Inversion of two new circulant matrices over Z m

Transcription

1 IOP Coferece Seres: Ert d Evroetl Scece PAPER OPEN ACCESS Iverso of two ew crcult trces over To cte ts rtcle: Yeg eg et l 7 IOP Cof Ser: Ert Evro Sc 8 Vew te rtcle ole for udtes d eceets Relted cotet - Ivertblty of soe crcult trces Busto d A Brr - Soe results o crcult d sew crcult tye trces wt -Fbocc sequeces Yg g - Ideotet crcult trces M Rdrs N Elul R Perul et l Ts cotet ws dowloded fro IP ddress o //9 t 8:35

2 MSETEE 7 IOP Publsg IOP Cof Seres: Ert d Evroetl Scece (7) do :88/755-35/8// Iverso of two ew crcult trces over Yeg eg Sugoog So * uwe Fu Te Uversty of Suwo Bogdeu Hwseog-s Gyeogg-do Kore E-l: zegyeg7@sco sso@suwocr fuzuwe@lyueduc Abstrct I ts er we cosder te roble of vertg RSFMLR crcult trx wt etres over We reset two dfferet lgorts Our lgorts requre dfferet degrees of owledge of d d ter costs rge fro log loglog to log logloglog oertos over Moreover for ec lgort we gve te cost ters of bt oertos Flly te exteded lgorts s used to solve te roble of vertg RSFMLR crcult trces over Itroducto Crcult trces ve ortt lctos vrous dscles cludg ge rocessg couctos sgl rocessg ecodg couter vso d tey ve bee ut o fr bss wt te wor of P Dvs [] d L Jg [] Te crcult trces log frutful subect of reserc [] ve recet yers bee exteded y drectos [34 6 8] Te f ( -crcult trces re oter turl exteso of ts well-studed clss d c be foud [9 4] Te f ( -crcult trx s wde lcto eseclly o te geerlzed cyclc codes [9] were s oc olyol wt o reeted roots ts slttg feld over feld Te roertes d structures of te x x -crcult trces wc re clled RSFMLR crcult trces re better t tose of te geerl f ( -crcult trces so tere re good lgorts for fdg te verse of te RSFMLR crcult trces I ts er we cosder te roble of vertg RSFMLR crcult wt etres over te rg I ts er we descrbe two lgorts for vertg RSFMLR crcult trx over wc trsfor te orgl roble to equvlet roble over te rg Our frst lgort ssues te fctorzto of s ow d requres log log ultlctos d log log ddtos over Ts corresods to te bt colexty boud O( log log ) (log ) log log log log were (d) ) deotes te bt colexty of ultlyg d -bt tegers Our secod lgort does ot requre te fctorzto of d ts cost s greter by fctor log t te revous cse Defto A row sew frst-us-lst rgt(rsfmlr) crcult trx wt te frst row ) over deoted by RSFMLRcrcfr ) s et squre trx of te for: ( ( Cotet fro ts wor y be used uder te ters of te Cretve Coos Attrbuto 3 lcece Ay furter dstrbuto of ts wor ust t ttrbuto to te utor(s) d te ttle of te wor ourl ctto d DOI Publsed uder lcece by IOP Publsg Ltd

3 MSETEE 7 IOP Publsg IOP Cof Seres: Ert d Evroetl Scece (7) do :88/755-35/8// It c be see tt te trx over () wt rbtrry frst row d te followg rule for obtg y oter row fro te revous oe: Get te st row by us te lst eleet of t row to te frst eleet of te t rowd t row d te sftg te eleets of te t row (cyclclly) oe osto to te rgt Obvously te RSFMLR crcult trx over s -crcult trx [9] d tt s te eter te exteso of crcult trx over fles of ttered trces We defe s te bsc RSFMLR crcult trx over ( ) tes te lst eleet of te x x [3] or ts secl cse d tey re two dfferet tt s g x x It s esly verfed tt ( RSFMLRcrcfr( ) () ) x s o reeted roots over g x x x d bot te l olyol d te crcterstc olyol of te trx ( ) I ddto s oderogtory d stsfes ( ) ( ) ( ) I trx ( ) over Tus A ( ) RSFMLRcrcfr ( ) I vew of te structure of te owers of te bsc RSFMLR crcult t s cler tt A RSFMLRcrcfr( ) ( ) s RSFMLR crcult trx over s d (3) f d oly f A f ) for soe ( ( ) olyol f ( over Te olyol f ( x ) x wll be clled te rereseter of te RSFMLR crcult trx A over By Defto d Equto (3) t s cler tt A s RSFMLR crcult trx over f d oly f coutes wt ( ) tt s A I ddto to te lgebrc roertes tt c be esly derved fro te A ( ) ( ) reresetto (3) we eto tt RSFMLR crcult trces ve very ce structure Te roduct of two RSFMLR crcult trces s RSFMLR crcult trx d A s RSFMLR crcult trx too Furter ore let ] { A A f ( ) f ( [ ]} [ ( ) ( ) x It s route to rove tt ] s couttve rg wt te trx ddto d ultlcto [ ( ) Defto A row sew lst-us-frst left (RSLMFL) crcult trx wt te frst row ( ) over deoted by RSLMFLcrcfr ( ) s et squre trx of te for: A

