Talk:Kunerth's algorithm - Knowledge (XXG)

2097:

square) this is a square*2 that will make the modulus be a multiple of RSA260. In:= sq = square /. y1 Out= {\ 1310074348496985087958227895934029051306378242573836062042042519166787\ 1151633616076909466726506246372628441650527102960316106219092081078278\ 2697034448186243471410872828623439969212125671314423500663717755571283\ 62798348048332822650444386978185719854262751334260} make the modulus and ensure it is a multiple of RSA260 In:= mod = ( sq - 1) ((RSA260 - 4)^2 - 3^2)^2 + sq (2 3 (RSA260 - 4))^2 + 31 sq ((RSA260 - 4)^2 + 3^2)^2 Out= {\ 1002358041414149256141274976883084858155451228382361129174692948058420\ 3599822966056879419332122392602397619194035843768768709541019270410555\ 6433865054516535536095433805068898049589154927235644010444905709641657\ 5664206624064244511391622161730259643289116736731448910028590146704947\ 3161788407261817986244137659489378985393013818434265570008614736512776\ 3545618430509765369662358402759029772411007313696489988416538699895793\ 2062486200292789019019944056514768367832365326072002803950843223356312\ 4166954570325030969756112101858031090970125818758607383491205471097319\ 2814060103436411355386558984309895156603054428229565301682429394578997\ 6942467702973118538396177479356022104478800105886098857119937058438326\ 5882291105445591594288115650278960467162451625164324314977683968541369\ 2081124697290064794747604396221843134077697331584288261403440452001303\ 0621478551564289565584772940068021559707725584052454208322434397897947\ 8871328222771961549263669750618180963391086959070362771744490889921681\ 5532811395394495346475476550388193475366787875136485329364485382293503\ 2066672551241188787138210654359505034219224134083222406474511360502171\ 8752529498150360411729367426811841015002456997874675195860846076561970\ 3321287953956776844475358091709648072725163365318816992866389964393118\ 360635870348680478314650421801249969664} modulus is multiple of RSA260 In:= Mod Out= {0} make the W variable in Kunerth's method In:= w1 = (((((1 (PowerMod[1, 1, RSA260] ((RSA260 - 4)^2 - 3^2)))^2 + (sq - 1) ((RSA260 - 4)^2 - 3^2)^2 + sq (2 3 (RSA260 - 4))^2) + (31) sq ((RSA260 - 4)^2 + 3^2)^2)/(( 2 sq))))^(1/2) Out= {\ 1955908211597675891140626252676009606866002222291707872958133034542665\ 8136981765170159272094426125800150518378794141701672722015496988945017\ 1997008661915134996647219327852545616155276501393879302125479496989475\ 0334488015844182921809722048018066266159436371799686019397863583980927\ 2518900637313552476000541370758002609991296119013933393490928922710232\ 5559298675115198076553162501416832541260653853680479089845834176986483\ 8838927812903169227782727641691137148863931697158492481780996676539958\ 573959447769493551280410632136} make the V variable in Kunerth's method In:= v1 = 1 (PowerMod ((RSA260 - 4)^2 - 3^2)) Out= 48897705289941897278515656316900240171650055557292696823953325\ 8635666453424544129253981802360653145003762959469853542541818050387424\ 7236254299925216547878374916180483196313640403881912534846982553136987\ 4247368758362200396104573045243051200451656653985909294992150484946589\ 5995231812972515932838811900013534268950065249782402975348334837273223\ 0677558138982466877879951913829062535420813531516346342011977246145854\ 4246620970973195322579230694568191042278428721598292428962312044524916\ 9134989643489861942373387820102658016 make the alpha variable in Kunerth's method In:= alpha1 = w1 v1 Out= {\ 9563942330488047218386506393582693870800543854518223556531938397032154\ 1513629873770693320222737701341332987395341333172862906303488542274050\ 1272240451561275108917114877394156536173935478378886386825609860440424\ 1811179004685535773787094974401423437736952868908225434566640747860098\ 3877982231006470294327231117537963340934886813968052527174610604544112\ 6446792134946419268141842319276017601204137720022890992661139113325319\ 9391187378239462665467103121813253475519495308197690941869365894281561\ 6546655629994820101244915485320888366429132996054452031763750688999910\ 0373429755223816490881797241928409251675605491355049233197310323467611\ 4507049622544668047364440656457744805636700155250482094869502994691294\ 5655709430147160222472286490593100993873738142913137560438652412311116\ 9071561761293061288370201477187340962214738792899421099831906504908793\ 4049878071431293908772739046972354503073523173903098439865856028936635\ 2054596141074097522698141038711562563733125043318767121833784330024745\ 9723593764289212411359121432052711136049324198680387602176} make the beta variable in Kunerth's method In:= beta1 = 0 Out= 0 Factor according to the Kunerth equation In:= Timing[ xx = Factor[ alpha1^2 x^2 + (2 alpha1 beta1 - mod) x + (beta1^2 - (( sq 2)))]] Out= {0.047995, {24 \ (-10917286237474875732985232466116908760886485354781967183683687659723\ 2259596946800640912222720885386438570347087725858002634218492434008985\ 6522475287068218695595090606905195333076767713927620195838864314629760\ 696899862337361068554203698914848809987855229277855 + 99624399275917158524859441599819727820838998484564828713874358302\ 4182724100311185111388752320184388972218618701472220550655273994672315\ 3546888252504703763282384553279972855797251811827899780066529433436046\ 2544185533114632140997643615572650014827476426592384460681610069174456\ 8760248728981572984065565908657474353784801405070978833880491402193797\ 3345067154084739025200709810857492458516679209767916905114506886865763\ 8054160324868523327736098615657518888057036661409460392613977805894732\ 0996005694329479112709387967869602085974849513712416680466426530556215\ 0115212210198057401771836041971430019188274680404183177939653719160316\ 5185624546036599462386474619066113157595981140178763390492757655425891\ 7895506568954194884391578400791041573151722439675871412894520897220137\ 7513691449660551498955665193174860972081406774420167692152421536477155\ 4631290704450732796429043374529801725175915272569126118811847037056819\ 4526903003619973509088619920670091682213546952296180311014237393098563\ 2257178903551037678692408139606017931677877067086204510377344 x) (1 + 38255769321952188873546025574330775483202175418072894226127753588\ 1286166054519495082773280890950805365331949581365332691451625213954169\ 0962005088961806245100435668459509576626144695741913515545547302439441\ 7616967244716018742143095148379897605693750947811475632901738266562991\ 4403935511928924025881177308924470151853363739547255872210108698442418\ 1764505787168539785677072567369277104070404816550880091563970644556453\ 3012797564749512957850661868412487253013902077981232790763767477463577\ 1262466186622519979280404979661955366092589219798400430317146322271925\ 4348561498721987748251542014260906299905715927568724500065038550325026\ 1977636230718926163787972661803864209441588335772596861907540016340415\ 0831690723197030341675413672388164780717689573069608682548987484686765\ 5924095055824438568891331527432670885448437516532998801582268785244811\ 5725056768515490056522517526092931916368473123518846795163769664618412\ 8232485683379058765290151535590994486530507835318613310707507511324791\ 9192921089786515845081406147362745535924783600832486911115922496 x)}} construct the root In:= y26 = Mod[alpha1 (-1 PowerMod[ 382557693219521888735460255743307754832021754180728942261277535\ 8812861660545194950827732808909508053653319495813653326914516252139541\ 6909620050889618062451004356684595095766261446957419135155455473024394\ 4176169672447160187421430951483798976056937509478114756329017382665629\ 9144039355119289240258811773089244701518533637395472558722101086984424\ 1817645057871685397856770725673692771040704048165508800915639706445564\ 5330127975647495129578506618684124872530139020779812327907637674774635\ 7712624661866225199792804049796619553660925892197984004303171463222719\ 2543485614987219877482515420142609062999057159275687245000650385503250\ 2619776362307189261637879726618038642094415883357725968619075400163404\ 1508316907231970303416754136723881647807176895730696086825489874846867\ 6559240950558244385688913315274326708854484375165329988015822687852448\ 1157250567685154900565225175260929319163684731235188467951637696646184\ 1282324856833790587652901515355909944865305078353186133107075075113247\ 919192921089786515845081406147362745535924783600832486911115922496 , \ -1, RSA260]) + beta1, RSA260] Out= {\ 2056492774246258978481140928717439476267924323729711431680478593678980\ 5335503861847456838645709864337149887306929202047588790250241599127012\ 8969941829374979202315995831917966362338217358971075796467647080228363\ 19356002506455439387143691264600758653834684346125} y26 is root of 2 sq In:= Mod == Mod Out= True the root is not one of the low inverses Table], -2, RSA260], RSA260] == x}, {x, 1, 100}] // Grid

1618:

1)^2) - 2 (RSA260^2 - 1)^2 - (2 1 RSA260)^2)/(-1))^(1/2) Out= 48897705289941897278515656316900240171650055557292696823953325\ 8635666453424544129253981802360653145003762959469853542541818050387424\ 7236254299925216547878374916180483196313640403881912534846982553136987\ 4247368758362200396104573045243051200451656653985909295009840745370213\ 3326714061654556142686232796729554029073220559768956005931866053289115\ 1695936264562194804751978316341066629080189896385088450648277821022198\ 3317787938356586115384593121867887478906081852529924976333801078939072\ 1301760772016869260727399301258323602 In:= v = (RSA260^2 - 1) Out= 48897705289941897278515656316900240171650055557292696823953325\ 8635666453424544129253981802360653145003762959469853542541818050387424\ 7236254299925216547878374916180483196313640403881912534846982553136987\ 4247368758362200396104573045243051200451656653985909295009840745370213\ 3326714061654556142686232796729554029073220559768956005931866053289115\ 1695936264562194804751978316341066629080189896385088450648277821022198\ 3317787938356586115384593121867887478906081852529924976333801078939072\ 1301760772016869260727399301258323600 In:= mod = -2 (RSA260^2 - 1)^2 - (2 1 RSA260)^2 Out= -4781971165244023609193253196791346935400271927259111778265969\ 1985160770756814936885346660111368850670666493697670666586431453151744\ 2711370250636120225780637554458557438697078268086967739189443193412804\ 9302202120905589502342767886893547487200711718868476434457572769846107\ 0779502776905976101427525915982884866302577443003583425421519721143731\ 1581794064984719510629844926745913046485178575326488301648428704124794\ 3077087313577288574842779067916119983682237916726558930943954396419758\ 4938176174479898752099204847165614981647667561896461244066743182500661\ 0726271597370219506195692307132921311229935371896691967627882162994221\ 5254113226582281041880468938995205313881101168610438216301537080461862\ 9021037268827123481880256026406850793159040334688592213201131483445273\ 6638821994514114746077142302601864843165084705043074367971789265654275\ 2125200928636991668349302806176203387917056163741814786488245113405979\ 3092448626793345719672585727633127359247751769700773062998076924975661\ 8909418568298155559461293692441859519326899815869330273769667214404 In:= alpha = v w Out= 23909855826220118045966265983956734677001359636295558891329845\ 9925803853784074684426733300556844253353332468488353332932157265758721\ 3556851253180601128903187772292787193485391340434838695947215967064024\ 6511010604527947511713839434467737436003558594342382172287863849230535\ 3897513884529880507137629579914424331512887215017917127107598605718655\ 7908970324923597553149224633729565232425892876632441508242143520623971\ 5385436567886442874213895339580599918411189583632794654719771982098792\ 4690880872399493760496024235828074908238337809482306220333715912503305\ 3631357986851097530978461535664606556149676859483459838139410814971107\ 6270566132911405209402344694976026569405505843052191081507685402309314\ 5105186344135617409401280132034253965795201673442961066005657417226368\ 3194109972570573730385711513009324215825423525215371839858946328271376\ 0626004643184958341746514030881016939585280818709073932441225567029896\ 5462243133966728598362928638165636796238758848503865314990384624878309\ 454709284149077779730646846220929759663449907934665136884833607200 In:= beta = 0 Out= 0 In:= Timing[ xx = Factor] Out= {0.015001, (1 + 239098558262201180459662659839567346770013596362955588913298459925\ 8038537840746844267333005568442533533324684883533329321572657587213556\ 8512531806011289031877722927871934853913404348386959472159670640246511\ 0106045279475117138394344677374360035585943423821722878638492305353897\ 5138845298805071376295799144243315128872150179171271075986057186557908\ 9703249235975531492246337295652324258928766324415082421435206239715385\ 4365678864428742138953395805999184111895836327946547197719820987924690\ 8808723994937604960242358280749072603837036834682747634027861653005597\ 0279757396389916136708704918932270811859768983802020921977520710323678\ 6721622028968457922672100313014834070015120946235831840757383830417024\ 4099616286480128906527345493065906279576655208696051007963932173351863\ 3127907115717622366374703427899345923324023616471839243552476899116514\ 0508519837719685358562078470982142940369226140343661144042344482927482\ 9661162765989427987241984992810374716271261961734525530268753001878176\ 58685649551103709068064778326238119416169413210338282316960000 x) (1 \ + 23909855826220118045966265983956734677001359636295558891329845992580\ 3853784074684426733300556844253353332468488353332932157265758721355685\ 1253180601128903187772292787193485391340434838695947215967064024651101\ 0604527947511713839434467737436003558594342382172287863849230535389751\ 3884529880507137629579914424331512887215017917127107598605718655790897\ 0324923597553149224633729565232425892876632441508242143520623971538543\ 6567886442874213895339580599918411189583632794654719771982098792469088\ 0872399493760496024235828074909216291915281144165904029038841310166568\ 7997962556070343252200837319885218167742068539474186623877871182886246\ 0103619913573012422179920751655404010174009758579831295066235587318596\ 2726642586805911907529519201340962445681365052526910518441235401452490\ 7154429575698534785555675858497058514648069096495793537408852840473795\ 8434386144714957171853914935664876524714804113498768046899611500344152\ 0151656858253927133077831992555005890570811557177427742374456431091759\ 882648604455752225627663533281207483646456119935487350254404 x)} In:= y25 = Mod[alpha (-1 PowerMod[ 239098558262201180459662659839567346770013596362955588913298459\ 9258038537840746844267333005568442533533324684883533329321572657587213\ 5568512531806011289031877722927871934853913404348386959472159670640246\ 5110106045279475117138394344677374360035585943423821722878638492305353\ 8975138845298805071376295799144243315128872150179171271075986057186557\ 9089703249235975531492246337295652324258928766324415082421435206239715\ 3854365678864428742138953395805999184111895836327946547197719820987924\ 6908808723994937604960242358280749092162919152811441659040290388413101\ 6656879979625560703432522008373198852181677420685394741866238778711828\ 8624601036199135730124221799207516554040101740097585798312950662355873\ 1859627266425868059119075295192013409624456813650525269105184412354014\ 5249071544295756985347855556758584970585146480690964957935374088528404\ 7379584343861447149571718539149356648765247148041134987680468996115003\ 4415201516568582539271330778319925550058905708115571774277423744564310\ 91759882648604455752225627663533281207483646456119935487350254404, \ -1, (1 + RSA260^4)/2]) + beta, (1 + RSA260^4)/2] Out= 11954927913110059022983132991978367338500679818147779445664922\ 9962901926892037342213366650278422126676666234244176666466078632879360\ 6778425626590300564451593886146393596742695670217419347973607983532012\ 3255505302263973755856919717233868718001779297171191086143931924615267\ 6948756942264940253568814789957212165756443607508958563553799302859327\ 8954485162461798776574612316864782616212946438316220754121071760311985\ 7692718283943221437106947669790299959205594791816397327359885991049396\ 2345440436199746880248012117914037453630191851841734137381701393082650\ 2798513987869819495806835435245946613540592988449190101046098876035516\ 1839336081101448422896133605015650741703500756047311791592037869191520\ 8512204980814324006445326367274653295313978832760434802550398196608667\ 5931656395355785881118318735171394967296166201180823591962177623844955\ 8257025425991885984267928103923549107147018461307017183057202117224146\ 3741483058138299471399362099249640518735813563098086726276513437650093\ 908829342824775551854534032389163119059708084706605169141158480000 show square root of -1 found In:= Mod Out= 1

