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GACTF2020 Writeups

This was an interesting challenge. Some of the other crypto challenges were copies of previous CTF tasks, not sure if this one was too, but I hadn't see it before.


elgaml_rsa

crypto

#!/usr/bin/env python
# coding=utf-8
from Crypto.Util.number import *
from gmpy2 import *
from binascii import *
from key import secret


size = 4096
next_state = getRandomInteger(size // 2)


def keygen(size):
    q = getPrime(size)
    k = 2
    while True:
        p = q * k + 1
        if isPrime(p):
        	break
        k += 1
    g = 2
    while True:
        if pow(g, q, p) == 1:
            break
        g += 1
    A = getRandomInteger(size) % q
    B = getRandomInteger(size) % q
    d = getRandomInteger(size) % q
    h = pow(g, d, p)
    return (g, h, A, B, p, q), (d,)


def rand(A, B, M):
    global next_state
    next_state, ret = (B * next_state + A) % M, next_state
    return ret


def enc(pubkey, m):
    g, h, A, B, p, q = pubkey
    assert 0 < m <= p
    r = rand(A, B, q)
    c1 = pow(g, r, p)
    c2 = (m * pow(h, r, p)) % p
    return (c1, c2)


pubkey, privkey = keygen(size)

c1, c2 = enc(pubkey, secret)
c11, c22 = enc(pubkey, secret)

print(pubkey)
print(c1, c2)
print(c11, c22)

with open('flag','rb') as f:
    flag = f.read().strip(b'gactf{').strip(b'}')

assert(len(flag)==36)
flag = bytes_to_long(flag)

e=0x1296
c = pow(flag,e,secret)
print(c)