4 MSETEE 7 IOP Publsg IOP Cof Seres: Ert d Evroetl Scece (7) do :88/755-35/8// Le Let ˆ I be te Te ( ) () RSLMFLcrcfr ( = RSFMLRcrcfr ) Iˆ () RSLMFLcrcfr ( ) Iˆ =RSFMLRcrcfr ) ( 3 3 trx of te couter detty Assug A s veble over we cosder te roble of coutg RSFMLR crcult trx B b ( ) suc tt AB I It s turl to rereseter wt RSFMLR crcult trx A ( te olyol (over te rg ) f ( x ) fdg olyol x Coutg te verse of g ( x ) b x suc tt ) A (4) s clerly equvlet to f ( (od x x ) (5) Te cogruece odulo x x follows fro te equlty ( ) ( ) I Hece te roble of vertg RSFMLR crcult trx s equvlet to verso te rg /x x Te followg teore sttes ecessry d suffcet codto for te vertblty of RSFMLR crcult trx over Teore Let deote te re owers fctorzto of d let ) f (x deote te olyol over rereseter to RSFMLR crcult trx A Te trx A s vertble f d oly f for we ve gcd( f ( x x ) Proof If A s vertble by (5) we ve tt tere exsts t ( suc tt for f ( t( ( x x ) Hece gcd( f ( x x ) s cled Te roof tt te bove codto s suffcet for vertblty s costructve d wll be gve Secto (Les d 3) 3

5 MSETEE 7 IOP Publsg IOP Cof Seres: Ert d Evroetl Scece (7) do :88/755-35/8// Revew of bt colexty results [3] I te followg we wll gve te cost of ec lgort ters of uber of bt oertos I our lyss we use te followg well-ow results (see for exle [5] or [6]) Addtos d subtrctos te bt oertos We deote by ( d O( d log d log log d) te uber of bt oertos requred by te Scöge-Strsse Olog lgort [8] for ultlcto of tegers odulo d Hece ultlcto betwee eleets of tes log O(log loglog loglog ) bt oertos Coutg te verse of eleet tes (log)loglog bt oertos usg odfed exteded Euclde lgort (see [5] Teore 8) Te se lgort returs we x s ot vertble Te su of two olyols f ( (od x x ) of degree t ost c be trvlly couted O ( log ) bt oertos Te roduct of two suc olyols c be couted ultlctos d O ( logloglog) ddtos/ subtrctos (see [6] Teore 7) Terefore te sytotc cost of olyol ultlcto s O( ( ) ) bt oertos were x O( log ) gcd( x ) ( ) lolog) log log loglog (6) Gve two olyols ( b( ( re) of degree t ost we c coute d( gcd( ( b( ) O( ( )) bt oertos were ( ) ( )log (log )loglog (7) Te se lgort lso returs d ( s( b( t( d( x Te boud (7) follows by strgtforwrd odfcto of te olyol gcd lgort descrbed [5] (Secto 89: te ter (log )loglog coes fro te fct tt we ust coute te verse of O () eleets of ) s( Iverso x x Fctorzto of Kow t( suc tt ) I ts secto we cosder te roble of coutg te verse of RSFMLR crcult trx over we te fctorzto of te odulus s ow We cosder te equvlet roble of vertg olyol f ( over x x d we sow tt we c coute te verse by cobg ow tecques (Cese rederg te exteded Euclde lgort d Newto-Hesel lftg) We strt by sowg tt t suffces to fd te verse of f ( odulo te re owers Le Let d let ) f (x be olyol Gve g ( g ( ) x suc tt f ( g ( (od x x ) for we c fd [ x] wc stsfes (5) t te cost of O( (log) (log)loglog) bt oertos f Proof Te roof s costructve Sce f ( g ( (od x x ) we ve g ( ( x x (od ) Let / Clerly for (od ) ( Sce gcd( ) we c fd suc tt (od ) Let 4