620:

2941402001986512729726569746599085900330031400051170742204560859276357\ 9537571859542988389587092292384910067030341246205457845664136645406842\ 14361293017694020846391065875914794251435144458199 Out= 22112825529529666435281085255026230927612089502470015394413748\ 3191288229414020019865127297265697465990859003300314000511707422045608\ 5927635795375718595429883895870922923849100670303412462054578456641366\ 4540684214361293017694020846391065875914794251435144458199 In:= w = (((((RSA260^2 + 1)^2) + (2 (RSA260^2 + 1))^2 - 5 (2 1 RSA260)^2))/5)^(1/2) Out= 4889770528994189727851565631690024017165005555729269682395332\ 5863566645342454412925398180236065314500376295946985354254181805038742\ 4723625429992521654787837491618048319631364040388191253484698255313698\ 7424736875836220039610457304524305120045165665398590929500984074537021\ 3332671406165455614268623279672955402907322055976895600593186605328911\ 5169593626456219480475197831634106662908018989638508845064827782102219\ 8331778793835658611538459312186788747890608185252992497633380107893907\ 21301760772016869260727399301258323600 In:= v = RSA260^2 + 1 Out= 4889770528994189727851565631690024017165005555729269682395332\ 5863566645342454412925398180236065314500376295946985354254181805038742\ 4723625429992521654787837491618048319631364040388191253484698255313698\ 7424736875836220039610457304524305120045165665398590929500984074537021\ 3332671406165455614268623279672955402907322055976895600593186605328911\ 5169593626456219480475197831634106662908018989638508845064827782102219\ 8331778793835658611538459312186788747890608185252992497633380107893907\ 21301760772016869260727399301258323602 In:= mod = (2 (RSA260^2 + 1))^2 - 5 (2 1 RSA260)^2 Out= 9563942330488047218386506393582693870800543854518223556531938\ 3970321541513629873770693320222737701341332987395341333172862906303488\ 5422740501272240451561275108917114877394156536173935478378886386825609\ 8604404241811179004685535773787094974401423437736952868915145539692214\ 1559005553811952202855051831965769732605154886007166850843039442287462\ 3163588129969439021259689853491826092970357150652976603296857408249588\ 6154174627154577149685558135832239967364475833453117861887908792839516\ 9876352348959797504198409694331229962708562660313619720791298489198519\ 2631945188073563888772510215162214625018776214242336153627436488248397\ 8953090070739457065594891387023575576163099738646335933608908362567962\ 1193608708150065325854135614322733161217734264623929549859346352485127\ 4479165679912828311264840179603809117590314736373913942382823882959671\ 7783229582553271512772210918532055540659365989826605833180397427262996\ 3397331042972613646006648791466441263235224506016931008533739200204450\ 7996533545599151062793114696228097909389597200609931755924234545596 In:= alpha = v w Out= 2390985582622011804596626598395673467700135963629555889132984\ 5992580385378407468442673330055684425335333246848835333293215726575872\ 1355685125318060112890318777229278719348539134043483869594721596706402\ 4651101060452794751171383943446773743600355859434238217228786384923053\ 5389751388452988050713762957991442433151288721501791712710759860571865\ 5790897032492359755314922463372956523242589287663244150824214352062397\ 1538543656788644287421389533958059991841118958363279465471977198209879\ 2469088087239949376049602423582807490823833780948230622033371591250330\ 5363135798685109753097846153566460655614967685948345983813941081497110\ 7627056613291140520940234469497602656940550584305219108150768540230931\ 4510518634413561740940128013203425396579520167344296106600565741722636\ 8319410997257057373038571151300932421582542352521537183985894632827137\ 6062600464318495834174651403088101693958528081870907393244122556702989\ 6546224313396672859836292863816563679623875884850386531499038462487830\ 9454709284149077779730646846220929759663449907934665136884833607200 In:= beta = 0 Out= 0 In:= Timing[ xx = Factor] Out= {0.016756, (1 + 239098558262201180459662659839567346770013596362955588913298459925\ 8038537840746844267333005568442533533324684883533329321572657587213556\ 8512531806011289031877722927871934853913404348386959472159670640246511\ 0106045279475117138394344677374360035585943423821722878638492305353897\ 5138845298805071376295799144243315128872150179171271075986057186557908\ 9703249235975531492246337295652324258928766324415082421435206239715385\ 4365678864428742138953395805999184111895836327946547197719820987924690\ 8808723994937604960242358280749072603837036834682747634027861653005597\ 0279757396389916136708704918932270811859768983802020921977520710323678\ 6721622028968457922672100313014834070015120946235831840757383830417024\ 4099616286480128906527345493065906279576655208696051007963932173351863\ 3127907115717622366374703427899345923324023616471839243552476899116514\ 0508519837719685358562078470982142940369226140343661144042344482927482\ 9661162765989427987241984992810374716271261961734525530268753001878176\ 58685649551103709068064778326238119416169413210338282316960000 x) (-5 \ + 23909855826220118045966265983956734677001359636295558891329845992580\ 3853784074684426733300556844253353332468488353332932157265758721355685\ 1253180601128903187772292787193485391340434838695947215967064024651101\ 0604527947511713839434467737436003558594342382172287863849230535389751\ 3884529880507137629579914424331512887215017917127107598605718655790897\ 0324923597553149224633729565232425892876632441508242143520623971538543\ 6567886442874213895339580599918411189583632794654719771982098792469088\ 0872399493760496024235828074909216291915281144165904029038841310166568\ 7997962556070343252200837319885218167742068539474186623877871182886246\ 0103619913573012422179920751655404010174009758579831295066235587318596\ 2726642586805911907529519201340962445681365052526910518441235401452490\ 7154429575698534785555675858497058514648069096495793537408852840473795\ 8434386144714957171853914935664876524714804113498768046899611500344152\ 0151656858253927133077831992555005890570811557177427742374456431091759\ 882648604455752225627663533281207483646456119935487350254404 x)} In:= y25 = Mod[alpha (-1 PowerMod[ 239098558262201180459662659839567346770013596362955588913298459\ 9258038537840746844267333005568442533533324684883533329321572657587213\ 5568512531806011289031877722927871934853913404348386959472159670640246\ 5110106045279475117138394344677374360035585943423821722878638492305353\ 8975138845298805071376295799144243315128872150179171271075986057186557\ 9089703249235975531492246337295652324258928766324415082421435206239715\ 3854365678864428742138953395805999184111895836327946547197719820987924\ 6908808723994937604960242358280749072603837036834682747634027861653005\ 5970279757396389916136708704918932270811859768983802020921977520710323\ 6786721622028968457922672100313014834070015120946235831840757383830417\ 0244099616286480128906527345493065906279576655208696051007963932173351\ 8633127907115717622366374703427899345923324023616471839243552476899116\ 5140508519837719685358562078470982142940369226140343661144042344482927\ 4829661162765989427987241984992810374716271261961734525530268753001878\ 17658685649551103709068064778326238119416169413210338282316960000, \ -1, (-1 + RSA260 + RSA260^2)]) + beta , (-1 + RSA260^2 + RSA260)] Out= 4422565105905933287056217051005246185522417900494003078882749\ 6638257645882804003973025459453139493198171800660062800102341484409121\ 7185527159075143719085976779174184584769820134060682492410915691328273\ 29081368428722586035388041692782131751829588502870288916399 In:= Mod Out= 5

1443:

2211282552952966643528108525502623092761208950247001539441374831912882\ 2941402001986512729726569746599085900330031400051170742204560859276357\ 9537571859542988389587092292384910067030341246205457845664136645406842\ 143612930176940208463910658759147942514351444582} In:= q = GCD Out= {\ 7370941843176555478427028418342076975870696500823338464804582773042940\ 9804673339955042432421899155330286334433438000170569140681869530921193\ 1791906198476627965290307641283033556767804154018192818880455484689473\ 812043100589800694879702195863826475047838148606600} show that these two numbers equate to modulus In:= p q == (-1 + RSA260^2)/3 Out= {\ 1629923509664729909283855210563341339055001851909756560798444195452221\ 5114151470975132726745355104833458765315661784751393935012914157454180\ 9997507218262612497206016106543788013462730417828232751771232914157895\ 8612073346536819101508101706681721888466196976500328024845673777755713\ 5388485204756207759890985134302440685325631866864395535109637172319787\ 5485406493491732610544702220969339663212836281688275927367406611059293\ 1278552870512819770728929582630202728417664165877793369297969071005869\ 24005623086909133100419441200} == \ 1629923509664729909283855210563341339055001851909756560798444195452221\ 5114151470975132726745355104833458765315661784751393935012914157454180\ 9997507218262612497206016106543788013462730417828232751771232914157895\ 8612073346536819101508101706681721888466196976500328024845673777755713\ 5388485204756207759890985134302440685325631866864395535109637172319787\ 5485406493491732610544702220969339663212836281688275927367406611059293\ 1278552870512819770728929582630202728417664165877793369297969071005869\ 24005623086909133100419441200 Now show the Kunerth work Use Complex numbers in the denominator square is 8+6 I) root is 3+ I In:= w10 = ((((8 + 6 I + (8 + 6 I) ( RSA260^2 - 1))))/(8 + 6 I))^(1/2) Out= 22112825529529666435281085255026230927612089502470015394413748\ 3191288229414020019865127297265697465990859003300314000511707422045608\ 5927635795375718595429883895870922923849100670303412462054578456641366\ 4540684214361293017694020846391065875914794251435144458199 In:= v10 = 3 + I Out= 3 + I In:= mod = (8 + 6 I) (RSA260^2 - 1) Out= 3911816423195351782281252505352019213732004444583415745916266\ 0690853316273963530340318544188852251600301036757588283403345444030993\ 9778900343994017323830269993294438655705091232310553002787758604250958\ 9939789500668976031688365843619444096036132532318872743600787259629617\ 0666137124932364491414898623738364322325857644781516480474549284263129\ 2135674901164975584380158265307285330326415191710807076051862225681775\ 8665423035068526889230767449749430998312486548202393998106704086315125\ 770414086176134954085819194410066588800 + 293386231739651383671093937901401441029900333343756180943719955181399\ 8720547264775523890814163918870022577756819121255250908302324548341752\ 5799551299287270249497082899177881842423291475209081895318821924548421\ 2550173202376627438271458307202709939923915455770059044472221279996028\ 4369927336856117396780377324174439323358613736035591196319734691017561\ 7587373168828511869898046399774481139378310530703889666926133189990672\ 7630139516692307558731207324873436491115179549858002806473634432781056\ 4632101215564364395807549941600 I In:= alpha = v10 w10 Out= 6633847658858899930584325576507869278283626850741004618324124\ 4957386468824206005959538189179709239797257700990094200153512226613682\ 5778290738612715578628965168761276877154730201091023738616373536992409\ 93622052643083879053082062539173197627744382754305433374597 + 221128255295296664352810852550262309276120895024700153944137483191288\ 2294140200198651272972656974659908590033003140005117074220456085927635\ 7953757185954298838958709229238491006703034124620545784566413664540684\ 214361293017694020846391065875914794251435144458199 I In:= beta = 0 Out= 0 In:= Timing[ xx = Factor[ alpha^2 x^2 + (2 alpha beta - mod) x + (beta^2 - (8 + 6 I))]] Out= {0.078202, (8 + 6 I) (-1 + x) (1 + 488977052899418972785156563169002401716500555572926968239533258635\ 6664534245441292539818023606531450037629594698535425418180503874247236\ 2542999252165478783749161804831963136404038819125348469825531369874247\ 3687583622003961045730452430512004516566539859092950098407453702133326\ 7140616545561426862327967295540290732205597689560059318660532891151695\ 9362645621948047519783163410666290801898963850884506482778210221983317\ 7879383565861153845931218678874789060818525299249763338010789390721301\ 760772016869260727399301258323601 x)} In:= y221 = Mod[alpha (-1 PowerMod[ 889770528994189727851565631690024017165005555729269682395332586\ 3566645342454412925398180236065314500376295946985354254181805038742472\ 3625429992521654787837491618048319631364040388191253484698255313698742\ 4736875836220039610457304524305120045165665398590929500984074537021333\ 2671406165455614268623279672955402907322055976895600593186605328911516\ 9593626456219480475197831634106662908018989638508845064827782102219833\ 1778793835658611538459312186788747890608185252992497633380107893907213\ 01760772016869260727399301258323601, -1, ( RSA260^2 - 1)/3]) + beta , ( RSA260^2 - 1)/3] Out= 4196011502678228522651394674668421599244198885219399131026904\ 3289669343577155055101920106196150772487107982269688214060323403496639\ 0066796195613254878752266922653202654974394238033891827146108341535789\ 7064021932851214217179647002005696802218962975079798254077514295596533\ 6989462233404330808446291888976180740040612923112825511380322657407122\ 7936139526603078738677239607595847651818136269895877292246515919558991\ 1613999769739099050764205062230787783633643396360497705733443250472625\ 3412595543910269408426978715859232603 + 683174886644184253849664892676727832993140580143898824633711542782971\ 6490622325495108245788323394027389854514210202052475425120859352707380\ 9852944235379279729823778790680409145755373200180947861974937294932705\ 0632398256084927933902890462301206061991392267302693151468442382550111\ 9576306095200278982929534402768833992545638139334142600283449817227060\ 9381904789119818857101428995383717763400808003637642506441101909066430\ 2750820925196413425650669453664522356017904645483712564781743868377274\ 89305297498445370605426224601 I This above is supposed to be the square root of 8+6 I or 3+I In:= yyy = Solve[3 + II == 4196011502678228522651394674668421599244198885219399131026904328966\ 9343577155055101920106196150772487107982269688214060323403496639006679\ 6195613254878752266922653202654974394238033891827146108341535789706402\ 1932851214217179647002005696802218962975079798254077514295596533698946\ 2233404330808446291888976180740040612923112825511380322657407122793613\ 9526603078738677239607595847651818136269895877292246515919558991161399\ 9769739099050764205062230787783633643396360497705733443250472625341259\ 5543910269408426978715859232603 + 683174886644184253849664892676727832993140580143898824633711542782\ 9716490622325495108245788323394027389854514210202052475425120859352707\ 3809852944235379279729823778790680409145755373200180947861974937294932\ 7050632398256084927933902890462301206061991392267302693151468442382550\ 1119576306095200278982929534402768833992545638139334142600283449817227\ 0609381904789119818857101428995383717763400808003637642506441101909066\ 4302750820925196413425650669453664522356017904645483712564781743868377\ 27489305297498445370605426224601 II, {II}, Modulus -: -->

