Joe is an overworked and under-appreciated programmer at This Secure World company. He is asked to write an RSA key generation algorithm...

Joe is an overworked and under-appreciated programmer at "This Secure World" company. He is asked to write an RSA key generation algorithm that performs better than the competition. To increase the efficiency of his algorithm he decides instead of generating two random prime factors for the modulus part of every RSA key pair, to reuse one of the previous factors and only generate one new random prime number for the new pair.

For instance, RSA moduli (n values) generated by Joe's algorithm would be of the form::

n1 = p1 × q1

n2 = p2 × q1

n3 = p2 × q2

n4 = p3 × q2

. . .

This has increased the efficiency of his algorithm by reducing the time required to test the primality of the randomly generated numbers and for the first time in quite a while he is praised by his supervisor for the surprisingly good performing algorithm. The company is going to embed this algorithm in all of their hardware and software products. You are tasked with evaluation of the security of Joe's approach by either approving or rejecting Joe's idea. You are given 20 public keys (modulus n) where some are generated using Joe's algorithm and others using a different algorithm. For all 20 samples, the value of e is 65537. Also, encryptions of 20 randomly generated plaintexts with respect to each public key are provided (c values in sample file).

• If you accept Joe's idea, first clearly state it. Then, you must explain in detail why his method is secure and why the plaintexts cannot be recovered.

• If you reject Joe's idea, first clearly state it. Then, you must demonstrate in detail how you can recover the plaintexts for keys generated using Joe's algorithm.

Also, provide all the recovered plaintexts in decimal format (base 10).

Plaintexts:

Pair 1

n=193765551361183401239207555249755400140519897453010533905950690837860828020980125726807862823713249849740039582554958092558999122674305977576569682730442603540319182943620304523686454588551425540403591950153090745281138723004485819093766104395198530460977646776063911557363334688060524431950462886603935522191

c=97499625229098299370915011118616207269882667596583244584929391517284710377071479929121898206079211480089650142489595033992731600286042973142916255101603036948982508449834654645769220850776268099268035589878857495001023619101622182314534839453862938921309952353999537180757873940181505086046514399470593180755

Pair 2

n=193765551361183401239207555249755400140519897453010533905950690837860828020980125726807862823713249849740039582554958092558999122674305977576569682730511479549490538467957450975137242174938680086274764599967779421612822607651037696175012230129288442784965445880288222842765206349996520862260669810042904755331

c=96359885319239204081269127232490992401527456309110927701552951264653065546397769140461112801759859518621493548144577373298119200402280828860449770482862419467862584625947178705168882393700370760090605619825016560574688102776016777996715389791008637373732294764875949322075043672865249466245671210566765472398

Pair 3

n=193765551361183401239207555249755400140519897453010533905950690837860828020980125726807862823713249849740039582554958092558999122674305977576569682730573200694330027230065556748324952797016506815537654561422575216045863501855324483357357687183644681920891456637792385381057829419726554850133640201823107094597

c=116607043646753723628130375735876229854236638652688866205136391943163595987932190087503979666926608431529512185983948310793380741029462515527845084455089271232148070414098457717640104572239855259597767203632738967823548423866418162307624771371216190684681349029640216469063773531654206961191900693977019185181

Pair 4

n=193765551361183401239207555249755400140519897453010533905950690837860828020980125726807862823713249849740039582554958092558999122674305977576569682730602516150133356804490381005955380981521672149734236938243459280790108906274410629908106341346133233932677694494198286667851754585457599410002940076674058998527

c=4221040776377023533309422379866496191928024742117345663553065142114357921495153525441055944214823399855361752623409910801650609256270313259666752524587838129719686329408930497130473483131158534679015378110913666477817345054341454965069543592130205480164624560640265345846093961466485677028657534715650031777

Pair 5

n=193765551361183401239207555249755400140519897453010533905950690837860828020980125726807862823713249849740039582554958092558999122674305977576569682730623563144043439575872306113997739678089483158901014029294350404196233812011190427431720759719201937941139608852643549130165341883931169863242437422720897867319

c=136343179443469262288642415876060165741628039871798557672413893637439186307020836684038434876700162678301626684522917850895622084897352735593977705818920176605461969456590234532225969551351876449006788535830883701880072667574538177873721072923967711985078690576703084786670517050059520564598352688960939268619

