# Description Franklin gave birth to the amazing show blacklist but before he leaves the stage he left us some few words as well as a certain machine that generates encrypted messages 
![image](https://user-images.githubusercontent.com/101048320/195287069-904494c2-c95d-4b60-936a-6970f5bc35fe.png)
## Files - `main.py` - `message.py`
# Solution - Looking at the challenge files, we see that `main.py` is very similar to the one supplied with the challenge **the_messager** but with a slight twist, Here's how the code in `main.py` looks like : ```pythonfrom Crypto.Util.number import bytes_to_long, getStrongPrimefrom math import gcdfrom flag import FLAGfrom Crypto.Random import get_random_bytes
def encrypt_message(m):    return pow(m,e,N)
def advanced_encrypt(a,m):	return encrypt_message(pow(a,3,N)+(m << 24))
e = 3p = getStrongPrime(512)q = getStrongPrime(512)
# generate secure keysresult = 0while (result !=1):    p = getStrongPrime(512)    q = getStrongPrime(512)    result = gcd(e,(p-1)*(q-1)) 	
N = p * q
print("N = " + str(N))print("e = " + str(e))
rand = bytes_to_long(get_random_bytes(64))
ct = []ct.append(encrypt_message(rand << 24))
for car in FLAG:	ct.append(advanced_encrypt(car,rand))
print("ct = "+str(ct))```- After examining the code, we notice that strange `advanced_encryption` function that does some operations to the ciphertext after encrypting, it basically mixes a 512 bit random number into the process. Or, to be precise, if $r$ is that random number then $r = r<<24 = r*2^{24}$ is used for encryption.- Looking at the output file, we can see that we have an array of ciphertexts, and by examining `main.py` we surely know it's encrypting the flag character by character- To better understand how the `advanced_encryption` works, we can understand it like this : 
1 - if we want to encrypt the letter `a`, we'll normally do it in RSA like this : $c = a^{3}$ mod $N$ (with $e = 3$)
2 - some additional operations are done to produce an output that we can call $l$ : $l = (c + r')^{3} = c^{3} + 3c^{2}r' + 3cr'^{2} + r'^{3}$ 
3 - Since we know the flag format which is `CyberErudites{}`, we know $l$ and $c$ for the first 14 ciphertexts and we also know $r'^{3}$ mod $N$ which is the first element in our list. If we can calculate $r'$ we can simply encrypt all letters and recover the flag. 
- This is known as the Franklin-Reiter attack on RSA, it basically states that if two messages differ only by a known fixed difference between the two messages and are encrypted under the same modulus $N$, then it is possible to recover both of them.
- If we wanna get $r$, we can solve two equations with our knowledge of the first two ciphertexts and $r'^{3}$, treating the exponents of $r'$ as independent variables we have a system of two linear modular equations in two variables which is solvable. I wrote a sage script that solves for $r'$ : ```sageN = 128704452311502431858930198880251272310127835853066867118127724648453996065794849896361864026440048456920428841973494939542251652347755395656512696329757941393301819624888067640984628166587498928291226622894829126692225620665358415985778838076183290137030890396001916620456369124216429276076622486278042629001e = 3R = 21340757543584301785921441484183053451553315439245254915339588451884106542258661009436759738472587801036386643847752005362980150928908869053740830266273664899424683013780904331345502086236995074501779725358484854206059302399319323859279240268722523450455802058257892548941510959997370995292748578655762731064l1 = 53066819955389743890197631647873076075338086201977617516688228878534943391813622173359672220491899999289257725082621279332126787067021987817619363964027754585057494857755310178293620211144789491527192983726079040683600073569676641124270473179040250808117008272524876858340200385005503388452491343904776677382l2 = 7842029648140254898731712025732394370883533642138819492816448948307815782380138847628158108013809453236401089035015649397623608296202122635822677717636589547775619483739816443584071749358123933593122063285229582290924379314987624399741427190797914523635723048501174115183499642950146958355891757875557441498
F = Zmod(N)
p1 = ord('C')p2 = ord('y')c1 = F(power_mod(p1, e, N))c2 = F(power_mod(p2, e, N))
d1 = F(l1 - c1^3 - R)k11 = F(3*c1^2)k21 = F(3*c1)
d2 = F(l2 - c2^3 - R)k12 = F(3*c2^2)k22 = F(3*c2)
a = k11*k22-k12*k21b = d1*k22 - d2*k21
r = b*a^(-1)
#Sanity checkassert(k21*r^2+k11*r == d1)assert(k22*r^2+k12*r == d2)
print(r)
# r' = 166948911880587234600972597325398559800623586442106754544249387904660171481281804594820145380464642946591165741209919048255667796045110331101490851949349850578944```- Now that we have $r'$, we only have to reconstruct the flag, this python script does exactly that : ```pythonciphertexts = [21340757543584301785921441484183053451553315439245254915339588451884106542258661009436759738472587801036386643847752005362980150928908869053740830266273664899424683013780904331345502086236995074501779725358484854206059302399319323859279240268722523450455802058257892548941510959997370995292748578655762731064, 53066819955389743890197631647873076075338086201977617516688228878534943391813622173359672220491899999289257725082621279332126787067021987817619363964027754585057494857755310178293620211144789491527192983726079040683600073569676641124270473179040250808117008272524876858340200385005503388452491343904776677382, 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15712112799522387502102713193722467424402631384092196807057434496598674428309494459811523180782790044819285433524468899851335245998768088831658170418466392438399410802792501309056236805241160176754059850691703432150262405441430237695612809894131939658419704673281367391393311992247965290741727023994779433141, 81111299336050275750472008224957667125146586306759086248084295752602324720839543288370663615750857287552903922536031859099788116379487391773903754451992919367596305768339444201446994649849712459520992343839084870637197983082887171667909318521242800685401171176726353172069040622627821647795603423551447271897, 85675609516128041505642974246105781207878835838608550975500845934782884530583531039538004477522284615403428772115964969520508967026139845649101029319923434926707385504059806901530960679704218584706660751338176542291587725130564020356976235189119834080273889731614886626208044719903723301580316649222174716532, 