550:

0452794751171383943446773743600355859434238217228786384923053538975138\ 8452988050713762957991442433151288721501791712710759860571865579089703\ 2492359755314922463372956523242589287663244150824214352062397153854365\ 6788644287421389533958059991841118958363279465471977198209879246908808\ 7239949376049602423582807490726038370368346827476340278616530055970279\ 7573963899161367087049189322708118597689838020209219775207103236786721\ 6220289684579226721003130148340700151209462358318407573838304170244099\ 6162864801289065273454930659062795766552086960510079639321733518633127\ 9071157176223663747034278993459233240236164718392435524768991165140508\ 5198377196853585620784709821429403692261403436611440423444829274829661\ 1627659894279872419849928103747162712619617345255302687530018781765868\ 5649551103709068064778326238119416169413210338282316959996 Out= 2390985582622011804596626598395673467700135963629555889132984\ 5992580385378407468442673330055684425335333246848835333293215726575872\ 1355685125318060112890318777229278719348539134043483869594721596706402\ 4651101060452794751171383943446773743600355859434238217228786384923053\ 5389751388452988050713762957991442433151288721501791712710759860571865\ 5790897032492359755314922463372956523242589287663244150824214352062397\ 1538543656788644287421389533958059991841118958363279465471977198209879\ 2469088087239949376049602423582807490726038370368346827476340278616530\ 0559702797573963899161367087049189322708118597689838020209219775207103\ 2367867216220289684579226721003130148340700151209462358318407573838304\ 1702440996162864801289065273454930659062795766552086960510079639321733\ 5186331279071157176223663747034278993459233240236164718392435524768991\ 1651405085198377196853585620784709821429403692261403436611440423444829\ 2748296611627659894279872419849928103747162712619617345255302687530018\ 7817658685649551103709068064778326238119416169413210338282316959996 set beta In:= beta = -1 Out= -1 find x value via factoring In:= Timing[ xxx1 = Factor[(alpha^2 x^2 + (2 alpha beta - (mod)) x + (beta^2 - ( 5)))]] Out= {0.016567, 4 (-1 + 5977463956555029511491566495989183669250339909073889722832461\ 4981450963446018671106683325139211063338333117122088333233039316439680\ 3389212813295150282225796943073196798371347835108709673986803991766006\ 1627752651131986877928459858616934359000889648585595543071965962307633\ 8474378471132470126784407394978606082878221803754479281776899651429663\ 9477242581230899388287306158432391308106473219158110377060535880155992\ 8846359141971610718553473834895149979602797395908198663679942995524698\ 1172720218099873440124006058957018726326118873021448095905694133372322\ 7382091988379180478221022385036616642236051052932055232499442906567720\ 4623721055196470029643028060035462827852498212544872146634214102632624\ 0215714299153677304967349484894852960073367412419171670822768586299817\ 2300429606748391956484622346252430343031537539163549468013793271631745\ 7072535817395349805528635140444815190927887282605988808365190392321271\ 2881103020224084907917578829791642379984340920395197431919577844035986\ 135889372162624437916477625473279798757837173092575185269320916401 x) \ (1 + 23909855826220118045966265983956734677001359636295558891329845992\ 5803853784074684426733300556844253353332468488353332932157265758721355\ 6851253180601128903187772292787193485391340434838695947215967064024651\ 1010604527947511713839434467737436003558594342382172287863849230535389\ 7513884529880507137629579914424331512887215017917127107598605718655790\ 8970324923597553149224633729565232425892876632441508242143520623971538\ 5436567886442874213895339580599918411189583632794654719771982098792469\ 0880872399493760496024235828074909216291915281144165904029038841310166\ 5687997962556070343252200837319885218167742068539474186623877871182886\ 2460103619913573012422179920751655404010174009758579831295066235587318\ 5962726642586805911907529519201340962445681365052526910518441235401452\ 4907154429575698534785555675858497058514648069096495793537408852840473\ 7958434386144714957171853914935664876524714804113498768046899611500344\ 1520151656858253927133077831992555005890570811557177427742374456431091\ 759882648604455752225627663533281207483646456119935487350254404 x)} find the root of 5 mod (modulus/4) In:= y20 = Mod[alpha (-1 PowerMod[ 239098558262201180459662659839567346770013596362955588913298459\ 9258038537840746844267333005568442533533324684883533329321572657587213\ 5568512531806011289031877722927871934853913404348386959472159670640246\ 5110106045279475117138394344677374360035585943423821722878638492305353\ 8975138845298805071376295799144243315128872150179171271075986057186557\ 9089703249235975531492246337295652324258928766324415082421435206239715\ 3854365678864428742138953395805999184111895836327946547197719820987924\ 6908808723994937604960242358280749092162919152811441659040290388413101\ 6656879979625560703432522008373198852181677420685394741866238778711828\ 8624601036199135730124221799207516554040101740097585798312950662355873\ 1859627266425868059119075295192013409624456813650525269105184412354014\ 5249071544295756985347855556758584970585146480690964957935374088528404\ 7379584343861447149571718539149356648765247148041134987680468996115003\ 4415201516568582539271330778319925550058905708115571774277423744564310\ 91759882648604455752225627663533281207483646456119935487350254404, -1, mod/4]) + beta, mod/4] Out= 2988731978277514755745783247994591834625169954536944861416230\ 7490725481723009335553341662569605531669166558561044166616519658219840\ 1694606406647575141112898471536598399185673917554354836993401995883003\ 0813876325565993438964229929308467179500444824292797771535982981153816\ 9237189235566235063392203697489303041439110901877239640888449825714831\ 9738621290615449694143653079216195654053236609579055188530267940077996\ 4423179570985805359276736917447574989801398697954099331839971497762349\ 0586360109049936720062003029478509364752234858433835709504605896985419\ 1746832262245710365578296023423967480854323210666782024826442680496482\ 5273688229999561105687889943052909678673818644078483258092972755196505\ 3239118771150663921813444263360465964683455219082984459381783457164587\ 4562760223895074176484556492459333378519541844219077299900607141760752\ 5218192819399602759231636532652525524062214972457433699463679861605741\ 8156876663858503144250621629353649297988759508947597992420495265082440\ 728151908643043067493304332580870621887973411643651640363750009905 verify root of 5 has been found In:= Mod Out= 5 show modulus is 1 off a multiple of RSA260 In:= GCD Out= 2211282552952966643528108525502623092761208950247001539441374\ 8319128822941402001986512729726569746599085900330031400051170742204560\ 8592763579537571859542988389587092292384910067030341246205457845664136\ 64540684214361293017694020846391065875914794251435144458199

1421:

8976878592977149419479354614619245503351517062310272892283206832270342\ 107180646508847010423195532937957397125717572229100 (-1 + 21625328517416900126428104365616202681730512043317208497357835087\ 5115234334109727661599420938600827151972897057921869929279481109202762\ 1510231145048713904694037946176059303576904834489105264234756986854257\ 0480281930784281722166730228879106835688621747839473279256785140670373\ 9441262638608267650972445085046983089688818364421080826758536129212203\ 9439550387809976052732454334373014462180471746794241046025164633407892\ 5782907333402733086683619910034389961807309434406429093250276188205284\ 2681146568095613828264777415531460988278822459756560975953602795309923\ 4987638237604385391803435271975466760682517268394958721388380363711294\ 9592377814223481525533669279783257630786410356098789722654346057740612\ 9973632350018706640058943255166425091562242735389525056302129269945612\ 05119513745600 x) (1 + 23909855826220118045966265983956734677001359636295558891329845992\ 5803853784074684426733300556844253353332468488353332932157265758721355\ 6851253180601128903187772292787193485391340434838695947215967064024651\ 1010604527947511713839434467737436003558594342382172287863849230535389\ 7513884529880507137629579914424331512887215017917127107598605718655790\ 8970324923597553149224633729565232425892876632441508242143520623971538\ 5436567886442874213895339580599918411189583632794654719771982098792469\ 0880872399493760496024235828074909216291915281144165904029038841310166\ 5687997962556070343252200837319885218167742068539474186623877871182886\ 2460103619913573012422179920751655404010174009758579831295066235587318\ 5962726642586805911907529519201340962445681365052526910518441235401452\ 4907154429575698534785555675858497058514648069096495793537408852840473\ 7958434386144714957171853914935664876524714804113498768046899611500344\ 1520151656858253927133077831992555005890570811557177427742374456431091\ 759882648604455752225627663533281207483646456119935487350254404 x)}} this is the root In:= y201 = Mod[alpha (-1 PowerMod[ 239098558262201180459662659839567346770013596362955588913298459\ 9258038537840746844267333005568442533533324684883533329321572657587213\ 5568512531806011289031877722927871934853913404348386959472159670640246\ 5110106045279475117138394344677374360035585943423821722878638492305353\ 8975138845298805071376295799144243315128872150179171271075986057186557\ 9089703249235975531492246337295652324258928766324415082421435206239715\ 3854365678864428742138953395805999184111895836327946547197719820987924\ 6908808723994937604960242358280749092162919152811441659040290388413101\ 6656879979625560703432522008373198852181677420685394741866238778711828\ 8624601036199135730124221799207516554040101740097585798312950662355873\ 1859627266425868059119075295192013409624456813650525269105184412354014\ 5249071544295756985347855556758584970585146480690964957935374088528404\ 7379584343861447149571718539149356648765247148041134987680468996115003\ 4415201516568582539271330778319925550058905708115571774277423744564310\ 91759882648604455752225627663533281207483646456119935487350254404 , \ -1, mod/16]) + beta, mod/16] Out= {2703166064677112515803513045702025335216314005414651062169729\ 3859389404291763715957699927617325103393996612132240233741159935138650\ 3452688778893131089238086754743272007412947113104311138158029344623356\ 7821310035241348035215270841278609888354461077718479934159907098142583\ 7967430157829826033456371555635630872886211102295552635103344817016151\ 5254929943798476247006591556791796626807772558968349280130753145579175\ 9865722863416675341635835452488754298745225913679300803636656284523525\ 6605335143321011951728533097176941432734098980455117902298399626624544\ 8950754059254824048943696116824953374492155414757875006326502034875418\ 2068864062926803520561793989089611086180726894271661768194686407876463\ 0799761874666855067252839436229599224208251745430393794863993095538317\ 17276357511447300} show an interesting relation between the square and the root mod (1+RSA260^4) In:= Mod, RSA260] == Mod, RSA260] Out= True show the square is equal to 1/2 mod RSA260 In:= Mod Out= {\ 1105641276476483321764054262751311546380604475123500769720687415956441\ 1470701000993256364863284873299542950165015700025585371102280429638178\ 9768785929771494194793546146192455033515170623102728922832068322703421\ 07180646508847010423195532937957397125717572229100} In:= PowerMod Out= 1105641276476483321764054262751311546380604475123500769720687\ 4159564411470701000993256364863284873299542950165015700025585371102280\ 4296381789768785929771494194793546146192455033515170623102728922832068\ 32270342107180646508847010423195532937957397125717572229100