Pair 6

n=193765551361183401239207555249755400140519897453010533905950690837860828020980125726807862823713249849740039582554958092558999122674305977576569682730767551308729878853104206456319272983180380697803885874420288089720675627448525232712320987001624341554585403908038916134247185148091628043341220853612793492303

c=81204187126951921384367188446149364563257577360012174913889497240203645929901535395383411838401057446242987902021104621613120810852036461962295023683311640077960036440865753025019353761613917638132779975170552615780615826848216174505689502499752850786375159865794324131851606796159364821742973765134044150271

Pair 7

n=193765551361183401239207555249755400140519897453010533905950690837860828020980125726807862823713249849740039582554958092558999122674305977576569682730841549866683582565185101875865026178256097182414062472321437039473956049999266901783441521440270658028780864707969481775556067475264578128937231363920366214179

c=121362003580460503685131949448138888030226532296375411989347859416494403155780011137700607120936252024609797987952081713302135056337526433576508586152291635975971322126271877395438894072138885834209945278455765583189843040046130745420912280166592769645885840732501455588061656622252240556216147175462060351260

Pair 8

n=193765551361183401239207555249755400140519897453010533905950690837860828020980125726807862823713249849740039582554958092558999122674305977576569682730818303517589774213195806286955807247523131319114778251385598298674863065488405617481565808025571758760175443425824251410434578858564272138917416173511477021681

c=134316691473295140689180847955330885229770021418881974794889043448597625619841327966532129537982994595340927433306411333295732982117744750665730799800115861219744963359540520573317131307427379398871295809849721475709900299020158866506861350282312438916641869568040969248721648277080696787419979114561266195866

Pair 9

n=193765551361183401239207555249755400140519897453010533905950690837860828020980125726807862823713249849740039582554958092558999122674305977576569682730636536555633517051353757304933902578354826982011752289122016096666146783007671096275641816547045874538948143364595258531538531938281280751085831289464055511251

c=88816737848815388688859679546903919370689078160537448053858703529730031199224016745607082416213385921042341787494022480051096407267308385459631636084009033161620323986555194852922131198121882377436067935103849938414006171237884719123695983395126517824870477903948721204808371798633639411624156499506857866683

Pair 10

n=193765551361183401239207555249755400140519897453010533905950690837860828020980125726807862823713249849740039582554958092558999122674305977576569682730916411462747355808896737610687648579143456711050919083215413535292831541436040546678943070388923759984804790250111427226827543990602449754282480522015588704277

c=150312514974789990254935504359112806965440220477121895510752036347033154915916663045941701346201856527621322942292787841628231434215142389806531608621426978336065854308718679150773328319071060009080534334790077950920005249398351553906683456797285256182206102465084120283507984008622071451227426415049634396176

Pair 11

n=193765551361183401239207555249755400140519897453010533905950690837860828020980125726807862823713249849740039582554958092558999122674305977576569682731538272180961005100375654460346598586068317176339675541950539227783059080381080675362772783615425931197785982081582098682778072411242850831016147477900485463447

c=58879325345923995727182420602772187792799670207804797016376517168474560976632335519837884502204219992422655311407088666452059854900569941165915051550931589650882505147738054446394678069863333640268864291908255004217835689283772195338579883647056008692479205832340006349859266319383002830051495055572390708787

Pair 12

n=193765551361183401239207555249755400140519897453010533905950690837860828020980125726807862823713249849740039582554958092558999122674305977576569682730936344859096058539464804247405014752083341211796173576856204599259478938932779085074323921652306289178004169457316199611770042014024508543329411870917108169869

c=16857225327213013265523273757440335963532962784981814095407902031153118251195296612705759465249281038376086174450447746731526292607977065314154009021268267716978060835186841586724151737647591980290936612413314842523361723263768157696681103670352373768792359330600937685571908381387819985405018910225186014638