'''
(64, 20780474293866131933862287255555455030374045530251095666369283432462810652525427691721304254462237017759426334882916049495889504594009770520464330762959016951546575145462307317894496460531556164410953148497518290028088712258668119908424719634615388291047264345913823034842970773368033086339559642636137744915369016200318607505245388543117271999254001335968170458213311497321117529280041082988824396151908188525489700796405858374530511663128135580173535828610757421885017756967359800260945101629391819567689637211651895850980281588310001703030835140810350079586117727295930982227960023008919855063362680952572131899713036162496210570179135414385493625653887326331804734492999669859178883625968923995256089541092732093557828711090165978111405804292224573421621191633338981450285157302862932940500093263243947199483627384074436212584903719549495723188986249991756615160063402163694830775734815955316835349073419639199758694671594404540029322234383581399522064379382021595719397223893607631901725607641744479242212920789603071407011348514876465946739422799404447033886982709934951857043468867599886808040297253407019293845393488573199993494540736979501538494818953066676387396633085444166742111351272947318154970048596923974504404380927416,381708384013002636433122354335926692336601617676926968898013795392655866091031994182231300776082068899737198812550890687137401508001940273120074129003646740029985994207331055782501249765461466584992023925230840665949906632198101999859500790212575156806509743006505701713970313218378023375414073482678391588622628187074299412433043817050683291933161425628794715547714403106509123476151979947917391393412645765122604354225100743870995206588381507456741290978603333889874930706693482317045587451791046190740066378574507676759649633128467597518860457862695484980816243465255252173136325137506270232696551053565764269904166529146681242149565805411556997705869017810058040591799368604689685951672647151912084757145775011455841321798009615242210532962596268560546312347194505828381287458597164561168495886248727096685588006006546480425238352362436971564880004651450568881978647634212713224047524921068489089978106015561713311517385843649234918915312918406300092887256427518081655040845694063162361305181653495296888548335581623440628369113672677762811628569895443681117805669311418808453448653593363449729346566117394039637974118545192223391804824330785203598209547944921552416944717704983567183156966157476160974179852872170934972858836781, 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22453152428394606088943774662302202422153960447312197242455569997061450760047528966506441566524526853033293751967810594140854923226151650938525170137514600880805050554663943082815906368514404641025604294366860158614875990137728517294116316762556975898260341247898716976970617032432937955286668439404344848291545559658276642393602228734034612184252718870517184522197911024934717798077233461582581868193787894582258952888627951734023431879183670191669298358680498935709882003781077406338523907367346789138656618116933889244879390502610424899510803955442942977900239324993614041517202681723278532043123236878326249862092754098907790488065071040929376826057117979467292302080257950897426644056280125378364535680687786693270799850067554410612758531513629387059863596252391870417805469348504325636998643613777442875034648277326137337187208698790044216186340918785533348914782916443684223939292156374009910727628455699988827744627946732973050302225074511302818542457166115156873282388208050368257145196360795205915922021436321478999333704296935314435099278218817835181668673690398237608806925783268658459276024403522197373931280630406781239277149660631726005650281230352429449777583372994535668183455158366753076284188469366650872129189188954 43014070623103292976184689427105928199147965783937089428385491282216272444365283117905989848749181365267234218854446028762485581571306912831438529312945584194862759018753822869158358589009967858962779001877279323670583250081922847895176784620706892400531521013763514799201763745183044164104035196613753228089053191247345267210940633115873824974769014722055077799856934880737833185923456222178802890352524448603177600618056088876614578821438861413350348617677694184716464398433385655237392506169213931196012955755770139080945778856434397878091389138231179032165240858902248596578661346637175321979963481071276143651366658999282369341715121067180208904222607402081278785172060423774990737421479393799816962905617768730613118540383524977589624740822007170019678257890472150388700140797012249510238048211606905864478903986719042623036582247378794747826919609810170285750234710357000132728486804956251659199729624366338890451927509641664833932346738489065555140914795441664133418400958836523237094378780782603370403266584597230077617701944349882551595000276212025361451916984278427199657858078227972152325611502177407278626854076690433960994875505694991760829975244773421088756265358618620304273856739824192560105567929686528910876014823965
44608013650780329531109538179331822570135966195550223402070161598185644434770862399901412649297405576993241936895024146617797344461666950233511185666728221978895618296351596298775136672055923289389665757911807359016802923641460730946632818202168872788458468347348232090434225957209352582129271299425216944747679759179672053700778261669489727413942176315455298932909861450580984970798585428405552956293406458215448336032446056582649917889593692072954837453042473056600004374540260728744840291557410530843899434261407426178111249646964369037637633211541371271197280517635042592551789127833173333234479307397045400972018112295715569721800138429266922182854791647269058891183615748790947697069153882008589561211157334372972499729485252727694797282796590848738021218670314348696571534824112823542032680872158338607531482751162776081697861530593966973408267629391304447962348239015723944462941622104654613580650653767954875700171277625377411714660734830575428948037611098387735691682421457169239064406477687024462745934438832041378409566709345316139490973699111690159932212499398783638503401404235464154454535785113901174157533110165240552563496872266944710590644159214078372821774380931384935459500471791384494332187743455497646590556880486 1522687118053736861056969003430796399893176012103287854193029873658554858136827462259124651478772173757025568892302195402291455992394789178299309535616987072965579421593018021114875919558287237186455725647948113582053830294196737121722225985979525758513270467145143657376879463222508333678601013048368285045304095162002671773027175721352146624828522742775630450642598352531401961139587698492307980703004014415044157520489051036928610329089210652248586776563084702121754645966572787310495034532000331181441756647075157708515223727039422012212657926685677978199476498472278242756713492597894691133614639186351240949930682742853869969584053534079567976916119132941681011705563769739500572388748003030282448776889448778821320406706124477759144963675421171984971802471706213665007068221227826503520100160877838988068439450319825803900962680452544694163391017757486433728437439479064627991529036015041631361993854848621812502254104495342939127025063439806000291665982509626369261288950954463853496165866013918850578916279480441428849258614763017063059042143901251569352672219853411840562939930258505979216301725869416361119563043278577212326582324332541993877436871165282813371607548756276989941992463786598573454058633853167962827467271628
255310806360822158306697936064463902328816816156848194779397173946813224291656351345682266227949792774097276485816149202739762582969208376195999403112665514848825884325279574067341653685838880693150001066940379902609411551128810484902428845412055387955258568610350610226605230048821754213270699317153844590496606931431733319116866235538921198147193538906156906954406577796507390570080177313707462469835954564824944706687157852157673146976402325057144745208116022973614795377968986322754779469798013426261911408914756488145211933799442123449261969392169406969410065018032795960230701484816708147958190769470879211953704222809883281592308316942052671516609231501663363123562942
'''