379:

917120890958410008482087841589586151217626232826371030111673791\ 2293875357204494206813995960346123827602284689018796275566146945366585\ 6281877724015699174823011835439795839630828791036440083780601718315985\ 2007245661902885039480713346440762676258074691033908734633 RSA260) x \ + (beta^2 - 67 1368837150684194042510578868044158434653173481830404522554737\ 0018349067697320140607185068597531528100898932371669845187412159619946\ 7343467494375306061634682640010077078988917200940626225807828498255766\ 39468896242063048056914759495383197503850593653212202196296)]] Out= {0.001808, \ (-91712089095841000848208784158958615121762623282637103011167379122938\ 7535720449420681399596034612382760228468901879627556614694536431201232\ 2123145506129523736880675164292257452463021957129124509383136348444160\ 48218224219813288886190674232757989774765217547151832 + 625 x) (1 + 221128255295296664352810852550262309276120895024700153944137483191\ 2882294140200198651272972656974659908590033003140005117074220456634122\ 2244687734384959634600580814240815308187625607064670916864685251180403\ 609859378794669961340279885370629131877531341079359556 x)} make square root of Y*67 mod X*RSA260 In:= y20 = Mod[alpha (-1 PowerMod[ 221128255295296664352810852550262309276120895024700153944137483\ 1912882294140200198651272972656974659908590033003140005117074220456634\ 1222244687734384959634600580814240815308187625607064670916864685251180\ 403609859378794669961340279885370629131877531341079359556, -1, 917120890958410008482087841589586151217626232826371030111673791\ 2293875357204494206813995960346123827602284689018796275566146945366585\ 6281877724015699174823011835439795839630828791036440083780601718315985\ 2007245661902885039480713346440762676258074691033908734633 RSA260]) + beta, 9171208909584100084820878415895861512176262328263710301116737\ 9122938753572044942068139959603461238276022846890187962755661469453665\ 8562818777240156991748230118354397958396308287910364400837806017183159\ 852007245661902885039480713346440762676258074691033908734633 RSA260] Out= 1135687518070006846887464059143685218548224789823900288712062\ 0161349230186311372609095609924686501885032258079250035399140494699580\ 4282128173250630469829918963238326756881815299910190655332367589825870\ 0152887712366916712677369184923841399860759273069677450787072142484552\ 7826122846536736649568463943702815871332771256125696257928782086877055\ 6128632769667392696780362313599985334892513732085200490148901318534771\ 9136356155445220411201535935080066343567341587283621854313017633376135\ 5086629145306692049420010258738365460786 ensure square root has been found Mod[y20^2, 91712089095841000848208784158958615121762623282637103011167379122938\ 7535720449420681399596034612382760228468901879627556614694536658562818\ 7772401569917482301183543979583963082879103644008378060171831598520072\ 45661902885039480713346440762676258074691033908734633 RSA260] == 67 136883715068419404251057886804415843465317348183040452255473700183\ 4906769732014060718506859753152810089893237166984518741215961994673434\ 6749437530606163468264001007707898891720094062622580782849825576639468\ 896242063048056914759495383197503850593653212202196296 = True now reduce modulus to RSA260 Mod == Mod[67 13688371506841940425105788680441584346531734818304045225547370\ 0183490676973201406071850685975315281008989323716698451874121596199467\ 3434674943753060616346826400100770789889172009406262258078284982557663\ 9468896242063048056914759495383197503850593653212202196296, RSA260] = True show square root Mod = 1347984549432556028783466692801644423562572831345407611132640\ 8461433033037038763662887179059324590157900214282015315896277432597863\ 4445457453370674239475817222859934247260381018593945056022964392877868\ 89939930371632801365916099130785038894308954176871426817131 show square Mod = 1049504424769368463556334613351068318553056322510039894071011\ 0145105795122967339234377677245252772197706555370592225458614264149435\ 9709015362105059888261212907604380304476139714777966034700737710906745\ 98247995429411206087834031488640531845483210456376624365673 show square is 67*Y mod RSA260 Mod[ 67 136883715068419404251057886804415843465317348183040452255473700183\ 4906769732014060718506859753152810089893237166984518741215961994673434\ 6749437530606163468264001007707898891720094062622580782849825576639468\ 896242063048056914759495383197503850593653212202196296, RSA260] = 1049504424769368463556334613351068318553056322510039894071011\ 0145105795122967339234377677245252772197706555370592225458614264149435\ 9709015362105059888261212907604380304476139714777966034700737710906745\ 98247995429411206087834031488640531845483210456376624365673

1449:

3551425435933295995634800199913176295840866753995725042617284527271288\ 2876111753197304245324752440247287871763001543655724356680812147048631\ 9783152116217946231941914101811179140199174955690607787148196387869661\ 9541883734114559588640512801} In:= Mod Out= {1} so square root of 1 is found, can now get factors of (-1+RSA260^2)/3 In:= GCD Out= {400} In:= ty = GCD Out= {\ 8149617548323649546419276052816706695275009259548782803992220977261107\ 5570757354875663633726775524167293826578308923756969675064570787270904\ 9987536091313062486030080532718940067313652089141163758856164570789479\ 3060366732684095507540508533408609442330984882501640124228368888778567\ 6942426023781038799454925671512203426628159334321977675548185861598937\ 7427032467458663052723511104846698316064181408441379636837033055296465\ 6392764352564098853644647913151013642088320829388966846489845355029346\ 20028115434545665502097206} show factorisation of (-1+RSA260^2)/3 is found In:= ty 400 == 2 (RSA260^2 - 1)/3 Out= {\ 3259847019329459818567710421126682678110003703819513121596888390904443\ 0228302941950265453490710209666917530631323569502787870025828314908361\ 9995014436525224994412032213087576026925460835656465503542465828315791\ 7224146693073638203016203413363443776932393953000656049691347555511427\ 0776970409512415519781970268604881370651263733728791070219274344639575\ 0970812986983465221089404441938679326425672563376551854734813222118586\ 2557105741025639541457859165260405456835328331755586738595938142011738\ 48011246173818266200838882400} == \ 3259847019329459818567710421126682678110003703819513121596888390904443\ 0228302941950265453490710209666917530631323569502787870025828314908361\ 9995014436525224994412032213087576026925460835656465503542465828315791\ 7224146693073638203016203413363443776932393953000656049691347555511427\ 0776970409512415519781970268604881370651263733728791070219274344639575\ 0970812986983465221089404441938679326425672563376551854734813222118586\ 2557105741025639541457859165260405456835328331755586738595938142011738\ 48011246173818266200838882400 how let's take the two square roots of 1 we know , ui and RSA260 In:= ui2 = Mod, (-1 + RSA260^2)/3] Out= {\ 1083899133927045389673763715024621990471576231519988112930965389975727\ 3050910728198463263285661144714250078934915086859676966783587914707030\ 3648342300144637310642000710851619028952715727855774779927869887915000\ 7477028775446984702502887634943345055830020989372497008267077765543196\ 6924817158539785399118554867309583614366713278582528890647710068319686\ 0628048719086101855980826925773868671475676850964749734513371097515471\ 2207945273980958117193491966991239150261101263466588977652972492010439\ 13397863438000322076634470199} show this number is a root of 1 for the modulus given In:= Mod Out= {1} get two factors In:= p = GCD Out= {\ 2211282552952966643528108525502623092761208950247001539441374831912882\ 2941402001986512729726569746599085900330031400051170742204560859276357\ 9537571859542988389587092292384910067030341246205457845664136645406842\ 143612930176940208463910658759147942514351444582} In:= q = GCD Out= {\ 7370941843176555478427028418342076975870696500823338464804582773042940\ 9804673339955042432421899155330286334433438000170569140681869530921193\ 1791906198476627965290307641283033556767804154018192818880455484689473\ 812043100589800694879702195863826475047838148606600} show factorisation has been found In:= p q == (-1 + RSA260^2)/3 Out= {\ 1629923509664729909283855210563341339055001851909756560798444195452221\ 5114151470975132726745355104833458765315661784751393935012914157454180\ 9997507218262612497206016106543788013462730417828232751771232914157895\ 8612073346536819101508101706681721888466196976500328024845673777755713\ 5388485204756207759890985134302440685325631866864395535109637172319787\ 5485406493491732610544702220969339663212836281688275927367406611059293\ 1278552870512819770728929582630202728417664165877793369297969071005869\ 24005623086909133100419441200} == \ 1629923509664729909283855210563341339055001851909756560798444195452221\ 5114151470975132726745355104833458765315661784751393935012914157454180\ 9997507218262612497206016106543788013462730417828232751771232914157895\ 8612073346536819101508101706681721888466196976500328024845673777755713\ 5388485204756207759890985134302440685325631866864395535109637172319787\ 5485406493491732610544702220969339663212836281688275927367406611059293\ 1278552870512819770728929582630202728417664165877793369297969071005869\ 24005623086909133100419441200

1365:

5378407468442673330055684425335333246848835333293215726575872135568512\ 5318060112890318777229278719348539134043483869594721596706402465110106\ 0452794751171383943446773743600355859434238217228678258280466454474506\ 7047769769903628871465840267290863853610037337134588190023227271092598\ 5562318397716277610476863026778615232203106043273058626818827630384175\ 9479841322242544292233604728629381109020566613070567544288470636075157\ 5795994034265171336190833826812096818624770027548542741892092041279704\ 3877187671645165702086770947823616335358481804813828489302226736083919\ 8037064379650707007910654350339367136086946104320051390020952377229881\ 9786493773024231171608856573158644157537927320149589850044116033963226\ 7387534225217176326179777658723207656613354776250973890776137426683101\ 1159663177479992648516657821624546868880590666965892418956501837057699\ 8523003495518762124028306405552801995847971288509008824915391663323682\ 2140993187141745318730151746076326575176657135023818410802 show the root of 2 has been found In:= Mod[ 239098558262201180459662659839567346770013596362955588913298459925803\ 8537840746844267333005568442533533324684883533329321572657587213556851\ 2531806011289031877722927871934853913404348386959472159670640246511010\ 6045279475117138394344677374360035585943423821722867825828046645447450\ 6704776976990362887146584026729086385361003733713458819002322727109259\ 8556231839771627761047686302677861523220310604327305862681882763038417\ 5947984132224254429223360472862938110902056661307056754428847063607515\ 7579599403426517133619083382681209681862477002754854274189209204127970\ 4387718767164516570208677094782361633535848180481382848930222673608391\ 9803706437965070700791065435033936713608694610432005139002095237722988\ 1978649377302423117160885657315864415753792732014958985004411603396322\ 6738753422521717632617977765872320765661335477625097389077613742668310\ 1115966317747999264851665782162454686888059066696589241895650183705769\ 9852300349551876212402830640555280199584797128850900882491539166332368\ 22140993187141745318730151746076326575176657135023818410802^2, (1 + RSA260^4)] Out= 2 get an answer for x^4+1 mod (1+RSA260^4) === 0 In:= x7 = Mod[RSA260^2 (1 + RSA260^2) PowerMod[ 239098558262201180459662659839567346770013596362955588913298459925\ 8038537840746844267333005568442533533324684883533329321572657587213556\ 8512531806011289031877722927871934853913404348386959472159670640246511\ 0106045279475117138394344677374360035585943423821722867825828046645447\ 4506704776976990362887146584026729086385361003733713458819002322727109\ 2598556231839771627761047686302677861523220310604327305862681882763038\ 4175947984132224254429223360472862938110902056661307056754428847063607\ 5157579599403426517133619083382681209681862477002754854274189209204127\ 9704387718767164516570208677094782361633535848180481382848930222673608\ 3919803706437965070700791065435033936713608694610432005139002095237722\ 9881978649377302423117160885657315864415753792732014958985004411603396\ 3226738753422521717632617977765872320765661335477625097389077613742668\ 3101115966317747999264851665782162454686888059066696589241895650183705\ 7699852300349551876212402830640555280199584797128850900882491539166332\ 36822140993187141745318730151746076326575176657135023818410802, -1, \ (1 + RSA260^4)/2], ( 1 + RSA260^4)/2] Out= 1081266425870845006321405218280810134086525602165860424867891\ 7543755761716705486383079971046930041357598644852896093496463974055460\ 1381075511557252435695234701897308802965178845241724455263211737849342\ 7128524014096539214086108336511443955341784431087391973664011736962323\ 4605944848288493582384950338754807922081452680451479689707791351350589\ 8641790001125921948840266217415254076141289527461586948306601183448218\ 2730038307171502099790738219814627067967915896841594568823420876009806\ 3687864509758916785207979778729869523169934220354650821198140419950934\ 4133353470102445210155739906936350120886108678774436571169599678678981\ 1757295785441382581952396103400801305159693249335218141169941996812851\ 5159132934420996270177387886126548768806950194545184798859978189208672\ 802253296159654599 In:= Mod Out= 0

1223:

