# 4k-rsa

Author: [roerohan](https://github.com/roerohan)

RSA using multiple primes.

# Requirements

- RSA

# Source

- [4k-rsa-public-key.txt](./4k-rsa-public-key.txt)

```
Only n00bz use 2048-bit RSA. True gamers use keys that are at least 4k bits long, no matter how many primes it takes...
```

# Exploitation

This is like any classic RSA exploit, except `n` is made up of multiple primes. We can facotrize `n` using `http://factordb.com`. The only difference is, `phi` is calculated as `(p-1)*(q-1)*(r-1)...`, where `p,q,r...` are the primes.

```python
from Crypto.Util.number import inverse

n = 5028492424316659784848610571868499830635784588253436599431884204425304126574506051458282629520844349077718907065343861952658055912723193332988900049704385076586516440137002407618568563003151764276775720948938528351773075093802636408325577864234115127871390168096496816499360494036227508350983216047669122408034583867561383118909895952974973292619495653073541886055538702432092425858482003930575665792421982301721054750712657799039327522613062264704797422340254020326514065801221180376851065029216809710795296030568379075073865984532498070572310229403940699763425130520414160563102491810814915288755251220179858773367510455580835421154668619370583787024315600566549750956030977653030065606416521363336014610142446739352985652335981500656145027999377047563266566792989553932335258615049158885853966867137798471757467768769820421797075336546511982769835420524203920252434351263053140580327108189404503020910499228438500946012560331269890809392427093030932508389051070445428793625564099729529982492671019322403728879286539821165627370580739998221464217677185178817064155665872550466352067822943073454133105879256544996546945106521271564937390984619840428052621074566596529317714264401833493628083147272364024196348602285804117877
e = 65537
c = 3832859959626457027225709485375429656323178255126603075378663780948519393653566439532625900633433079271626752658882846798954519528892785678004898021308530304423348642816494504358742617536632005629162742485616912893249757928177819654147103963601401967984760746606313579479677305115496544265504651189209247851288266375913337224758155404252271964193376588771249685826128994580590505359435624950249807274946356672459398383788496965366601700031989073183091240557732312196619073008044278694422846488276936308964833729880247375177623028647353720525241938501891398515151145843765402243620785039625653437188509517271172952425644502621053148500664229099057389473617140142440892790010206026311228529465208203622927292280981837484316872937109663262395217006401614037278579063175500228717845448302693565927904414274956989419660185597039288048513697701561336476305496225188756278588808894723873597304279725821713301598203214138796642705887647813388102769640891356064278925539661743499697835930523006188666242622981619269625586780392541257657243483709067962183896469871277059132186393541650668579736405549322908665664807483683884964791989381083279779609467287234180135259393984011170607244611693425554675508988981095977187966503676074747171

x = [9353689450544968301 , 9431486459129385713 , 9563871376496945939 , 9734621099746950389 , 9736426554597289187 , 10035211751896066517 , 10040518276351167659 , 10181432127731860643 , 10207091564737615283 , 10435329529687076341 , 10498390163702844413 , 10795203922067072869 , 11172074163972443279 , 11177660664692929397 , 11485099149552071347 , 11616532426455948319 , 11964233629849590781 , 11992188644420662609 , 12084363952563914161 , 12264277362666379411 , 12284357139600907033 , 12726850839407946047 , 13115347801685269351 , 13330028326583914849 , 13447718068162387333 , 13554661643603143669 , 13558122110214876367 , 13579057804448354623 , 13716062103239551021 , 13789440402687036193 , 13856162412093479449 , 13857614679626144761 , 14296909550165083981, 14302754311314161101 , 14636284106789671351 , 14764546515788021591 , 14893589315557698913 , 15067220807972526163 , 15241351646164982941 , 15407706505172751449 , 15524931816063806341 , 15525253577632484267 , 15549005882626828981 , 15687871802768704433 , 15720375559558820789 , 15734713257994215871 , 15742065469952258753 , 15861836139507191959 , 16136191597900016651 , 16154675571631982029 , 16175693991682950929 , 16418126406213832189 , 16568399117655835211 , 16618761350345493811 , 16663643217910267123 , 16750888032920189263 , 16796967566363355967 , 16842398522466619901 , 17472599467110501143 , 17616950931512191043 , 17825248785173311981 , 18268960885156297373 , 18311624754015021467 , 18415126952549973977]

phi = 1
for i in x:
phi *= i-1

d = inverse(e, phi)

m = pow(c,d,n)

t = hex(m)[2:]

# print(''.join(chr(int(t[i:i+2],16)) for i in range(0,len(t),2)))
# OR

print(bytes.fromhex(t))
```

Run this with `python`.

```bash
$ python 4k-rsa.py
b'flag{t0000_m4nyyyy_pr1m355555}'
```

The flag is:

```
flag{t0000_m4nyyyy_pr1m355555}
```

Original writeup (https://github.com/csivitu/CTF-Write-ups/tree/master/redpwnCTF%202020/crypto/4k-rsa).