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Chosen recurrence relation for attack: k_next = 43511367934207785493205740240378275743609768675817908100722913788393695639657*k_prev + 78730372042131832072287759890573273005645422101261818863735891162602450516391 mod n
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Generating artificial signatures...

Created artificial signature 1 between original signatures 1 and 2
  r_new = 23298015045193904612586625097013013238434742231554167744515936879399869488397, s_new = 110955400697554132573244702028682455501659791587247294823293562059184441715205, z_new = 114348426784376972262212387491392248456909105810912478725596391450242197501624
Created artificial signature 2 between original signatures 1 and 2
  r_new = 23298015045193904612586625097013013238434742231554167744515936879399869488397, s_new = 65922615754394972779235513816836497423895401971455841314960028650904449284001, z_new = 95651824604975196373056940850546687721428079367574184939506225302834722908348
Created artificial signature 3 between original signatures 2 and 3
  r_new = 34559190504341613833450353695106665408056903224372678973840446297532206243167, s_new = 68474226316309826745487865238288822884863264116278386354529472763624044908668, z_new = 14119843862881681218543777296966920285776233738925951248823701482312902925979
Created artificial signature 4 between original signatures 2 and 3
  r_new = 34559190504341613833450353695106665408056903224372678973840446297532206243167, s_new = 87856688150857517286884451144948438861914677933532768283681731806952275948583, z_new = 50454272941657236491777748105016448688256465114149558571605332890217841528754
Created artificial signature 5 between original signatures 3 and 4
  r_new = 19464743338216844258300736972864389034943655834701603668204034612723902071488, s_new = 64530538090822446044806789889239841870632092828991942322079830579050702743505, z_new = 100696216182135497006398112131941561404271020258497146408493916851894222017995
Created artificial signature 6 between original signatures 3 and 4
  r_new = 19464743338216844258300736972864389034943655834701603668204034612723902071488, s_new = 47235015937058937548982965514420387033624898602564791643518102370593631469157, z_new = 90587031616179313788031102499266527866257203397777530630520324049364149405545