2153046474983581941267398767559165543946077062914571196477686542167660\ 429831652624386837205668074719 In:= r5p = PowerMod] Out= 42130598235734700400412863678358400343851829547637867333716817\ 8958752526454143408751791721631733642988693659529746447756284068354472\ 9257728658437929345467149708089989426050768748178317699688401942080486\ 0786622076634145117791469050888307772631985482470951248377653103007701\ 43407447960378245283882437800 In:= r5a = Mod, p] Out= 74640729477180716247627684292179290699996155359095614039045928\ 3664551788789539765622495050223714750558144166223649797290735606820082\ 6199082321507752022020976821328882125140877084626461381029771910017334\ 1158543413234982265232728363239981606046378055808331606577259262342152\ 86619550292382541244775256259 In:= Mod Out= 0 In:= np = r5a^4 - 3 r5a^2 + 1 Out= \ 3103869838918094728884305995599976746373014716905007833273463661659975\ 7874489453218577997915701531597502839203811906752595136260883862051195\ 0658532357495564886020678883564860716187031269171482659261544788426532\ 0530173905940599236646422938784819182242004349923496333627042180751321\ 1975805647007865761874801550350638442367369367979682891989299371296604\ 9036724928519156687566200631838797781418115462658676686205077129284117\ 6045796269373062308220483058587895436470192379188556261616694027357712\ 6767391763106589376996923833640639310742418038147928733141777246369946\ 0744757335940195420952098015156697270852511421434049776171310300583051\ 3639892880989235785628870836915056657154042060429444645393006482631065\ 5220513401556019838834534472239253169617995837629758167239666420993770\ 9281375585236710279230584056173386527732972000884312106585215710776656\ 3978273511398769247955581181381472395212574006308066870622897481731283\ 3510819097373182273322546812327204205142254204478344459167307435979740\ 6252690006632099504504051458836008843497360416156187511842595352038793\ 4543044754351424425877001272963004568523466224175032773792526289116620\ 0568336967114120165672550441659487736918062146518195311333334118971437\ 64555494331319 In:= n5root = (4 np + 5)^(1/2) Out= \ 1114247699377134855726317142179837991669774193500027654225864197927618\ 7646938837316951141336216915442127166483212586214421913115995296844607\ 3946658155463253315027850184444739515341546021953572982444114714925658\ 0519870859653135390989665115182357512121444199702727957702373225396420\ 7216365056061158130605226296647748179164483946460250464229475581745544\ 0036374343860347460160207493425647713512071331927925811700396550562457\ 3846126215346561195211149928753609216034043894350314242828278541445301\ 7773780965470434846252578303623858451376965385995516573524516571330514\ 2198625602789862461515560148629318237350159 In:= Mod Out= 5 In:= Mod Out= 5 In:= Mod Out= 5 show that np/p is a large composite number that can't be factored In:= np/p Out= \ 2896728797231704193216809955936005179085860357124327763928880605469952\ 9890127008436789926639374159700090035187081671639709475400680850173904\ 0058210935814740529974851954492933178526173822540769424561928057025308\ 1257357592242862948979902547430700868992340623025119956027913426964695\ 8275605565192054755733867646312586793389844610210453703079127599181132\ 3778275022514076127981575818233734927751691268453164338584834775401646\ 9904577932281754673944241802200714626359063831683748837550009899452654\ 4352368244112477441025374597126682484717899443437339805422403580723893\ 3002641386756501799284720230157622349756874028010246670229789729282372\ 3986813886707581994241628069211456142529347675422400609607277288095115\ 1341531104295685383087888884085673305395919840016723826438909922393694\ 6141812381089445179866616739862690808926728095173508565815687725169736\ 666681638317401920585688059230128574752301883525710237230351401 In:= PrimeQ Out= False

1414:

546:

4723625429992521654787837491618048319631364040388191253484698255313698\ 7424736875836220039610457304524305120045165665398590929500984074537021\ 3332671406165455614268623279672955402907322055976895600593186605328911\ 5169593626456219480475197831634106662908018989638508845064827782102219\ 8331778793835658611538459312186788747890608185252992497633380107893907\ 21301760772016869260727399301258323602 set v In:= v = 2 (RSA260^2 - 1^2) Out= 9779541057988379455703131263380048034330011111458539364790665\ 1727133290684908825850796360472130629000752591893970708508363610077484\ 9447250859985043309575674983236096639262728080776382506969396510627397\ 4849473751672440079220914609048610240090331330797181859001968149074042\ 6665342812330911228537246559345910805814644111953791201186373210657823\ 0339187252912438960950395663268213325816037979277017690129655564204439\ 6663557587671317223076918624373577495781216370505984995266760215787814\ 42603521544033738521454798602516647200 make the modulus a variable. (Please note that probably the modular square root of 5 can be solved for any modulus that is x^4+18x^2+1 by this method, no matter how big) In:= mod = (RSA260^2 - 1^2)^2 + 5 (((2 1 RSA260)^2)) Out= 2390985582622011804596626598395673467700135963629555889132984\ 5992580385378407468442673330055684425335333246848835333293215726575872\ 1355685125318060112890318777229278719348539134043483869594721596706402\ 4651101060452794751171383943446773743600355859434238217228786384923053\ 5389751388452988050713762957991442433151288721501791712710759860571865\ 5790897032492359755314922463372956523242589287663244150824214352062397\ 1538543656788644287421389533958059991841118958363279465471977198209879\ 2469088087239949376049602423582807491703992476167184773046653404954534\ 8594032808685422438526157752221902651776609480274917656256432838107178\ 4959761186928798048189304205947855234339204482167029856642017237764576\ 9783217378669834197799692670939878034230039774474178421414940663330766\ 6517128460930159144372737789700813274692324363089889374327026605350455\ 5763358876399563570064243443818628546720647588356443002938261756026433\ 0727573629317789549844076859516283862514294434927309207692660437108140\ 4188164670644817863924855879204361453559753554627758324307483432020 solve for alpha and beta In:= Solve Out= {{a -: -->

1388:

0452794751171383943446773743600355859434238217228678258280466454474506\ 7047769769903628871465840267290863853610037337134588190023227271092598\ 5562318397716277610476863026778615232203106043273058626818827630384175\ 9479841322242544292233604728629381109020566613070567544288470636075157\ 5795994034265171336190833826812096818624770027548542741892092041279704\ 3877187671645165702086770947823616335358481804813828489302226736083919\ 8037064379650707007910654350339367136086946104320051390020952377229881\ 9786493773024231171608856573158644157537927320149589850044116033963226\ 7387534225217176326179777658723207656613354776250973890776137426683101\ 1159663177479992648516657821624546868880590666965892418956501837057699\ 8523003495518762124028306405552801995847971288509008824915391663323682\ 2140993187141745318730151746076326575176657135023818410802 Out= 23909855826220118045966265983956734677001359636295558891329845\ 9925803853784074684426733300556844253353332468488353332932157265758721\ 3556851253180601128903187772292787193485391340434838695947215967064024\ 6511010604527947511713839434467737436003558594342382172286782582804664\ 5447450670477697699036288714658402672908638536100373371345881900232272\ 7109259855623183977162776104768630267786152322031060432730586268188276\ 3038417594798413222425442922336047286293811090205666130705675442884706\ 3607515757959940342651713361908338268120968186247700275485427418920920\ 4127970438771876716451657020867709478236163353584818048138284893022267\ 3608391980370643796507070079106543503393671360869461043200513900209523\ 7722988197864937730242311716088565731586441575379273201495898500441160\ 3396322673875342252171763261797776587232076566133547762509738907761374\ 2668310111596631774799926485166578216245468688805906669658924189565018\ 3705769985230034955187621240283064055528019958479712885090088249153916\ 633236822140993187141745318730151746076326575176657135023818410802 show this is the root of 2 In:= Mod Out= 2 Now show bridge between square and root In:= Mod == Mod)^2, ( 1 + RSA260^4)/2] Out= True where RSA260^2 is I

375:= 0) || (C \ Integers && C <= -1)]}} Make the new quadratic In:= Expand[((67 z + 25)^2 + 917120890958410008482087841589586151217626232826371030111673791229\ 3875357204494206813995960346123827602284689018796275566146945366585628\ 1877724015699174823011835439795839630828791036440083780601718315985200\ 7245661902885039480713346440762676258074691033908734633 RSA260)/(67 \ 1368837150684194042510578868044158434653173481830404522554737001834906\ 7697320140607185068597531528100898932371669845187412159619946734346749\ 4375306061634682640010077078988917200940626225807828498255766394688962\ 42063048056914759495383197503850593653212202196296)] Out= 22112825529529666435281085255026230927612089502470015394413748\ 3191288229414020019865127297265697465990859003300314000511707422045663\ 4122224468773438495963460058081424081530818762560706467091686468525118\ 0403609859378794669961340279885370629131877531341079359556 + (25 z)/ 68441857534209702125528943402207921732658674091520226127736850091745\ 3384866007030359253429876576405044946618583492259370607980997336717337\ 4718765303081734132000503853949445860047031311290391424912788319734448\ 121031524028457379747691598751925296826606101098148 + (67 z^2)/ 13688371506841940425105788680441584346531734818304045225547370018349\ 0676973201406071850685975315281008989323716698451874121596199467343467\ 4943753060616346826400100770789889172009406262258078284982557663946889\ 6242063048056914759495383197503850593653212202196296 Don't worry about the fractions to Z^{2} and Z, since Z is zero set W and V v = 25 = 25 w = 221128255295296664352810852550262309276120895024700153944137483191288\ 2294140200198651272972656974659908590033003140005117074220456634122224\ 4687734384959634600580814240815308187625607064670916864685251180403609\ 859378794669961340279885370629131877531341079359556^(1/2) = 47024276208709120795061378748873252735571493245640595713707096\ 29848608672008763225036307999162383887826987432678940935201933498834 Solve for alpha and beta In:= Solve Out= {{a -: --> 1416:

0106045279475117138394344677374360035585943423821722878638492305353897\ 5138845298805071376295799144243315128872150179171271075986057186557908\ 9703249235975531492246337295652324258928766324415082421435206239715385\ 4365678864428742138953395805999184111895836327946547197719820987924690\ 8808723994937604960242358280749082383378094823062203337159125033053631\ 3579868510975309784615356646065561496768594834598381394108149711076270\ 5661329114052094023446949760265694055058430521910815076854023093145105\ 1863441356174094012801320342539657952016734429610660056574172263683194\ 1099725705737303857115130093242158254235252153718398589463282713760626\ 0046431849583417465140308810169395852808187090739324412255670298965462\ 2431339667285983629286381656367962387588485038653149903846248783094547\ 09284149077779730646846220929759663449907934665136884833607200 + 23909855826220118045966265983956734677001359636295558891329845992\ 5803853784074684426733300556844253353332468488353332932157265758721355\ 6851253180601128903187772292787193485391340434838695947215967064024651\ 1010604527947511713839434467737436003558594342382172287863849230535389\ 7513884529880507137629579914424331512887215017917127107598605718655790\ 8970324923597553149224633729565232425892876632441508242143520623971538\ 5436567886442874213895339580599918411189583632794654719771982098792469\ 0880872399493760496024235828074909216291915281144165904029038841310166\ 5687997962556070343252200837319885218167742068539474186623877871182886\ 2460103619913573012422179920751655404010174009758579831295066235587318\ 5962726642586805911907529519201340962445681365052526910518441235401452\ 4907154429575698534785555675858497058514648069096495793537408852840473\ 7958434386144714957171853914935664876524714804113498768046899611500344\ 1520151656858253927133077831992555005890570811557177427742374456431091\ 759882648604455752225627663533281207483646456119935487350254404 C, C \ Integers], b -: -->

548:

0106045279475117138394344677374360035585943423821722878638492305353897\ 5138845298805071376295799144243315128872150179171271075986057186557908\ 9703249235975531492246337295652324258928766324415082421435206239715385\ 4365678864428742138953395805999184111895836327946547197719820987924690\ 8808723994937604960242358280749072603837036834682747634027861653005597\ 0279757396389916136708704918932270811859768983802020921977520710323678\ 6721622028968457922672100313014834070015120946235831840757383830417024\ 4099616286480128906527345493065906279576655208696051007963932173351863\ 3127907115717622366374703427899345923324023616471839243552476899116514\ 0508519837719685358562078470982142940369226140343661144042344482927482\ 9661162765989427987241984992810374716271261961734525530268753001878176\ 58685649551103709068064778326238119416169413210338282316959996 + 23909855826220118045966265983956734677001359636295558891329845992\ 5803853784074684426733300556844253353332468488353332932157265758721355\ 6851253180601128903187772292787193485391340434838695947215967064024651\ 1010604527947511713839434467737436003558594342382172287863849230535389\ 7513884529880507137629579914424331512887215017917127107598605718655790\ 8970324923597553149224633729565232425892876632441508242143520623971538\ 5436567886442874213895339580599918411189583632794654719771982098792469\ 0880872399493760496024235828074909216291915281144165904029038841310166\ 5687997962556070343252200837319885218167742068539474186623877871182886\ 2460103619913573012422179920751655404010174009758579831295066235587318\ 5962726642586805911907529519201340962445681365052526910518441235401452\ 4907154429575698534785555675858497058514648069096495793537408852840473\ 7958434386144714957171853914935664876524714804113498768046899611500344\ 1520151656858253927133077831992555005890570811557177427742374456431091\ 759882648604455752225627663533281207483646456119935487350254404 C, C \ Integers], b -: -->