6 MSETEE 7 IOP Publsg IOP Cof Seres: Ert d Evroetl Scece (7) do :88/755-35/8// ( ( By costructo for we ve (od ) d ( ( (od ) Hece for we ve f ( f ( g ( f ( (od ) ( x (od ) - x (od ) ( x x We coclude tt f ( ( x x (od ) or equvletly f ( (od x x ) Te coutto of cossts (oe for ec coeffcet) lctos of Cese rederg Obvously te coutto of sould be doe oly oce Sce teger dvso s te se sytotc cost s ultlcto we c coute O( (log )) bt oertos Sce ec te s obted troug verso coutg tes O( (log ) loglog ) bt oertos Flly gve g ( we c coute tess follows usg te equlty (log ) loglog (logb) loglogb O( (log )) bt oertos Te (lo b)) loglo b) I vew of Le we c restrct ourselves to te roble of vertg olyol over x x we s re ower Next le sows ow to solve ts rtculr roble Le 3 Let f ( be olyol vertble If gcd( f ( x x ) te f ( s x x I ts cse te verse of f ( c be couted O( ( ) ( )) bt oertos were ( ) d ( ) re defed by (7) d (6) resectvely Proof If gcd( f ( x x ) by Bezout s le tere exst t ( suc tt f ( s( x x t( (od ) Next we cosder te sequece g ( s( g ( g ( [ g ( ] f ( od x x ow s Newto-Hesel lftg It s strgtforwrd to verfy by ducto tt f ( ( (od x x ) Hece te verse of f ( x x s log ( g Te coutto of s( tes O( ( )) bt oertos For coutg te sequece g ( g ( ) log x we observe tt t suffces to coute ec g odulo cost of obtg te wole sequece s s( Hece te 5

7 MSETEE 7 IOP Publsg IOP Cof Seres: Ert d Evroetl Scece (7) do :88/755-35/8// O( ( )) ( 4 ) ( O( ( )) bt oertos Note tt fro Les d 3 we fd tt te codto gve Teore s deed suffcet codto for vertblty of RSFMLR crcult trx Cobg te bove les we obt Algort for te verso of olyol over x x Te cost of te lgort s ( ) ( f ( )) log )) T ( ) O( (log ) (log ) loglog bt oertos I order to get ore geble exresso we boud wt log I ddto we use te equltes ( ) ( b ) ( b ) d ( ) ( b ) ( b ) We get T ( ) O( log (log ) (log )loglog ( ) ( )) O( log (log )) ( ) log ) Note tt f O() te dot ter s ( )log Tt s te cost of vertg f ( s sytotclly bouded by te cost of executg log ultlctos Iverse ( f ( ) {Coutes te verse let for do 3 f gcd( f ( x x ) [ x ] te 4 coute g ( suc tt of te olyol f ( x x } f ( g ( (od x x ) 5 usg Newto-Hesel lftg (Le 3) 6 else 7 retur f ( s ot vertble 8 edf 9 edfor coute usg Cese rederg (Le ) Algort Iverso x x Fctorzto of ow 3 A Geerl Iverso Algort x x Te lgort descrbed Secto reles o te fct tt te fctorzto of te odulus s ow If ts s ot te cse d te fctorzto ust be couted befored te crese te rug te y be sgfct sce te fstest ow fctorzto lgorts requre te d wt 6

