734011792215065774609983390432428036137453401404511730410595609138140441384381140434709123509507412098963062352514519801938315793583554812274182187166556207679813626740486834909299435103532087886743107402138337819127106393630155591187052192592768360568615482279489728303155692177202699660408675290448360087377516035018720139480608360145681977445078359006122398216982658270012457757918360559874257031971071466702054740672492979859050720556486468907749247317125767439522125033157280673514896222594860231916321612316373414230977836316684012399810022699866047249292314143910277891224901227676543229301854239805048852751753718452085934738230534394786918582999166978975731235786980193716535038972364925871721576464369926284473469322169345932697475574026851062545326839136910259737680110603316183269062287676145272111842764815038653589575448189991669189039680679391320058642386385904979297205209960026834027355463625741637831547936201579980499257524969290528523228793264610947645811882323798981455817077587582259794354346200056385025658887609323422245456200058754051911510696552898584742736617473091204698952845833647824992896981893505822744013992729729002741594330169868768031061441416662299284
```

Bit length of `n` is 7168.

```
>>> n = 3704315853248399520931286149673707861739862840174474318850719536726710629666865320437415173536083044248674656366508763974563867624182606586607673025903533434817835086465783354244510555600264634130273148019420890567027535547227590914410411615928952211983453962602151306110276831526240787368162679547412294786291064966782230443213405903174988451410995303610907533223263661657022610213941595505402486344210348467569618138176118521585956414022200085910233519821468163410050367827176390736810757738681372850691835129931503547843319099082181577780559763017889775357876998458807632809093618617485403908641927128567353193360668090395039602867558019689535059386138345918370693234076176881713132466541413701890144073499822831857803210562795039757846366860142062528042547185814880309533531009787005636747054029265024429693545769057677993257539297043796211840831462670392733400569034198413996091467275311168017215341595275384890539948027166145573735873646315369542533040843746818921047007048215557270397023005318175507358018082722469914231819553975967320629679189163059501755587530922829237056878993018295219879385447112790139581486354382509344797829578897288431682124840984604266323302273393857383811762984514580342125613255178652414822963564869351218173787050628467442637469971056029077563975073591774597269174531187818103532433894598280645766078102633738987646941950999789200401692007650522695229851082972906186018526013063835926631066293946238364751626692658502230496793712404951869930251498399478950443234400311619244641598277508121436749410796181780508379156355104722456111706582162269020945596481112763783522150990226114867335614536816138186566447583639848730080522260064509836147032605761376505234474572072334454737916929928957071355381652814660308963367515589390876610261412033928965154188453244893992884372548886256587154030720268721740548470139041848679945758053674161556432495785493981806766135281532135845427813009741588593121034452658170351001464780761847912542795779189019845240208570375066538167569500377796309764656262247487848676594103588076166586891212714090941313565555203618668077425758661000673064635747737323512221035673339963530741105199538223119
>>> n.bit_length()
7168
```
This is very large compared to normal `n` of RSA.

After some trials, I found that `n` was divisible by 7.

```
>>> 7168 % 7
0
```

Moreover, when divided by 7, it was a good number of 1024.

```
>>> 7168 / 7
1024.0
```

When 1024-bits prime numbers are randomly generated, it is rarely that their product is just multiple of 1024 bits. So I thought those prime numbers were close.

```
#!/usr/bin/env python3
# -*- coding: utf-8 -*-
import sympy
import gmpy2
import numpy as np
import binascii

n = (snip)
e = 65537
c = (snip)

def next_prime(p):
while True:
if gmpy2.is_prime(p):
return p
p += 1

def prev_prime(p):
while True:
if gmpy2.is_prime(p):
return p
p -= 1

def main():
p0 = int(gmpy2.iroot(n, 7)[0])
p = []
p1 = p0
for _ in range(4):
while True:
if n % p1 == 0:
break
p1 = next_prime(p1)
p.append(p1)
p1 += 1
p1 = p0
for _ in range(3):
while True:
if n % p1 == 0:
break
p1 = prev_prime(p1)
p.append(p1)
p1 -= 1
assert n == np.prod(p)

phi = 1
for p0 in p:
phi *= p0 - 1
d = gmpy2.invert(e, phi)
m = pow(c, d, n)
print(binascii.unhexlify(hex(m)[2:].encode()).decode())

if __name__ == '__main__':
main()
```