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QUESTION

# 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 t

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).

Pair1

n=193765551361183401239207555249755400140519897453010533905950690837860828020980125726807862823713249849740039582554958092558999122674305977576569682730442603540319182943620304523686454588551425540403591950153090745281138723004485819093766104395198530460977646776063911557363334688060524431950462886603935522191

c=97499625229098299370915011118616207269882667596583244584929391517284710377071479929121898206079211480089650142489595033992731600286042973142916255101603036948982508449834654645769220850776268099268035589878857495001023619101622182314534839453862938921309952353999537180757873940181505086046514399470593180755Pair 2n=193765551361183401239207555249755400140519897453010533905950690837860828020980125726807862823713249849740039582554958092558999122674305977576569682730511479549490538467957450975137242174938680086274764599967779421612822607651037696175012230129288442784965445880288222842765206349996520862260669810042904755331c=96359885319239204081269127232490992401527456309110927701552951264653065546397769140461112801759859518621493548144577373298119200402280828860449770482862419467862584625947178705168882393700370760090605619825016560574688102776016777996715389791008637373732294764875949322075043672865249466245671210566765472398

Pair3

n=193765551361183401239207555249755400140519897453010533905950690837860828020980125726807862823713249849740039582554958092558999122674305977576569682730573200694330027230065556748324952797016506815537654561422575216045863501855324483357357687183644681920891456637792385381057829419726554850133640201823107094597

c=116607043646753723628130375735876229854236638652688866205136391943163595987932190087503979666926608431529512185983948310793380741029462515527845084455089271232148070414098457717640104572239855259597767203632738967823548423866418162307624771371216190684681349029640216469063773531654206961191900693977019185181

Pair4

n=193765551361183401239207555249755400140519897453010533905950690837860828020980125726807862823713249849740039582554958092558999122674305977576569682730602516150133356804490381005955380981521672149734236938243459280790108906274410629908106341346133233932677694494198286667851754585457599410002940076674058998527

c=4221040776377023533309422379866496191928024742117345663553065142114357921495153525441055944214823399855361752623409910801650609256270313259666752524587838129719686329408930497130473483131158534679015378110913666477817345054341454965069543592130205480164624560640265345846093961466485677028657534715650031777

Pair 5

n=193765551361183401239207555249755400140519897453010533905950690837860828020980125726807862823713249849740039582554958092558999122674305977576569682730623563144043439575872306113997739678089483158901014029294350404196233812011190427431720759719201937941139608852643549130165341883931169863242437422720897867319

c=136343179443469262288642415876060165741628039871798557672413893637439186307020836684038434876700162678301626684522917850895622084897352735593977705818920176605461969456590234532225969551351876449006788535830883701880072667574538177873721072923967711985078690576703084786670517050059520564598352688960939268619Pair 6

n=193765551361183401239207555249755400140519897453010533905950690837860828020980125726807862823713249849740039582554958092558999122674305977576569682730767551308729878853104206456319272983180380697803885874420288089720675627448525232712320987001624341554585403908038916134247185148091628043341220853612793492303

c=81204187126951921384367188446149364563257577360012174913889497240203645929901535395383411838401057446242987902021104621613120810852036461962295023683311640077960036440865753025019353761613917638132779975170552615780615826848216174505689502499752850786375159865794324131851606796159364821742973765134044150271

Pair7

n=193765551361183401239207555249755400140519897453010533905950690837860828020980125726807862823713249849740039582554958092558999122674305977576569682730841549866683582565185101875865026178256097182414062472321437039473956049999266901783441521440270658028780864707969481775556067475264578128937231363920366214179

c=121362003580460503685131949448138888030226532296375411989347859416494403155780011137700607120936252024609797987952081713302135056337526433576508586152291635975971322126271877395438894072138885834209945278455765583189843040046130745420912280166592769645885840732501455588061656622252240556216147175462060351260Pair8

n=193765551361183401239207555249755400140519897453010533905950690837860828020980125726807862823713249849740039582554958092558999122674305977576569682730818303517589774213195806286955807247523131319114778251385598298674863065488405617481565808025571758760175443425824251410434578858564272138917416173511477021681

c=134316691473295140689180847955330885229770021418881974794889043448597625619841327966532129537982994595340927433306411333295732982117744750665730799800115861219744963359540520573317131307427379398871295809849721475709900299020158866506861350282312438916641869568040969248721648277080696787419979114561266195866

Pair9

n=193765551361183401239207555249755400140519897453010533905950690837860828020980125726807862823713249849740039582554958092558999122674305977576569682730636536555633517051353757304933902578354826982011752289122016096666146783007671096275641816547045874538948143364595258531538531938281280751085831289464055511251

c=88816737848815388688859679546903919370689078160537448053858703529730031199224016745607082416213385921042341787494022480051096407267308385459631636084009033161620323986555194852922131198121882377436067935103849938414006171237884719123695983395126517824870477903948721204808371798633639411624156499506857866683

Pair10

n=193765551361183401239207555249755400140519897453010533905950690837860828020980125726807862823713249849740039582554958092558999122674305977576569682730916411462747355808896737610687648579143456711050919083215413535292831541436040546678943070388923759984804790250111427226827543990602449754282480522015588704277

c=150312514974789990254935504359112806965440220477121895510752036347033154915916663045941701346201856527621322942292787841628231434215142389806531608621426978336065854308718679150773328319071060009080534334790077950920005249398351553906683456797285256182206102465084120283507984008622071451227426415049634396176