85675609516128041505642974246105781207878835838608550975500845934782884530583531039538004477522284615403428772115964969520508967026139845649101029319923434926707385504059806901530960679704218584706660751338176542291587725130564020356976235189119834080273889731614886626208044719903723301580316649222174716532, 61799162491846407044403618488152290977719649337271367195813541845489917481067315542645819191562014741305490739984114955413967671171269164971415239796230389202065094904892080733323413509954520339080338852975324810347271929340532300585500631891040710031375781370408675545957886220089645926354912765277207193798, 15712112799522387502102713193722467424402631384092196807057434496598674428309494459811523180782790044819285433524468899851335245998768088831658170418466392438399410802792501309056236805241160176754059850691703432150262405441430237695612809894131939658419704673281367391393311992247965290741727023994779433141, 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85675609516128041505642974246105781207878835838608550975500845934782884530583531039538004477522284615403428772115964969520508967026139845649101029319923434926707385504059806901530960679704218584706660751338176542291587725130564020356976235189119834080273889731614886626208044719903723301580316649222174716532, 25192663003159777174615451629938213843222366842683940183362551204469496974891644208321342646667233572676576472164648007266602388592839440014310951184481049521012297732473729920930726303235436945383858287317594336428793167109440236357984389807244311551143246546400865347076405661274883099848046482994822149839, 49537543950196981196008705073490431322869824547419754434164989869148910207602945400222339540873326843877995376121122967451939219149806674439960220589193189585760172923124276530124072984487005587747672999647203434177959209139239195212611226079544263186906487384149209242751551401702465174463378551587903464013, 81604909815994086673421693572426298878862835900596359748090824964404410968235612905849185964777674389932839535035599455168277871014038156501897270840412741045214049490644167087296649367556864887777074100267203854799140784786464099851736527907291367077573005882823454864594785834874833652994406695607530576301, 82605108086464943169862550306200946889734302370750892864214984485030047777068048016807926063544535451667188130157336614116049572675082898787939202381661956285707961358162934993215182977800779844581722076853699116970869839871673603035667037900906095736260590599998045628715894324887335765024740238758306886601, 15712112799522387502102713193722467424402631384092196807057434496598674428309494459811523180782790044819285433524468899851335245998768088831658170418466392438399410802792501309056236805241160176754059850691703432150262405441430237695612809894131939658419704673281367391393311992247965290741727023994779433141, 87183215347967191590542176417724958857058607082507361242348562809358164198308247679047180368020902280813201721298429403561075113836349947551334588733174809438143792381327054583566337165561000865476978009557708693850414825902676261268716823948760442922777052374371288490023049947720611595532141510532644435250, 49537543950196981196008705073490431322869824547419754434164989869148910207602945400222339540873326843877995376121122967451939219149806674439960220589193189585760172923124276530124072984487005587747672999647203434177959209139239195212611226079544263186906487384149209242751551401702465174463378551587903464013, 25192663003159777174615451629938213843222366842683940183362551204469496974891644208321342646667233572676576472164648007266602388592839440014310951184481049521012297732473729920930726303235436945383858287317594336428793167109440236357984389807244311551143246546400865347076405661274883099848046482994822149839, 66766715720902966595178914295223547348495022878823462889622775721913448987561303310541276071530863698182890278728480864181198078543766251390078585132912767108062691265660663002406799106089377580272724589978697222635560956414236639690889939243054319449595599517712915493280854882719256210734111619403134480752, 23942811148204157202654637859911277560708533796069053111751942163750895868053622268645847217089096022452975956692323271828103232820789988209286647453449929108598776835367103056318289898885824280500932307262080931914632255743345269865080087850167325191747197921286245735954464811042826275305804860020107464497]
r = 166948911880587234600972597325398559800623586442106754544249387904660171481281804594820145380464642946591165741209919048255667796045110331101490851949349850578944
chars = "abcdefghijklmnopqrstuvwxyzABCDEFGHIJKLMNOPQRSTUVWXYZ{}-_#0123456789"
N = 128704452311502431858930198880251272310127835853066867118127724648453996065794849896361864026440048456920428841973494939542251652347755395656512696329757941393301819624888067640984628166587498928291226622894829126692225620665358415985778838076183290137030890396001916620456369124216429276076622486278042629001e = 3
def encrypt_message(m):    return pow(m,e,N)
def advanced_encrypt(a):    return encrypt_message(pow(a,3,N)+r)
# Sanity check                                                                                                                                                                          # assert(advanced_encrypt(ord('C')) == ciphertexts[0])# assert(advanced_encrypt(ord('y')) == ciphertexts[1])
# Build a dictionary of all ciphertexts                                                                                                                                                 ctpt = {}for char in chars:    ct = advanced_encrypt(ord(char))    ctpt[ct] = char
# Reconstruct the flag from the knows plaintext/ciphertext pairs                                                                                                                        flag = ''for ct in ciphertexts:    if(ct in ctpt):        flag += ctpt[ct]    else:        flag += "#"
print(flag)```- I ended up commenting out the assertion part cause it kept failing for some reason, then after executing the code we get our flag ! 
# Flag 
CyberErudites{Fr4nkl1n_W3_n33d_an0th3R_S3450N_A54P}