Total number of signatures (original + artificial): 10
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Generating polynomial to solve for the private key...
Nonces difference equation:
(((((((k12*k12-k23*k01)*k13*k23-(k23*k23-k34*k12)*k01*k02)*k14*k24*k34-((k23*k23-k34*k12)*k24*k34-(k34*k34-k45*k23)*k12*k13)*k01*k02*k03)*k15*k25*k35*k45-(((k23*k23-k34*k12)*k24*k34-(k34*k34-k45*k23)*k12*k13)*k25*k35*k45-((k34*k34-k45*k23)*k35*k45-(k45*k45-k56*k34)*k23*k24)*k12*k13*k14)*k01*k02*k03*k04)*k16*k26*k36*k46*k56-((((k23*k23-k34*k12)*k24*k34-(k34*k34-k45*k23)*k12*k13)*k25*k35*k45-((k34*k34-k45*k23)*k35*k45-(k45*k45-k56*k34)*k23*k24)*k12*k13*k14)*k26*k36*k46*k56-(((k34*k34-k45*k23)*k35*k45-(k45*k45-k56*k34)*k23*k24)*k36*k46*k56-((k45*k45-k56*k34)*k46*k56-(k56*k56-k67*k45)*k34*k35)*k23*k24*k25)*k12*k13*k14*k15)*k01*k02*k03*k04*k05)*k17*k27*k37*k47*k57*k67-(((((k23*k23-k34*k12)*k24*k34-(k34*k34-k45*k23)*k12*k13)*k25*k35*k45-((k34*k34-k45*k23)*k35*k45-(k45*k45-k56*k34)*k23*k24)*k12*k13*k14)*k26*k36*k46*k56-(((k34*k34-k45*k23)*k35*k45-(k45*k45-k56*k34)*k23*k24)*k36*k46*k56-((k45*k45-k56*k34)*k46*k56-(k56*k56-k67*k45)*k34*k35)*k23*k24*k25)*k12*k13*k14*k15)*k27*k37*k47*k57*k67-((((k34*k34-k45*k23)*k35*k45-(k45*k45-k56*k34)*k23*k24)*k36*k46*k56-((k45*k45-k56*k34)*k46*k56-(k56*k56-k67*k45)*k34*k35)*k23*k24*k25)*k37*k47*k57*k67-(((k45*k45-k56*k34)*k46*k56-(k56*k56-k67*k45)*k34*k35)*k47*k57*k67-((k56*k56-k67*k45)*k57*k67-(k67*k67-k78*k56)*k45*k46)*k34*k35*k36)*k23*k24*k25*k26)*k12*k13*k14*k15*k16)*k01*k02*k03*k04*k05*k06)*k18*k28*k38*k48*k58*k68*k78-((((((k23*k23-k34*k12)*k24*k34-(k34*k34-k45*k23)*k12*k13)*k25*k35*k45-((k34*k34-k45*k23)*k35*k45-(k45*k45-k56*k34)*k23*k24)*k12*k13*k14)*k26*k36*k46*k56-(((k34*k34-k45*k23)*k35*k45-(k45*k45-k56*k34)*k23*k24)*k36*k46*k56-((k45*k45-k56*k34)*k46*k56-(k56*k56-k67*k45)*k34*k35)*k23*k24*k25)*k12*k13*k14*k15)*k27*k37*k47*k57*k67-((((k34*k34-k45*k23)*k35*k45-(k45*k45-k56*k34)*k23*k24)*k36*k46*k56-((k45*k45-k56*k34)*k46*k56-(k56*k56-k67*k45)*k34*k35)*k23*k24*k25)*k37*k47*k57*k67-(((k45*k45-k56*k34)*k46*k56-(k56*k56-k67*k45)*k34*k35)*k47*k57*k67-((k56*k56-k67*k45)*k57*k67-(k67*k67-k78*k56)*k45*k46)*k34*k35*k36)*k23*k24*k25*k26)*k12*k13*k14*k15*k16)*k28*k38*k48*k58*k68*k78-(((((k34*k34-k45*k23)*k35*k45-(k45*k45-k56*k34)*k23*k24)*k36*k46*k56-((k45*k45-k56*k34)*k46*k56-(k56*k56-k67*k45)*k34*k35)*k23*k24*k25)*k37*k47*k57*k67-(((k45*k45-k56*k34)*k46*k56-(k56*k56-k67*k45)*k34*k35)*k47*k57*k67-((k56*k56-k67*k45)*k57*k67-(k67*k67-k78*k56)*k45*k46)*k34*k35*k36)*k23*k24*k25*k26)*k38*k48*k58*k68*k78-((((k45*k45-k56*k34)*k46*k56-(k56*k56-k67*k45)*k34*k35)*k47*k57*k67-((k56*k56-k67*k45)*k57*k67-(k67*k67-k78*k56)*k45*k46)*k34*k35*k36)*k48*k58*k68*k78-(((k56*k56-k67*k45)*k57*k67-(k67*k67-k78*k56)*k45*k46)*k58*k68*k78-((k67*k67-k78*k56)*k68*k78-(k78*k78-k89*k67)*k56*k57)*k45*k46*k47)*k34*k35*k36*k37)*k23*k24*k25*k26*k27)*k12*k13*k14*k15*k16*k17)*k01*k02*k03*k04*k05*k06*k07) = 0
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Polynomial in d:
10076378950690840322333303104212707217658418276379388800499402055322370197830*dd^29 + 9907754340187044042462333394731559996501271924517098312112490292652290220374*dd^28 + 25413118996992445384351078544952248707143401694174929858895015610527546784254*dd^27 + 54637476460606508381736836066201543629053576715917127510892537352262580074245*dd^26 + 112660205308610187075066405664391979505309077946149247712537995226455794436810*dd^25 + 41916469819176829829003617962051564034209293180602522782111440946370565287073*dd^24 + 79200169230764140950417025891323641817621003989352606693423673374557515131241*dd^23 + 6892011050106687223571788452179503704436289455822671952398809341012131076293*dd^22 + 87282091338451661877346906037796853908695573093119175349285366988383343671979*dd^21 + 11304997998728144123509981342384890488924738957001454069700450158894601413555*dd^20 + 43979018229079941203255031467982475320154780564095188717134688641454823445236*dd^19 + 75899029772163776616855855544521674685821741983222872578787604745827781681373*dd^18 + 111185857515521162710720049266785741959219058774265220255138138663908361369509*dd^17 + 90097802487495971394348001492670173807693450020911370453545320234648163713708*dd^16 + 91619490151160204957571842154554124171353946727089446509984910179141836095578*dd^15 + 33619806607511208853645921383885292007303237141080347372188866889704588006017*dd^14 + 98507260632833035704714521876291822323109177386128401580070683384149763516801*dd^13 + 30322200714711228447010355230180551241231394995885110745433945031787692193260*dd^12 + 7096479951060851309715641712092460614122972796251168231097030576464187818685*dd^11 + 40542171423306135701504689200828153305093321849084037823359721756352871767487*dd^10 + 69643050130527340465724621294234560230601973582750268941005387368948139703866*dd^9 + 87241409706175103059766549621140491981504711229707304358100814572131640990309*dd^8 + 61237513059092026569007272320630567118531728841589147251671505528852005631367*dd^7 + 1453725589557907817852160717443063393256773858393394886840189629397322782141*dd^6 + 48670189451632826179229665351532123528783509993012732650965262261411195940816*dd^5 + 1812573516712236544783329436796020444157609066755981011895459683138721491048*dd^4 + 110725644076326742608099456151642593983254672417300349789291108993460560776502*dd^3 + 2954658004812788114563058055514340074961271172864817687704379643190729804515*dd^2 + 13438906721576106075723197136896004432378483577017847622586678742441233468181*dd + 108712551478291873565295586254742667835744622857708672501710504063220089310291
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Finding roots of the polynomial...
Found 1 roots.
Roots of the polynomial:
Root: 103944520760024560506038323186664680005228038115110724833232378297114715088318, Multiplicity: 1
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SUCCESS! Private key found: d = 103944520760024560506038323186664680005228038115110724833232378297114715088318

Verifying the artificial signatures follow the specified recurrence relation:
Artificial signature 1:
  Expected k: 90360872727669507550394887425756877359698428223587058630502028768486693956925
  Actual k:   90360872727669507550394887425756877359698428223587058630502028768486693956925
  Match: True
Artificial signature 2:
  Expected k: 90360872727669507550394887425756877359698428223587058630502028768486693956925
  Actual k:   90360872727669507550394887425756877359698428223587058630502028768486693956925
  Match: True
Artificial signature 3:
  Expected k: 20707329528702014598452831463605667149461823117219811019611191842061086457883
  Actual k:   20707329528702014598452831463605667149461823117219811019611191842061086457883
  Match: True
Artificial signature 4:
  Expected k: 20707329528702014598452831463605667149461823117219811019611191842061086457883
  Actual k:   20707329528702014598452831463605667149461823117219811019611191842061086457883
  Match: True
Artificial signature 5:
  Expected k: 19741724224496760945367500974295943818662631917946907732377899462187157547640
  Actual k:   19741724224496760945367500974295943818662631917946907732377899462187157547640
  Match: True