Pair 13

n=193765551361183401239207555249755400140519897453010533905950690837860828020980125726807862823713249849740039582554958092558999122674305977576569682730981306360636777687324340027151587893560873989315148447288994499128383334124444922376754193705885597344229026188650986485627771494229291985104264005660064344971

c=71624743893724254978643781526410254295591953037544517198322560144956779737821262430966943012025846237626331129760141922011779738641587794917201686062067060264284806790413294220692235754532641074803296882888980758731513072968260635565859021471654350234903677872466051684541438165238699532917068473348594584088

Pair 14

n=193765551361183401239207555249755400140519897453010533905950690837860828020980125726807862823713249849740039582554958092558999122674305977576569682731073707118292286573854246685342842541588076025590985967392046931119029580341868398541934424632095952640637880362830650655493904091628472321482427618794366493641

c=191079146435602516649210100756921711835871045593651588842427293495695963078941770894432298822668535904846814719238794824646131992880865787089403309896795923418002572174955829025562826183573237796100520762404365818922915241366173611694152034842518757472838976767873306662414450293092024210539866578757056806424

Pair 15

n=193765551361183401239207555249755400140519897453010533905950690837860828020980125726807862823713249849740039582554958092558999122674305977576569682731092944516165126778913651777482088387260506458572788837651525457936003535188581149982169296451924622574298175100510116742079471318143336320409005219480058630039

c=96164929144694978762639177567555787226851401793968228132838253050960711532348079492337101366837431460289699964878057751660551816768000366480783227574914731282232029299226610576516508776555974541758507956832752137180068569585872255907445257473709629221033670378570273147028330924192781708107381925107112512193

Pair 16

n=193765551361183401239207555249755400140519897453010533905950690837860828020980125726807862823713249849740039582554958092558999122674305977576569682730851878484065382443733639197404331834905115548023684563300115090775109938925649580197807856382609929440340878235725027243172920501367348814323280987443358097221

c=161331351671201721423282206243110584071957974685055676226967631934174204774373530509851290796667846165385476049005183351048457841341565216279276136568857746870728879636759458878993221871012110768874975236804660817546685562256072757235562780091326625464955769269133606355566055059501996509200024605051161416239

Pair 17

n=193765551361183401239207555249755400140519897453010533905950690837860828020980125726807862823713249849740039582554958092558999122674305977576569682731543088490413973776895592348827191250759099325262760775749221984858799303519259121092382961364818637273796446610697218267408033525840638778648995415554072510321

c=165025535603172498279505331728719708361496318691538126472626230463672641564628267162454933683716468118051296992743098638543698689452979234393648532415741332644914088589286138250894323647782440749625447222035428279763187974015230134048867304925784330315011694706361602797347161121967435433969596786439265122557

Pair 18

n=193765551361183401239207555249755400140519897453010533905950690837860828020980125726807862823713249849740039582554958092558999122674305977576569682731643173071243004945041122459160894311818994269647104217981964826981840356592967285242374805347863598795516793328436263971134682121095249597955864408991342980431

c=13319708813460889642827401484101453935982035887033412395242259896662083245906859668384809876816411986205372701296427408187348395404205554840357747917708446479386511800842060558282157241931541414449695309357506444793244456783088489675872185163968988773362097253554916828454135619149513411007590367346450832438

Pair 19

n=193765551361183401239207555249755400140519897453010533905950690837860828020980125726807862823713249849740039582554958092558999122674305977576569682731744649649023761164203975658650838027413797349558350906977332743404228294966727023302658608218016278836315308985225922271575192310164249997503440898860312897137

c=168277777683859463243002952421965559741009209683879307535821915012301104514460296296585566986534598226748065783548323545163847878587427777704982605984373631725846258093154898446509531293163414334735056623068266867390504173620399992660330145554752817396547328277480522401826454096189005638445010659313821992329

Pair 20

n=193765551361183401239207555249755400140519897453010533905950690837860828020980125726807862823713249849740039582554958092558999122674305977576569682731776804778608609842704139018159997147170175280229815907193971959719141345397918380630402858510204576627021011477295073255665395127276649301063784066431967233207

c=9843235703784063725077472029114219770521198196798716723770289323061129632065825646584976153848366418643437107513083603705756101212732157389730475892127203122295602748285083193988222178306733931241052159607193946978863283853504111229303288856848143358672633038099728013528075496121949664915655364645564509340