490:= 0) || (C \ Integers && C <= -1)]}} Thus the square in this case is In:= y4 = \ 1049504424769368463556334613351068318553056322510039894071011014510579\ 5122967339234377677245252772197706555370592225458614264149435970901536\ 2105059888261212907604380304476139714777966034700737710906745982479954\ 29411206087834031488640531845483210456376624365673 Out= 1049504424769368463556334613351068318553056322510039894071011\ 0145105795122967339234377677245252772197706555370592225458614264149435\ 9709015362105059888261212907604380304476139714777966034700737710906745\ 98247995429411206087834031488640531845483210456376624365673 and the root is In:= y45 = Mod[alpha (-1 PowerMod[ 221128255295296664352810852550262309276120895024700153944137483\ 1912882294140200198651272972656974659908590033003140005117074220456634\ 1222244687734384959634600580814240815308187625607064670916864685251180\ 403609859378794669961340279885370629131877531341079359556 , -1, RSA260]) + beta, RSA260] Out= 1347984549432556028783466692801644423562572831345407611132640\ 8461433033037038763662887179059324590157900214282015315896277432597863\ 4445457453370674239475817222859934247260381018593945056022964392877868\ 89939930371632801365916099130785038894308954176871426817131 In:= Mod == Mod Out= True 159:

22112825529529666435281085255026230927612089502470015394413748\ 3191288229414020019865127297265697465990859003300314000511707422045608\ 5927635795375718595429883895870922923849100670303412462054578456641366\ 4540684214361293017694020846391065875914794251435144458199 Ensure F term make C+F a square in Associated Quadratic In:= f = (Floor + 1)^2 - bb Out= 61680487993617575466423211001635452497867860629177112238643415\ 1536338262777021476003752497731780456662275070549172392086402843049 Create Quadratic In:= Expand Out= 33004217208253233485494157097054076011361327615626888648378728\ 8345206312558238835619592980993578307449043288507931344047324510516450\ 5254225922019813202664210455889910867157854962940197688303572761354816\ 50763031441980751727213603623335755264318574943643805121 + 50 z + 67 z^2 proceed with algorithm In:= y14 = Mod) + beta, (f 67 + RSA260)] Out= 1415220833889898651857989456321678779367173728158080985242479\ 8924242446682497281271368147025004637823414976211220096032749275010946\ 9381712791946325774058734024268628937046348559896791469316269255010985\ 41461225897986427940404144662922132883373729786377668512709 verify that Out is square root of 67 mod 67*f+RSA260 In:= Mod Out= 67 show that resulting modulus is not a prime In:= PrimeQ Out= False

595:

8979579392137458719472234654521409245163585278762047542492940883636308\ 3678950219585711529988865922879855140362335939123011995218675993425843\ 3271082460543624565920975768082361934527646650410830443351245929703307\ 6287030748722050298768260679895617537321926364807588422393592601067756\ 5549095656672346035967606247938975172957634984222136270967969963570105\ 6553751939601616527321203518370764097038022075629587155481954685331444\ 4094066742732712236528258372338715068772988089009842280873698339792902\ 5160882616217173344613023417439569667402744047420335248014401881280214\ 5519185246374158133468389128849889359291370585485195400027702946545599\ 5777185658332199608310262912041769488861310563165769670482764013832200\ 5386500144368219885210862528421435063594967227053060657583055697144495\ 171977259983373194369016225 verify that square root of 5 mod x*RSA270+1 has been found In:= Mod Out= 5

1446:

5676942426023781038799454925671512203426628159334321977675548185861598\ 9377427032467458663052723511104846698316064181408441379636837033055296\ 4656392764352564098853644647913151013642088320829388966846489845355029\ 34620028115434545665502097203 + 814961754832364954641927605281670669527500925954878280399222097726\ 1107557075735487566363372677552416729382657830892375696967506457078727\ 0904998753609131306248603008053271894006731365208914116375885616457078\ 9479306036673268409550754050853340860944233098488250164012422836888877\ 8567694242602378103879945492567151220342662815933432197767554818586159\ 8937742703246745866305272351110484669831606418140844137963683703305529\ 6465639276435256409885364464791315101364208832082938896684648984535502\ 934620028115434545665502097206 C}} Solve for I (but instead we get square root of 9) In:= Mod /. yyy /. C -: -->

1257:

3455775675806644650385191815744751597420578815339758689329568471272697\ 7962106783779046149327395158869256074859912500865606397449960529827116\ 4411243978956599768491187076310593031748078213552442903120664495841764\ 0686416399318426826009739029277662686571694061227815713974218476677730\ 9160116220693559552200738391668049489625480467424903184317395220271193\ 9203555943986135421168957559877095295978323201888953681859832544993604\ 5225543394561502975804440555356733543797736990654096227907634817991350\ 7981534740325879183669889182467390198771279775847210108538722098532307\ 5655374868923845139515003637195406420774970715619765466278800746719990\ 1376858572442673550342594754331808347104358909323725678874383676044877\ 0364455695150440655288154991280286896110473371117639497577702180738743\ 9225451232988061301295313997 In:= Mod Out= 13

1419:

5389751388452988050713762957991442433151288721501791712710759860571865\ 5790897032492359755314922463372956523242589287663244150824214352062397\ 1538543656788644287421389533958059991841118958363279465471977198209879\ 2469088087239949376049602423582807490823833780948230622033371591250330\ 5363135798685109753097846153566460655614967685948345983813941081497110\ 7627056613291140520940234469497602656940550584305219108150768540230931\ 4510518634413561740940128013203425396579520167344296106600565741722636\ 8319410997257057373038571151300932421582542352521537183985894632827137\ 6062600464318495834174651403088101693958528081870907393244122556702989\ 6546224313396672859836292863816563679623875884850386531499038462487830\ 9454709284149077779730646846220929759663449907934665136884833607200} In:= beta = b /. x11 /. C -: -->

81: 1573: 71: 53: 1574:

https://www.amazon.com./Unlimited-Congruent-Squares-Kunerth-Algorithm-ebook/dp/B0D1KDGJ3M/ref=sr_1_4?dib=eyJ2IjoiMSJ9.vnAuzEUQeyJUFIgaUkfmv0zAY35k8z5Xx3sm9TnxXMTx54p7uId0BnwWX7Qy8YQ5i6o5QfbZVxmMvL-uoV3YMADJFNcGeKMwrbdX6HjG1P5zzwY9oTE63yF4DuCh9XaPp7FF5bXvnofwZSXB5MAacVIzL83PmcpVfV5B_f9NXPX86wXOHVo53JFxJgaUlkBvQ2uWpgewBfXPSE5ud09rdsFtBFkXpK7CtFaulWzZMKE.M4uyUgrzxxgep1jLdeebnSoW2nKtaHBP3TfUr0aMSI4&dib_tag=se&keywords=paul+cheffers&qid=1713497200&sr=8-4

1286:

1267657551565979868477989855361 x1) (1 + 1267657551565988875701940675225 x1)} get the square root of 5 In:= y23 = Mod) + beta, mod] Out= 950743163674505449088130199350 verify root of 5 has been found In:= Mod Out= 5 As such if Rabin's cipher is 5 mod P*Q where P*Q is X^4+18x^2+1 then the plaintext can be found, breaking the cipher but the modulus has not been factored. I don't know how to get the other root either.

22: 1471:

5912256789143394418465928855820910695963428632915745653033631345567701\ 3804896583170813346973298098213872136943300938448156734352029533373451\ 5619553741388032743245759604063496548832937573677282974966035 show the square root of 5 has been found In:= Mod Out= 5 A square root of 1 is found In:= Mod Out= 1 Generally, any modulus of X*R^2-1 will have the root of X as X*R.

1506:

engineer earlier in his career, as Kunerth was. It's possible that Kunerth actually existed, as an engineer, and that Petzval used his name for various reasons that are unknown, to publish Petzval's math on the modular square root algorithm. Petzval was a mathematician who could have produced the math presented in the papers authored by Kunerth.

371:

14815593104784876511638327120867574721500099966654910314257211373\ 8163113707393413309635289168017302213875532211210380342843972770594486\ 1890394078203792295518238914554134625648570915673332951429933911829087\ 041860578379242887409798752319832151835794599852317090252 C[ 1], (C \ Integers && C : -->

489:

22112825529529666435281085255026230927612089502470015394413748319\ 1288229414020019865127297265697465990859003300314000511707422045608592\ 7635795375718595429883895870922923849100670303412462054578456641366454\ 0684214361293017694020846391065875914794251435144458199 C[ 1], (C \ Integers && C : -->

486:

22112825529529666435281085255026230927612089502470015394413748319\ 1288229414020019865127297265697465990859003300314000511707422045663412\ 2224468773438495963460058081424081530818762560706467091686468525118040\ 3609859378794669961340279885370629131877531341079359556 C[ 1], (C \ Integers && C : -->

374:

1283:

Expand[((2 (x^2 - 1^2))^2 + 1 (x^2 - 1^2)^2 + 5 (((2 1 x)^2)))/(5)]^(1/2) Out= 1125902993852485 make the v variable In:= v = 2 (x^2 - 1) Out= 2251805987704966 make the modulus In:= mod = (x^2 - 1^2)^2 + 5 (((2 1 x)^2)) Out= 1267657551566006890149842314969 solve for alpha and beta In:= Solve Out= {{a -: -->

369:

RSA260 bb = (Floor + 1)^2 - RSA260 = 54819458867339771990053357616221050115768171809225729400503710\ 80118837515862925645017501652267319433494304753217083279905934901357 Find the X and Y of ((67 z+25)^2+X*RSA260)/(67*Y) In:= Solve[(( 25)^2 + x RSA260)/(y 67) == bb + RSA260, {x, y}, Integers] Out= {{x -: -->

1313:

ConditionalExpression, C \ Integers]}} In:= alpha = 1267769985869025060348644534085 Out= 1267769985869025060348644534085 In:= beta = -1 Out= -1 In:= Timing[ xx = Factor] Out= {0.002433, 3 (-2 + 422589995289673518845650218987 x1) (1 + 1267769985869029564160338411225 x1)} In:= y20 = Mod) +

960:

solves the equation it's easy to create two natural squares (both congruent to x) for the polynomial that makes up the new modulus of z*p (or z*p*q). This gives up to 4 different congruent squares(this only applies when b=1 in the above equation). Unfortunately, these two do not factor the modulus.

488:

ConditionalExpression[ 104950442476936846355633461335106831855305632251003989407101101451\ 0579512296733923437767724525277219770655537059222545861426414943597090\ 1536210505988826121290760438030447613971477796603470073771090674598247\ 995429411206087834031488640531845483210456376624365673 +

485:

ConditionalExpression[ 104950442476936846355633461335106831855305632251003989407101101451\ 0579512296733923437767724525277219770655537059222545861426414943857270\ 6745043047208403211494541014084968672595260474749330214565087861455459\ 241427372303571065761871140566881851095906049654992837 +

373:

ConditionalExpression[ 136883715068419404251057886804415843465317348183040452255473700183\ 4906769732014060718506859753152810089893237166984518741215961994673434\ 6749437530606163468264001007707898891720094062622580782849825576639468\ 896242063048056914759495383197503850593653212202196296 +

370:

ConditionalExpression[ 917120890958410008482087841589586151217626232826371030111673791229\ 3875357204494206813995960346123827602284689018796275566146945366585628\ 1877724015699174823011835439795839630828791036440083780601718315985200\ 7245661902885039480713346440762676258074691033908734633 +

368:

RSA260 = 2211282552952966643528108525502623092761208950247001539441374\ 8319128822941402001986512729726569746599085900330031400051170742204560\ 8592763579537571859542988389587092292384910067030341246205457845664136\ 64540684214361293017694020846391065875914794251435144458199 Find nearest square above

1545:

One Hypothesis I have come up with regarding Petzval and Kunerth is that Kunerth was a student of Petzval, and they both worked on the math together. Petzval was known to be nice to his students, so Petzval gave the naming rights to the math paper to Kunerth, which would undoubtedly have helped his

1617:

1505:

who was a very well known and important Austrian/Hungarian mathematician. In 1878 when this paper was published Petzval had retired and was becoming a recluse in Kahlenberg, outside Vienna. Petzval was a mathematician of great reknown and is important in the history of optics. He was also a good

1285:

ConditionalExpression, C \ Integers]}} set beta In:= beta = -1 Out= -1 set alpha In:= alpha = 1267657551565984372089965265285 Out= 1267657551565984372089965265285 factor for x1 variable In:= Timing[ xxx1 = Factor[(alpha^2 x1^2 + (2 alpha beta - (mod)) x1 + (beta^2 - \ (5)))]] Out= {0.001683, (-4 +

1448:

0 Out= {\ 5460243757376845196100914955387193485834256203897684478674788054764942\ 0632407427766694634596939601192086863807466978917169682293262427471506\ 3491649181179751865640153956921689845100146899724579718433630262428951\ 1350445710898343990052140717383768326361759871276098883233007155481640\

1328:

Thus a bridge between squares and roots is known via the Complex Square Root formula when using coefficients of Pythagorean Squares only returns natural square coefficients(see two Australian Innovation Patents 2017101124 and 2017101121) Thus roots can be found of modular squares instantly if the

657:

2413}} Note the relationship between the answer above 168 and 337, the root of 5 mod 89*29 In:= Mod Out= 5 take the root of 5 for this multiple of 89*29 In:= PowerMod Out= 56445 Note that the quotient of the square is always 4, no matter what the modulus is (as long as it equals x^4-3x^2+1===0 mod

2096:

1310074348496985087958227895934029051306378242573836062042042519166\ 7871151633616076909466726506246372628441650527102960316106219092081078\ 2782697034448186243471410872828623439969212125671314423500663717755571\ 28362798348048332822650444386978185719854262751334260}} make the square (or half the

2093:

Using the Pythagorean theorum come up with a quadratic number and determine a SQUARE that will ensure a modulus that RSA260*x In this case use (RSA260-4), and 3^2 and the inputs to the pythagorean triple. As such (RSA260-4)^2+3^2 is the resulting square. 32 makes the square (4(RSA260-4)^2+3^2)*2

1470:

This is the modulus for a quadratic with ((S+1)/2) being the second square in the pythagorean expansion take the square root of 5 for the modulus 5 RSA270^2-1 In:= Factor[ Expand] Out= 1/4 (-1 - 4 S + S^2) (-1 + 5 S^2) y25= 1165542651722037722638188284553402620728099062401527245214743059842479\

545:

Using the Pythagorean Theorum Solve the quadratic In:= w = Expand[((2 (RSA260^2 - 1^2 ))^2 + 1 (RSA260^2 - 1^2)^2 + 5 (((2 1 RSA260)^2)))/(5)]^(1/2) Out= 4889770528994189727851565631690024017165005555729269682395332\ 5863566645342454412925398180236065314500376295946985354254181805038742\

1310:

In:= x = 2^25 + 790 Out= 33555222 In:= w = (((2 (x^2 - 1))^2 + 3 (x^2 - 1)^2 + 7 (2 1 x)^2)/7)^(1/2) Out= 1125952923469285 In:= v = 2 (x^2 - 1) Out= 2251905846938566 In:= mod = 3 (x^2 - 1)^2 + 7 (2 1 x)^2 Out= 3803309957607106707727790742219 Notice how the modulus has mostly 3 mod 4 primes. In:=

1282:

In:= mod = (2^25 + 46)^2 (18 + (2^25 + 46)^2) + 1 set x in x^4+18x^2+1 In:= x = 2^25 + 46 Out= 33554478 Out= 1267657551566006890149842314969 show modulus is a p*q In:= FactorInteger Out= {{109, 1}, {11629885794183549450915984541, 1}} make the w variable (using the Pythagorean Theorum) In:= w =

1278:

In order to challenge Rabin's cryptosystem, which is claimed to be as hard as factoring, I am going to show a p*q modulus that is x^4+18x^2+1 where Kunerth can take 1 modular square root of 5 but not two. Therefore Kunerth cannot easily factor p*q but can take the modular square root of 5.