Pair11

n=193765551361183401239207555249755400140519897453010533905950690837860828020980125726807862823713249849740039582554958092558999122674305977576569682731538272180961005100375654460346598586068317176339675541950539227783059080381080675362772783615425931197785982081582098682778072411242850831016147477900485463447

c=58879325345923995727182420602772187792799670207804797016376517168474560976632335519837884502204219992422655311407088666452059854900569941165915051550931589650882505147738054446394678069863333640268864291908255004217835689283772195338579883647056008692479205832340006349859266319383002830051495055572390708787

Pair12

n=193765551361183401239207555249755400140519897453010533905950690837860828020980125726807862823713249849740039582554958092558999122674305977576569682730936344859096058539464804247405014752083341211796173576856204599259478938932779085074323921652306289178004169457316199611770042014024508543329411870917108169869

c=16857225327213013265523273757440335963532962784981814095407902031153118251195296612705759465249281038376086174450447746731526292607977065314154009021268267716978060835186841586724151737647591980290936612413314842523361723263768157696681103670352373768792359330600937685571908381387819985405018910225186014638

Pair13

n=193765551361183401239207555249755400140519897453010533905950690837860828020980125726807862823713249849740039582554958092558999122674305977576569682730981306360636777687324340027151587893560873989315148447288994499128383334124444922376754193705885597344229026188650986485627771494229291985104264005660064344971

c=71624743893724254978643781526410254295591953037544517198322560144956779737821262430966943012025846237626331129760141922011779738641587794917201686062067060264284806790413294220692235754532641074803296882888980758731513072968260635565859021471654350234903677872466051684541438165238699532917068473348594584088

Pair14

n=193765551361183401239207555249755400140519897453010533905950690837860828020980125726807862823713249849740039582554958092558999122674305977576569682731073707118292286573854246685342842541588076025590985967392046931119029580341868398541934424632095952640637880362830650655493904091628472321482427618794366493641

c=191079146435602516649210100756921711835871045593651588842427293495695963078941770894432298822668535904846814719238794824646131992880865787089403309896795923418002572174955829025562826183573237796100520762404365818922915241366173611694152034842518757472838976767873306662414450293092024210539866578757056806424

Pair15

n=193765551361183401239207555249755400140519897453010533905950690837860828020980125726807862823713249849740039582554958092558999122674305977576569682731092944516165126778913651777482088387260506458572788837651525457936003535188581149982169296451924622574298175100510116742079471318143336320409005219480058630039

c=96164929144694978762639177567555787226851401793968228132838253050960711532348079492337101366837431460289699964878057751660551816768000366480783227574914731282232029299226610576516508776555974541758507956832752137180068569585872255907445257473709629221033670378570273147028330924192781708107381925107112512193

Pair16

n=193765551361183401239207555249755400140519897453010533905950690837860828020980125726807862823713249849740039582554958092558999122674305977576569682730851878484065382443733639197404331834905115548023684563300115090775109938925649580197807856382609929440340878235725027243172920501367348814323280987443358097221

c=161331351671201721423282206243110584071957974685055676226967631934174204774373530509851290796667846165385476049005183351048457841341565216279276136568857746870728879636759458878993221871012110768874975236804660817546685562256072757235562780091326625464955769269133606355566055059501996509200024605051161416239

Pair17

n=193765551361183401239207555249755400140519897453010533905950690837860828020980125726807862823713249849740039582554958092558999122674305977576569682731543088490413973776895592348827191250759099325262760775749221984858799303519259121092382961364818637273796446610697218267408033525840638778648995415554072510321

c=165025535603172498279505331728719708361496318691538126472626230463672641564628267162454933683716468118051296992743098638543698689452979234393648532415741332644914088589286138250894323647782440749625447222035428279763187974015230134048867304925784330315011694706361602797347161121967435433969596786439265122557

Pair18

n=193765551361183401239207555249755400140519897453010533905950690837860828020980125726807862823713249849740039582554958092558999122674305977576569682731643173071243004945041122459160894311818994269647104217981964826981840356592967285242374805347863598795516793328436263971134682121095249597955864408991342980431

c=13319708813460889642827401484101453935982035887033412395242259896662083245906859668384809876816411986205372701296427408187348395404205554840357747917708446479386511800842060558282157241931541414449695309357506444793244456783088489675872185163968988773362097253554916828454135619149513411007590367346450832438

Pair19

n=193765551361183401239207555249755400140519897453010533905950690837860828020980125726807862823713249849740039582554958092558999122674305977576569682731744649649023761164203975658650838027413797349558350906977332743404228294966727023302658608218016278836315308985225922271575192310164249997503440898860312897137

c=168277777683859463243002952421965559741009209683879307535821915012301104514460296296585566986534598226748065783548323545163847878587427777704982605984373631725846258093154898446509531293163414334735056623068266867390504173620399992660330145554752817396547328277480522401826454096189005638445010659313821992329

Pair20

n=193765551361183401239207555249755400140519897453010533905950690837860828020980125726807862823713249849740039582554958092558999122674305977576569682731776804778608609842704139018159997147170175280229815907193971959719141345397918380630402858510204576627021011477295073255665395127276649301063784066431967233207

c=9843235703784063725077472029114219770521198196798716723770289323061129632065825646584976153848366418643437107513083603705756101212732157389730475892127203122295602748285083193988222178306733931241052159607193946978863283853504111229303288856848143358672633038099728013528075496121949664915655364645564509340