Solution

Part 1: ElGamal and the Broken RNG

The script encrypts secret twice using the probabilistic ElGamal encryption. The random value r in the ElGamal encryption is generated from a LCG which is easily broken. Having two encryptions of the same message is enough for us to recover the message completely. This will be the first part of the challenge.

Let r1r_1 and r2r_2 be the random values used in the first and second ElGamal encryption. r1r_1 is some securely generated random value, and r2Br1+A(modq)r_2 \equiv Br_1 + A \pmod q is generated from the LCG. We are given (c1,c2)(c_1, c_2) and (c11,c22)(c_{11}, c_{22}) where

c1gr1(modp)c2mhr1(modp)c11gr2gBr1+Amodq(modp)c22mhr2mhBr1+Amodq(modp)\begin{aligned} c_1 &\equiv g^{r_1} \pmod p \\ c_2 &\equiv mh^{r_1} \pmod p \\ c_{11} &\equiv g^{r_2} \equiv g^{Br_1 + A \mod q} \pmod p \\ c_{22} &\equiv mh^{r_2} \equiv mh^{Br_1 + A \mod q} \pmod p \end{aligned}

Notice that

c22mhBr1+Am(hr1)BhA(mhr1)(hr1)B1hAc2(hri)B1hA(modp)\begin{aligned} c_{22} &\equiv mh^{Br_1 + A} \\ &\equiv m(h^{r_1})^B h^A \\ &\equiv (mh^{r_1})(h^{r_1})^{B-1} h^A \\ &\equiv c_2 (h^{r_i})^{B-1} h^A \pmod p \end{aligned}

Therefore

hri(c22c21hA)B(modp)h^{r_i} \equiv (c_{22} c_2^{-1} h^{-A})^{B'} \pmod p

where B(B1)1(modp1)B' \equiv (B-1)^{-1} \pmod {p-1}.

Then, we can recover mm by computing

mc2(hr1)1(modp)m \equiv c_2 (h^{r_1})^{-1} \pmod p

This gives us the secret modulus:

N = 42044128297^6 * 232087313537^5 * 653551912583^15 * 28079229001363^14 * 104280142799213^6 * 802576647765917^7

Part 2: RSA and the Non-Coprime ee and φ(N)\varphi(N)

We've now recovered the secret modulus, and luckily for us, it's easily factored! All that's left is to decrypt the (almost) textbook RSA...

Except there's a slight issue! e = 4758 is even, and in fact, e divides phi(N). One of the first things we learn when studying RSA is that e and phi(N) need to be coprime for decryption to be possible, so is the challenge broken?

Clearly not. In fact this particular line of code gives us a hint as to how to proceed:

assert(len(flag)==36)

For clarity, write the modulus as

N=p1k1p2k2prkrN = p_1^{k_1}p_2^{k_2} \cdots p_r^{k_r}

Then

cme(modN)me(modp1k1p2k2prkr)\begin{aligned} c &\equiv m^e \pmod N \\ &\equiv m^e \pmod {p_1^{k_1} p_2^{k_2} \cdots p_r^{k_r}} \end{aligned}

This implies cme(modpiki)c \equiv m^e \pmod {p_i^{k_i}}. This is quite easy to show (sketch):

Suppose ab(modpq)a \equiv b \pmod {pq}, then ab=kpq=(kp)qa - b = kpq = (kp)q, so ab(modq)a \equiv b \pmod q.

Since the flag is 36 bytes, we can solve for mm modulo some factor(s) of NN such that their product is greater than 236×82^{36\times 8}.