1256:

y21=1968527137708915753781396172423958959752843397871385953905541559583361\ 7047928713223231088902575813035048789781731014443653341300985605006420\ 4150266696006320793043864946273912817379704904250761155345357266156831\ 9505939969785925196949573822667063211917988445136037366509246826500102\

594:

y20=3690988383204217038340117823294923049536581371008848663572890424218803\ 1964866337293558291692329649440716480840745652081850014939348009387038\ 2781750055011851486957246774263586532586946695470177166272544874044059\ 9073637443348609744280450917500743522346228334630070062204837799687691\

158:

In:= RSA260 = \ 2211282552952966643528108525502623092761208950247001539441374831912882\ 2941402001986512729726569746599085900330031400051170742204560859276357\ 9537571859542988389587092292384910067030341246205457845664136645406842\ 14361293017694020846391065875914794251435144458199 Out=

1639:

If you have x^2+y^2==p*o^2 then you can the square root of p quite easiiy if x^2 or y^2 is a square of the modulus. However, Kunerth's method is more broad than that. It can discover the modular square root of the equation x^2+y*z==p o^2, which is more general, where z is the modulus.

1418:

0 Out= {2390985582622011804596626598395673467700135963629555889132984\ 5992580385378407468442673330055684425335333246848835333293215726575872\ 1355685125318060112890318777229278719348539134043483869594721596706402\ 4651101060452794751171383943446773743600355859434238217228786384923053\

1445:

8149617548323649546419276052816706695275009259548782803992220977261\ 1075570757354875663633726775524167293826578308923756969675064570787270\ 9049987536091313062486030080532718940067313652089141163758856164570789\ 4793060366732684095507540508533408609442330984882501640124228368888778\

549:

ConditionalExpression, C \ Integers]}} set alpha In:= alpha = \ 2390985582622011804596626598395673467700135963629555889132984599258038\ 5378407468442673330055684425335333246848835333293215726575872135568512\ 5318060112890318777229278719348539134043483869594721596706402465110106\

378:

ConditionalExpression, C \ Integers]}} set alpha and beta alpha = 11756069052177280198765344687218313183892873311410148928426774\ 0746215216800219080625907699979059597195674685816973523380048337470850\ beta = 0 In:= Timing[ xx = Factor[ alpha^2 x^2 + (2 alpha beta -

1420:

0 Out= {0} In:= Timing[ xx = Factor[ alpha^2 x^2 + (2 alpha beta - mod) x + (beta^2 - ((Floor + 1)))]] Out= {0.01033, \ {110564127647648332176405426275131154638060447512350076972068741595644\ 1147070100099325636486328487329954295016501570002558537110228042963817\

392:

1347984549432556028783466692801644423562572831345407611132640\ 8461433033037038763662887179059324590157900214282015315896277432597863\ 4445457453370674239475817222859934247260381018593945056022964392877868\ 89939930371632801365916099130785038894308954176871426817131

385:

1049504424769368463556334613351068318553056322510039894071011\ 0145105795122967339234377677245252772197706555370592225458614264149435\ 9709015362105059888261212907604380304476139714777966034700737710906745\ 98247995429411206087834031488640531845483210456376624365673

619:

Factor the modulus to reveal the factors In:= Factor] See that (-1+R+R^2) is a factor, we will show the root of 5 for this modulus Out= 4 (-1 - R + R^2) (-1 + R + R^2) In:= RSA260 = \ 2211282552952966643528108525502623092761208950247001539441374831912882\

1442:

The following work will show two non trivial factors of (-1+RSA260^2)/3 They are get two factors, in this case the trivial factorisation is (RSA260-1)(RSA260+1), and in our case the factor of 100 has been taken off both factors. In:= p = GCD Out= {\

1364:

get the square root of 2 mod (1+RSA260^4) considering that RSA260^2 is I for this modulus (See Dickson's quote above for the source for this equation) In:= x6 = Reduce Out= x == \ 2390985582622011804596626598395673467700135963629555889132984599258038\

571:

Thanks to the Mathematica computing system. It's been a breeze to work with Kunerth's method with Mathematica. Thanks to Mr. Steven Wolfram and his team at Mathematica. The work shown here has been done on the Raspberry PI 4 version of Mathematica.

1387:

this is the root of 2 In:= x = \ 2390985582622011804596626598395673467700135963629555889132984599258038\ 5378407468442673330055684425335333246848835333293215726575872135568512\ 5318060112890318777229278719348539134043483869594721596706402465110106\

1413:

Get the square of PowerMod in the modulus of ((1+RSA260)/2)(1+RSA260^2)^2 show the root of this in modulus ((1+RSA260)/2)(1+RSA260^2)^2 Show what the modulus is algebraicly In:= Factor[ Expand] Out= (1 + RSA260) (1 + RSA260^2)^2 In:= RSA260 = \

1517:) 02:49, 25 October 2022 (UTC). Petrol, better known as , wrote several math books for high schools in Hungaria. It is more likely that Otto Petzval gave the lecture before a secondary school, but Otto was not known for independent math. 1415:

ConditionalExpression[ 239098558262201180459662659839567346770013596362955588913298459925\ 8038537840746844267333005568442533533324684883533329321572657587213556\ 8512531806011289031877722927871934853913404348386959472159670640246511\

547:

376:

ConditionalExpression[ 117560690521772801987653446872183131838928733114101489284267740746\ 215216800219080625907699979059597195674685816973523380048337470850 + 22112825529529666435281085255026230927612089502470015394413748319\

1778:

After some thought on the matter, it seems the novel aspect of Kunerth, is that joined to the Pythagorean theorum in the Quadratic there is a modulus of a certain formula that allows the root of a certain residue to be taken easily.

1496:

The w. M. Mr. Hofrath Petzval presented a treatise by Prof. Adolf Kunerth at the secondary school in Brunn, entitled: "Practical method for the numerical resolution of indefinite quadratic equations in whole and in rational numbers".

377:

1288229414020019865127297265697465990859003300314000511707422045663412\ 2224468773438495963460058081424081530818762560706467091686468525118040\ 3609859378794669961340279885370629131877531341079359556 C, C \ Integers], b -: -->

1222:

In:= p = NextPrime Out= 10715086071862673209484250490600018105614048117055336074437503\ 8837035105112493612249319837881569585812759467291755314682518714528569\ 2314043598457757469857480393456777482423098542107460506237114187795418\

2086:

I'd say that by changing the N and O variables in the equation given above that you can automatically generate the minus of two squares equaling the multiple of a modulus you are interested in, particularly a RSA number.

1942: 480:

So it is possible to take the modular square root of a square mod RSA260, but you cannot pick the number. You set the SQUARE term (which is the first item in the quadratic) and then solve via Mathematica such as

958: 504:

Kunerth's method is quite exciting since it doesn't have to factor the modulus. Follow the mathematica below and see how the square root of 5 mod x*RSA260+1 is found within seconds on a Raspberry PI 4 palmtop.

516:

seems to be the modulus. So if you factor N-1 and it has the factorisation of X^4+18*X^2 then you should be able to use Kunerth (see code below) to find the square root of 5 quickly of modulus x^4+18x^2+1.

310: 591:

Using the formula given above that any modulus of x^4+18x^2+1 will allow Kunerth's method to retrieve the modular square root of 5, here is the modular square root of 5 mod RSA270^4+18*RSA270^2+1

1061: 133: 2062: 1196: 2131:((RSA260^2-n)^2-o^2)^2, (2 o (RSA260-n))^2 and ((RSA260^2-n)^2+o^2)^2 where the division of the square was a low number mod RSA260 then you should be able to produce the root of a low square 1509:

This is a speculation. Against this argument, Petzval doesn't seem to have done any number theory. It's also possible that it was his brother, Petrol Petzval, who was also a mathematician.

361: 256: 1774: 1147: 437: 1314:

beta, mod] Out= 2852482468205330875260535658630 In:= Mod Out= 7 thus the cipher of 7 has had its root taken without factoring the modulus. the second root of 7 has not been found.

1710: 1654:

Upon further investigation it appears tha Kunerth's method is a another type of minus of 2 squares method, which can be quickly found by other means. Since the quadratic equals

1994: 1531:

Dickson's three volume "Theory of Numbers" only mentions Petzval once in the author's index and then only in a footnote on a page. Petzval was not known as a Number Theory man.

2094:

In:= y1 = Solve[( square - 1) ((RSA260 - 4)^2 - 3^2)^2 + square (2 3 (RSA260 - 4))^2 + 31 square ((RSA260 - 4)^2 + 3^2)^2 == 0, {square}, Modulus -: -->

678:

This means that you can find the root of 5 for mod (x^4-3x^2+1). When the original modulus is a prime, p, then the new modulus for which the root of 5 is found is therefore

2072:

Using a better method of setting the modulus to be a multiple of RSA260, I was able to find a square root that was not obviously a division of two easy numbers, like (2/5).

710: 736: 508:

The use of the Pythagorean Theorum to solve the quadratic shows what is in the notes in the Mathematica below. that for the modulus RSA260 and square of 5 that

442:

I believe this is the first time someone has been able to find a modular square root of a square of a modulus that is an rsa number that hasn't been factored yet.

2162: 127: 641:

taken from the Kunerth definition of the w variable In:= Factor] Out= 4 (-1 - x + x^2) (-1 + x + x^2) In:= Expand Out= 1 - 3 x^2 + x^4 In:= Solve Out= {{x -: -->

2083:

Setting n=1 returned only naiive divisions of squares like (4/16). Setting n=2 returned squares of (4/25). Setting n=3 returned a root that was (2/5+10).

1219:

We will show that knowing the root of 5 mod p where p is a 1000 bit prime can make knowing the root of 5 mod(x^4-3 x^2+1) easy when x is the (rootof5-1)/2.

1501:

So it appears that a Mr Petzval presented the treatise of Kunerth's. Hofrath is a German word for Counsellor, so this may be an honorific. There is a

2157: 103: 1785: 1342:

3) take the root of 4+5 (which is 3). This is the real coefficient of the root, root of 5-4 is 1, this is the imaginary coefficient of the root

396:

Thus a hard modular square root of a still to be factored (as of this date) RSA number has been found, although I am still unable to find

1447:

0 Out= {9} This is not the natural root of 9 so we can find the square root of 1 In:= ui = Mod, (RSA260^2 - 1)/3] /. yyy /. C -: -->

94: 58: 873: 1614:

the root found will be -RSA260^2, but the exercise shows that the algorithm does find the square root of -1 for a very big number.

1384:

We saw how 8+6*I was the square and 3+1*I was the root. Now we see how (4+3*I) is the square and (3+1*I)/(root of 2) is the root.

1381:

We show here that with the square root of 2 mod 1+RSA260^4 known we can take roots of polar coefficients of Pythagorean Triples.

1312:

ConditionalExpression[ 1267769985869025060348644534085 + 1267769985869029564160338411225 C, C \ Integers], b -: -->

1284:

ConditionalExpression[ 1267657551565984372089965265285 + 1267657551565988875701940675225 C, C \ Integers], b -: -->

167:

The expansion of the quadratic is quite fluid. We can easily set the following quadratic be a square and find the square root

675:

This means if you know the root of 5 mod p*q then you can know the root of 5 mod (x^4-3x^2+1) where (x^4-3x^2+1)===0 mod p*q.

33: 1311:

FactorInteger Out= {{3, 1}, {7, 1}, {71707, 1}, {1180987, 1}, {907027753, 1}, {2357844607, 1}} In:= Solve Out= {{a -: -->

1307:

The actual Rabin cipher specifies 3 mod 4 semiprimes. I show the root of 7 mod a modulus that has several 3 mod 4 primes.

263: 2114:

The root is 7/100 mod RSA260. The square always seems to be the division of two squares which in this case is 49/10000

964: 1999: 1348:

5) Once you know the modular imaginary number you can make a bridge between square and root for any Pythagorean Triple.

1152: 682:, so the root of y is also found, which is (x^4-3^x2+1)/p. This is shown for a 1000 bit prime in the next section. 493:

It doesn't seem possible to set the Y term though, so a bridge has been found, not a general square root algorithm.

2090:

This leads to a modular square that is the division of two natural squares, and thus the root can be found easily.

317: 172: 1715: 1231:

The square root of 13 mod 1+11 RSA270^2+RSA270^4 is (and this is partial confirmation of the formula given above

1066: 399: 1201:

It is possible to quickly find the square roots of two modular squares 1 apart, even for RSA numbers. Consult

39: 21: 1339:

2) double 3, and 4 to make 8 and 6. These are the polar coefficients of the square, thus 8+6*I is the square

1317:

This modulus has several 3 mod 4 primes and Kunerth will take the modular square root of 7 for this modulus.

1996:. Any modulus that fits this will reveal the root of 5 easily. It's really hard to do this with ordinary 1246:

then Kunerth with the Pythagorean Method below should be :able to take the square root of C mod N quickly.

533:

then Kunerth with the Pythagorean Method below should be able to take the square root of C mod N quickly.

102:

on Knowledge (XXG). If you would like to participate, please visit the project page, where you can join

1657: 80: 1947: 1208:

These two types of polynomials can be used as the definition of a modulus for the Kunerth method.

741:

The following two tables summarise the nature of the primes for which n*n+1 mod Y*P can be found.

86: 70: 52: 2136: 2119: 2104: 1645: 1625: 1551: 1536: 1522: 1514: 1478: 1456: 1428: 1396: 1372: 1293: 1264: 627: 602: 577: 557: 466: 449: 1586: 1211:

Examples showing unlimited congruent squares generated by the Kunerth method are shown here.

688: 715: 1502: 2151: 1576:, title="Unlimited Congruent Squares Using the Kunerth Modular Square Root Algorithm" 1598: 2132: 2115: 2100: 1641: 1621: 1547: 1532: 1518: 1510: 1474: 1452: 1424: 1392: 1368: 1355:

With I known, the root of 2 is found via Leonard Dickson's formula for (x^4+1)===0

1289: 1260: 1235:

If N is the modulus, C is the square and D is the natural :square just below C that

1202: 623: 598: 573: 553: 524:

If N is the modulus, C is the square and D is the natural square just below C that

462: 445: 2140: 2123: 2108: 1649: 1629: 1555: 1540: 1526: 1482: 1460: 1432: 1400: 1376: 1297: 1268: 631: 606: 581: 561: 470: 453: 99: 668:

2) the root squared of the root of 5 mod (x^4-3x^2+1) shows a quotient of four

76: 1937:{\displaystyle (2(x^{2}-1))^{2}+((x^{2}-1))^{2}+5(2*1*x)^{2}==5(x^{2}+1)^{2}} 1417:

ConditionalExpression, C \ Integers]}} In:= alpha = a /. x11 /. C -: -->

953:{\displaystyle y=\pm ({\sqrt {x}}\pm 1^{2})*({\sqrt {x-1}})^{-1}*b} 1329:

polar coefficients of the square are twice Pythagorean triples.

1438:

Nontrivial Factorisation of (-1+RSA260^2)/3 found using Kunerth

1325:

Working with modulus (1+RSA260^4) where the I term is RSA260^2

637:

root of 5 mod (x^4-3x^2+1) squared is equal to 5+4(x^4-3x^2+1)

15: 2033: 411: 335: 287: 2068:

A Way To Generate The Division Of Two Squares Mod a Modulus

616:

The square root of 5 mod RSA260^2+RSA260-1 is shown below.

1587:

https://mathshistory.st-andrews.ac.uk/Biographies/Petzval/

487:= 0) || (C \ Integers && C <= -1)], y -: --> 484:

In:= Solve + 1)^2, {x, y}, Integers] Out= {{x -: -->

372:= 0) || (C \ Integers && C <= -1)], y -: --> 1409:

The Root of (RSA260+1)/2 mod ((RSA260+1)/2)(1+RSA260^2)^2

520:

I haven't proven this but I'd say at this point that

305:{\displaystyle {\sqrt {-RSA260}}{\bmod {67}}\equiv 25} 2002: 1950: 1788: 1718: 1660: 1488:

Link between Adolf Kunerth and Joseph or Otto Petzval

1249:

using the Pythagorean method and Kunerth shown below.

1155: 1069: 967: 876: 718: 691: 536:

using the Pythagorean method and Kunerth shown below.

402: 320: 266: 175: 1274:

Taking the root of 5 mod p*q which is (x^4+18 x^2+1)

98:, a collaborative effort to improve the coverage of 1056:{\displaystyle (x-1)(y^{2}+b^{2})^{2}-x(2*b*y)^{2}} 2057:{\displaystyle 5*x^{2}-y^{2}{\bmod {p*q}}\equiv 0} 2056: 1988: 1936: 1768: 1704: 1191:{\displaystyle y=\pm {\sqrt {x}}\pm {\sqrt {x-1}}} 1190: 1141: 1055: 952: 730: 704: 476:Root Of A Modular Square Found, But Cannot Pick It 431: 355: 304: 250: 132:This article has not yet received a rating on the 1599:https://www.zobodat.at/pdf/SBAWW_78_0244-0248.pdf 1303:A modulus with 3 mod 4 primes shown as an example 685:Cipolla and Tonelli both find the roots of 5 for 672:seems to work no matter what the p*q modulus is. 459:This took around 1 second on a Raspberry PI 4 356:{\displaystyle {\sqrt {Y*67}}{\bmod {RSA260}}} 251:{\displaystyle ((67z+25)^{2}+X*RSA260)/(Y*67)} 1769:{\displaystyle p*o^{2}-x^{2}\mod {z}\equiv 0} 8: 1227:square root of 13 mod 1+11 RSA270^2+RSA270^4 1142:{\displaystyle (1)(y^{2}+1)^{2}-x(2*1y)^{2}} 432:{\displaystyle {\sqrt {67}}{\bmod {RSA260}}} 1361:(Theory of Algebraic Equations-p 43(1903)) 1336:1) take a Pythagorean Triple, such as 3,4,5 19: 2128:If you could come up with a combination of 961:This is the solution for the polynomial: 661:The two points of the above Mathematica, 163:Taking The Square Root of 67*Y Mod RSA260 151:Square Root of 67 mod 67*f+RSA260 is taken 47: 2036: 2032: 2026: 2013: 2001: 1974: 1955: 1949: 1928: 1912: 1893: 1859: 1840: 1821: 1802: 1787: 1742: 1729: 1717: 1696: 1665: 1659: 1175: 1165: 1154: 1133: 1102: 1086: 1068: 1047: 1013: 1003: 990: 966: 935: 918: 903: 889: 875: 717: 696: 690: 414: 410: 403: 401: 338: 334: 321: 319: 290: 286: 267: 265: 228: 198: 174: 1755: 1753: 1565: 1215:Showing this with a large prime modulus 112:Knowledge (XXG):WikiProject Mathematics 49: 1243:if Y equals (C-D)x^4+(4*C-2*(C-D))*x^2 612:Square root of 5 mod RSA260^2+RSA260-1 530:if Y equals (C-D)x^4+(4*C-2*(C-D))*x^2 2163:Unknown-priority mathematics articles 1944:then the formula for the modulus is 1321:Taking the root of 2 mod (1+RSA260^4) 7: 1635:x^2+y*z==p o^2 is not x^2+y^2==p o^2 785:((10 (x^2 + 1))^2 - 101 (2 1 x)^2)/4 382:Thus the square root mod RSA260 of 92:This article is within the scope of 1238:take the factorisation of N-(C-D)=Y 527:take the factorisation of N-(C-D)=Y 38:It is of interest to the following 1782:For instance, if the quadratic is 1705:{\displaystyle x^{2}+y*z==p*o^{2}} 1444:(RSA260^2 - 1)/3] Out= {{II -: --> 1063:. The solution for the polynomial 777:((6 (x^2 + 1))^2 - 37 (2 1 x)^2)/4 769:((4 (x^2 + 1))^2 - 17 (2 1 x)^2)/4 587:square root of 5 mod x*RSA270+1 is 14: 1466:Square Root of 5 mod 5*RSA270^2-1 761:((2 (x^2 + 1))^2 - 5 (2 1 x)^2)/4 665:1) 2*x+1 is the root of 5 mod p*q 365:Dropping into Mathematica we see 2158:Start-Class mathematics articles 1989:{\displaystyle X^{4}+18*x^{2}+1} 1492:Translating from the German at 314:Thus we can find ultimately the 115:Template:WikiProject Mathematics 79: 69: 51: 20: 500:Square Root of 5 mod X*RSA260+1 1925: 1905: 1890: 1871: 1856: 1852: 1833: 1830: 1818: 1814: 1795: 1789: 1401:20:01, 19 September 2022 (UTC) 1130: 1114: 1099: 1079: 1076: 1070: 1044: 1025: 1010: 983: 980: 968: 932: 915: 909: 886: 632:06:09, 20 September 2022 (UTC) 245: 233: 225: 195: 179: 176: 1: 2095:RSA260] Out= {{square -: --> 1748: 1650:16:17, 17 February 2023 (UTC) 1630:04:23, 29 November 2022 (UTC) 1610:Root of -1 mod (1+RSA260^4)/2 1358:E=(1+I)/(root of 2), E_1=E*I 454:14:06, 22 December 2021 (UTC) 106:and see a list of open tasks. 2075:The changing parameters are 1541:10:21, 25 October 2022 (UTC) 1527:00:52, 31 October 2022 (UTC) 1483:21:15, 18 October 2022 (UTC) 866:Congruent Squares Can Be Had 1556:19:34, 8 January 2023 (UTC) 1461:12:59, 9 October 2022 (UTC) 1433:19:07, 26 August 2022 (UTC) 1377:21:35, 25 August 2022 (UTC) 1352:In:= Mod == Mod Out= True 2179: 1345:4) Thus 3+1*I is the root. 155:Dropping into Mathematica 1712:this can be rewritten as 131: 64: 46: 2141:22:57, 11 May 2023 (UTC) 2124:10:16, 11 May 2023 (UTC) 2109:00:51, 11 May 2023 (UTC) 1298:06:22, 6 June 2022 (UTC) 1269:06:06, 17 May 2022 (UTC) 607:05:55, 17 May 2022 (UTC) 582:13:13, 14 May 2022 (UTC) 562:03:05, 14 May 2022 (UTC) 134:project's priority scale 471:20:19, 5 May 2022 (UTC) 95:WikiProject Mathematics 2058: 1990: 1938: 1770: 1706: 1499: 1192: 1143: 1057: 954: 753:Polynomial for Modulus 732: 706: 512:RSA260^4+18*RSA260^2+1 433: 357: 306: 252: 28:This article is rated 2059: 1991: 1939: 1771: 1707: 1494: 1332:To briefly explain, 1205:for how to do this. 1193: 1144: 1058: 955: 733: 707: 705:{\displaystyle p^{y}} 434: 358: 307: 253: 32:on Knowledge (XXG)'s 2000: 1948: 1786: 1716: 1658: 1153: 1067: 965: 874: 716: 689: 658:p*q) In:= N Out= 4. 400: 318: 264: 173: 118:mathematics articles 803:Quotient Multiplier 731:{\displaystyle y*p} 2054: 1986: 1934: 1766: 1756: 1754: 1749: 1702: 1188: 1139: 1053: 950: 728: 702: 429: 353: 302: 248: 87:Mathematics portal 34:content assessment 1186: 1170: 929: 894: 863: 862: 789: 788: 408: 332: 284: 148: 147: 144: 143: 140: 139: 2170: 2079:(RSA260-n)^2+o^2 2063: 2061: 2060: 2055: 2047: 2046: 2031: 2030: 2018: 2017: 1995: 1993: 1992: 1987: 1979: 1978: 1960: 1959: 1943: 1941: 1940: 1935: 1933: 1932: 1917: 1916: 1898: 1897: 1864: 1863: 1845: 1844: 1826: 1825: 1807: 1806: 1775: 1773: 1772: 1767: 1747: 1746: 1734: 1733: 1711: 1709: 1708: 1703: 1701: 1700: 1670: 1669: 1602: 1595: 1589: 1583: 1577: 1570: 1197: 1195: 1194: 1189: 1187: 1176: 1171: 1166: 1148: 1146: 1145: 1140: 1138: 1137: 1107: 1106: 1091: 1090: 1062: 1060: 1059: 1054: 1052: 1051: 1018: 1017: 1008: 1007: 995: 994: 959: 957: 956: 951: 943: 942: 930: 919: 908: 907: 895: 890: 856:(rootof101-1)/10 791: 790: 747: 746: 737: 735: 734: 729: 711: 709: 708: 703: 701: 700: 656:2412}, {x -: --> 655:1967}, {x -: --> 654:1949}, {x -: --> 653:1948}, {x -: --> 652:1879}, {x -: --> 651:1503}, {x -: --> 650:1415}, {x -: --> 649:1166}, {x -: --> 648:1078}, {x -: --> 438: 436: 435: 430: 428: 427: 409: 404: 389:is found to be 362: 360: 359: 354: 352: 351: 333: 322: 311: 309: 308: 303: 295: 294: 285: 268: 257: 255: 254: 249: 232: 203: 202: 120: 119: 116: 113: 110: 89: 84: 83: 73: 66: 65: 55: 48: 31: 25: 24: 16: 2178: 2177: 2173: 2172: 2171: 2169: 2168: 2167: 2148: 2147: 2098: 2070: 2022: 2009: 1998: 1997: 1970: 1951: 1946: 1945: 1924: 1908: 1889: 1855: 1836: 1817: 1798: 1784: 1783: 1738: 1725: 1714: 1713: 1692: 1661: 1656: 1655: 1637: 1619: 1612: 1607: 1606: 1605: 1596: 1592: 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