653551912583^15 works well here. Actually gcd(4758, phi(653551912583^15)) = 2, but it turns out that this doesn't cause much troubles for us because m^2 < 653551912583^15 (we'll see why this is soon enough).

So let N=65355191258315N = 653551912583^{15}. Then φ(N)=6535519125831565355191258314\varphi(N) = 653551912583^{15} - 653551912583^{14}.

Let g=gcd(e,φ(N))=2g = \gcd(e, \varphi(N)) = 2 and let d(eg)1(modφ(N))d \equiv (\frac{e}{g})^{-1} \pmod {\varphi(N)}.

dd exists since gcd(eg,φ(N))=1\gcd(\frac{e}{g}, \varphi(N)) = 1. It follows that d=ge+kφ(N)d = \frac{g}{e} + k\varphi(N) for some integer kk.

Then

cd(me)d(me)ge+kφ(N)mgmkφ(N)mg(modN)\begin{aligned} c^d &\equiv (m^e)^d \\ &\equiv (m^e)^{\frac{g}{e} + k\varphi(N)} \\ &\equiv m^g m ^{k\varphi(N)} \\ &\equiv m^g \pmod N \end{aligned}

But we know that mg=m2<Nm^g = m^2 < N, so the congruence is actually an equality. Therefore, we can just take the integer square root to recover mm completely.

And we're done!

Solve Script

pubkey = (64, 20780474293866131933862287255555455030374045530251095666369283432462810652525427691721304254462237017759426334882916049495889504594009770520464330762959016951546575145462307317894496460531556164410953148497518290028088712258668119908424719634615388291047264345913823034842970773368033086339559642636137744915369016200318607505245388543117271999254001335968170458213311497321117529280041082988824396151908188525489700796405858374530511663128135580173535828610757421885017756967359800260945101629391819567689637211651895850980281588310001703030835140810350079586117727295930982227960023008919855063362680952572131899713036162496210570179135414385493625653887326331804734492999669859178883625968923995256089541092732093557828711090165978111405804292224573421621191633338981450285157302862932940500093263243947199483627384074436212584903719549495723188986249991756615160063402163694830775734815955316835349073419639199758694671594404540029322234383581399522064379382021595719397223893607631901725607641744479242212920789603071407011348514876465946739422799404447033886982709934951857043468867599886808040297253407019293845393488573199993494540736979501538494818953066676387396633085444166742111351272947318154970048596923974504404380927416,381708384013002636433122354335926692336601617676926968898013795392655866091031994182231300776082068899737198812550890687137401508001940273120074129003646740029985994207331055782501249765461466584992023925230840665949906632198101999859500790212575156806509743006505701713970313218378023375414073482678391588622628187074299412433043817050683291933161425628794715547714403106509123476151979947917391393412645765122604354225100743870995206588381507456741290978603333889874930706693482317045587451791046190740066378574507676759649633128467597518860457862695484980816243465255252173136325137506270232696551053565764269904166529146681242149565805411556997705869017810058040591799368604689685951672647151912084757145775011455841321798009615242210532962596268560546312347194505828381287458597164561168495886248727096685588006006546480425238352362436971564880004651450568881978647634212713224047524921068489089978106015561713311517385843649234918915312918406300092887256427518081655040845694063162361305181653495296888548335581623440628369113672677762811628569895443681117805669311418808453448653593363449729346566117394039637974118545192223391804824330785203598209547944921552416944717704983567183156966157476160974179852872170934972858836781, 43505603305111680803563977459640944464105385282379422421980575558368972842607765670721003089672603097313103981235686237397257862849601241879471167930153646225844008471355704339980238211134686502225715019115869317427197390964362320090838527730898664248447107765413761830863605920486819969055546423684508389628123987801815335523239425432391765479775424040215970484578304273140784182204718527220185459004052198324217205275587873438839100176140844478555649819782992169483866083330097141949046245149094504794723574282964767779251414958958706784791732572996736294885028909774443354205765706938344147696599409331866916073619245974602504186911091638010576094381687164075770345437007237017511597686483438791354205795335973602676663054120116592544019595778273459615865899404356938805021081374452052068221554454819002242766042331745787877501397324349289129890310340865730592151597455837880210622067528365150318411999840330958344868422035016628605636595819056821789774656720527976088720077220560311329066034711257025398105457680968934892169828204426383391000448054370375803473806685234956611656904127231542272618726532676811311523606477787525194092155173656104677312540399404023913503117830278920886909662890420006539377593361467113937140744010, 45546439253566873046135178791096659917209492861617458211267719798885277887938347304820298843145674472870766332758636279557209016031713547700909688953215131457341421729182912313815167760908611069679571547056188204597788444236869052385801746036327394329618747230203387842029040907229766131476048451293016635093208306323063715706067371953108251630052089462662417303475933291290566336468462586635807978767551023772175046232601222877403596811965579296677734661047163959600195727625890255130528298615338776353196525781432396712546801275867984255488198557346322961570926619651871594687507639988384211396556450312496725130292814328927584109530411297121292814256252280901057553636074873996093840854880090055633725117055959590322998817024919326703993851904085347843055734589470195083739394356572346483715507513378038693666480211356659114879422949481825838585239350398353217171146754711645184363669060413784452297408480588454177322580570407385734077514306779834329286166444213819533159384145482066266175499414964502711323195573855494716545548783117249211738964334485427936770661680033789944683657229394946749668495670252890519005455449387321211576562318106342614124875224420621771672914794854566658369446093868181072056414942873885989261187792529, 948884151115976521794482891481180414941864434617030379401410829143443289332048902183756225898868218184807631932471589157441854500660698910435618519858648572027946286024644006537815995018929397284991073897003920929120592588268105258037536375756820715200390567295903913375605018900620127739084342735271179897775173048397160743876403582356421908959418530472133693822415276901886798676426303888245999557657312995253646796512525476612574933582616235347452805438482582491670744325539380315219339554486224507358260953779841598178058359913916338656004136611381728366060971242747324889323075833091337737428259381510348440214433631852658002281883568690026933630338589185438699034084893208251955017810001876159035939938665824798395808688019152639666538581335111413396994470613962397577904049095257218410739739862042472784718337736597064893321311447538038303859153133299025357732224056492608007576438758620509422862676678926128694220428550153869459948214724579881860128467587787906940820503030876380545322904478427139819233241121989473261365599648276025244561756968446415349388785000703957180909525612394723951426993130268552479280321862235858574511714960548804460934900508762953576519058226136805382696793622253772334508644643205958109608079011)
c1, c2 = 22453152428394606088943774662302202422153960447312197242455569997061450760047528966506441566524526853033293751967810594140854923226151650938525170137514600880805050554663943082815906368514404641025604294366860158614875990137728517294116316762556975898260341247898716976970617032432937955286668439404344848291545559658276642393602228734034612184252718870517184522197911024934717798077233461582581868193787894582258952888627951734023431879183670191669298358680498935709882003781077406338523907367346789138656618116933889244879390502610424899510803955442942977900239324993614041517202681723278532043123236878326249862092754098907790488065071040929376826057117979467292302080257950897426644056280125378364535680687786693270799850067554410612758531513629387059863596252391870417805469348504325636998643613777442875034648277326137337187208698790044216186340918785533348914782916443684223939292156374009910727628455699988827744627946732973050302225074511302818542457166115156873282388208050368257145196360795205915922021436321478999333704296935314435099278218817835181668673690398237608806925783268658459276024403522197373931280630406781239277149660631726005650281230352429449777583372994535668183455158366753076284188469366650872129189188954, 43014070623103292976184689427105928199147965783937089428385491282216272444365283117905989848749181365267234218854446028762485581571306912831438529312945584194862759018753822869158358589009967858962779001877279323670583250081922847895176784620706892400531521013763514799201763745183044164104035196613753228089053191247345267210940633115873824974769014722055077799856934880737833185923456222178802890352524448603177600618056088876614578821438861413350348617677694184716464398433385655237392506169213931196012955755770139080945778856434397878091389138231179032165240858902248596578661346637175321979963481071276143651366658999282369341715121067180208904222607402081278785172060423774990737421479393799816962905617768730613118540383524977589624740822007170019678257890472150388700140797012249510238048211606905864478903986719042623036582247378794747826919609810170285750234710357000132728486804956251659199729624366338890451927509641664833932346738489065555140914795441664133418400958836523237094378780782603370403266584597230077617701944349882551595000276212025361451916984278427199657858078227972152325611502177407278626854076690433960994875505694991760829975244773421088756265358618620304273856739824192560105567929686528910876014823965
c11, c22 = 44608013650780329531109538179331822570135966195550223402070161598185644434770862399901412649297405576993241936895024146617797344461666950233511185666728221978895618296351596298775136672055923289389665757911807359016802923641460730946632818202168872788458468347348232090434225957209352582129271299425216944747679759179672053700778261669489727413942176315455298932909861450580984970798585428405552956293406458215448336032446056582649917889593692072954837453042473056600004374540260728744840291557410530843899434261407426178111249646964369037637633211541371271197280517635042592551789127833173333234479307397045400972018112295715569721800138429266922182854791647269058891183615748790947697069153882008589561211157334372972499729485252727694797282796590848738021218670314348696571534824112823542032680872158338607531482751162776081697861530593966973408267629391304447962348239015723944462941622104654613580650653767954875700171277625377411714660734830575428948037611098387735691682421457169239064406477687024462745934438832041378409566709345316139490973699111690159932212499398783638503401404235464154454535785113901174157533110165240552563496872266944710590644159214078372821774380931384935459500471791384494332187743455497646590556880486, 1522687118053736861056969003430796399893176012103287854193029873658554858136827462259124651478772173757025568892302195402291455992394789178299309535616987072965579421593018021114875919558287237186455725647948113582053830294196737121722225985979525758513270467145143657376879463222508333678601013048368285045304095162002671773027175721352146624828522742775630450642598352531401961139587698492307980703004014415044157520489051036928610329089210652248586776563084702121754645966572787310495034532000331181441756647075157708515223727039422012212657926685677978199476498472278242756713492597894691133614639186351240949930682742853869969584053534079567976916119132941681011705563769739500572388748003030282448776889448778821320406706124477759144963675421171984971802471706213665007068221227826503520100160877838988068439450319825803900962680452544694163391017757486433728437439479064627991529036015041631361993854848621812502254104495342939127025063439806000291665982509626369261288950954463853496165866013918850578916279480441428849258614763017063059042143901251569352672219853411840562939930258505979216301725869416361119563043278577212326582324332541993877436871165282813371607548756276989941992463786598573454058633853167962827467271628
c = 255310806360822158306697936064463902328816816156848194779397173946813224291656351345682266227949792774097276485816149202739762582969208376195999403112665514848825884325279574067341653685838880693150001066940379902609411551128810484902428845412055387955258568610350610226605230048821754213270699317153844590496606931431733319116866235538921198147193538906156906954406577796507390570080177313707462469835954564824944706687157852157673146976402325057144745208116022973614795377968986322754779469798013426261911408914756488145211933799442123449261969392169406969410065018032795960230701484816708147958190769470879211953704222809883281592308316942052671516609231501663363123562942
e = 0x1296

from Crypto.Util.number import *
from gmpy2 import *

g, h, A, B, p, q = pubkey

hr1 = pow(c22 * invert(c2, p) * pow(h,-A, p), invert(B-1, p-1), p)
N = c2*invert(int(hr1), int(p)) % p

# fac = factor(N)
# print(fac)
fac = [(42044128297, 6), (232087313537, 5), (653551912583, 15), (28079229001363, 14), (104280142799213, 6), (802576647765917, 7)]
f,k = fac[2]
phi = f^k - f^(k-1)
g = gcd(e, phi)
d = inverse_mod(e//g, phi)
M = pow(c, d, f^k)
flag = iroot(int(M), g)[0]

print('gactf{' + long_to_bytes(flag).decode() + '}')

Flag: gactf{you_4re_good_at_b0th_el94mal_and_rs4}