8 MSETEE 7 IOP Publsg IOP Cof Seres: Ert d Evroetl Scece (7) do :88/755-35/8// exoetl f ( log (see for exle [7]) I ts secto we sow ow to coute te verse of wtout owg te fctorzto of te odulus Te uber of bt oertos of te ew lgort s oly fctor for O(log ) greter t te revous cse Our de cossts tryg to coute gcd( f ( x x ) usg te gcd lgort Suc lgort requres te verso of soe sclrs wc s ot roble but t s ot lwys ossble f s ot re Terefore te coutto of gcd( f ( x x ) y fl However f te gcd lgort tertes we ve solved te roble I fct togeter wt te lleged gcd ( f ( s( ( x x ) t( ( If de ( ) oe c esly rove tt f ( te lgort lso returs ( te s ot vertble s( s( t( s te verse of suc tt f ( If x x Note tt we ust force te gcd lgort to retur oc olyol If te coutto of gcd( f ( x x ) fls we use recurso I fct te gcd lgort fls y f t cot vert eleet Iverso s doe by usg te teger gcd lgort If s ot vertble te teger gcd lgort returs d gcd( y) wt Hece d s otrvl fctor of We use d to coute eter r suc tt gcd( ) d or sgle fctor suc tt d ( ) d I te frst cse we vert y f ( x x d x x d we use Cese rederg to get te desred result I te secod cse we vert f ( x x d we use oe ste of Newto- Hesel lftg to get te verse x x Te coutto of te fctors s doe by rocedure GetFctors wose correctess s rove by Les 4 d 5 Cobg tese rocedures togeter we get Algort Iverse ( f ( ) {Coutes te verse of te olyol ) f gcd( f ( x x ) te let s( 3 retur f (x x x t( suc tt s( f ( s( ( x x ) t( 4 else f gcd( f ( x x ) t( ( de ( ) te 5 retur f ( s ot vertble 6 else f gcd( f ( x x ) fls let d be suc tt 7 let ( ) 8 f 9 g ( ) f ( x ) ) g ( ) f ( x ) ) coute g ( usg Cese rederg (Le ) } d 7

9 MSETEE 7 IOP Publsg IOP Cof Seres: Ert d Evroetl Scece (7) do :88/755-35/8// else 3 g ( x ) Iverse 4 coute 5 edf 6 retur 7 edf GetFctors (d)!() 8 let log gcd( d ) 9 f ( / ) te retur ( / ) edf let e d ( f ( ) usg Newto-Hesel lftg (Le 3) 3 let log gcd( e ) 4 f ( / ) te 5 retur ( / ) 6 edf 7 let lc 8 retur ( ) Algort Iverso x x ( d e) Fctorzto of uow Te followg Le 4 d Le 5 roved [3] Le 4 Let be dvsor of d let log gcd( ) Te d gcd( / ) s dvsor of log log Le 5 Let be suc tt d gcd( ) gcd( ) Te lc( ) = / gcd( ) s suc tt / d / Teore If f ( s vertble x x Algort returs te verse O( ( )log ) bt oertos Proof Oe c esly rove te correctess of te lgort by ducto o te bse o te ducto beg te cse wc s re were te verse s couted by te gcd lgort To rove te boud o te uber of bt oertos we frst cosder te cost of te sgle stes By (7) we ow tt coutg gcd( f ( x x ) tes O( ( )) O( ( )log (log)loglog) bt oertos By Le we ow tt Cese rederg t Ste tes O( (log)) (log)loglog) bt oertos By Le 3 we ow tt Newto-Hesel lftg t Ste 4 tes O( ( )) bt oertos Flly t s strgtforwrd to verfy tt GetFctors coutes ( ) O( (log )loglog ) bt oertos We coclude tt rt fro te recursve clls te cost of te lgort s doted by te cost of te gcd coutto o tter wc s te outut of te gcd lgort Hece tere exsts costt c suc tt te totl uber of bt oertos stsfes te recurrece T ) c( ) T( ) T( ) were we ssue T ( ) f Let ( 8

10 MSETEE 7 IOP Publsg IOP Cof Seres: Ert d Evroetl Scece (7) do :88/755-35/8// deote te re fctorzto of Defe l( ) We ow sow tt T( ) cl( ) ( ) Sce l( ) lo ) ts wll rove te teore We rove te result by ducto o If te s re d te equlty olds sce te l() l( ) l( ) coutto s doe wtout y recursve cll Let By ducto we ve T( ) cl( ) ( ) T( ) cl( ) ( ) We ve T( ) c( ) c[ l( ) ][ ( ) ( )] wc les te tess sce ( ) ( ) I ddto by Le () () Algort d Algort t s esly to get two lgorts for vertg RSLMFL crcult trces over resectvely 4 Cocluso I ts er te roble of vertg RSFMLR crcult trx wt etres over studed Two dfferet lgorts re reseted Furterore for ec lgort te cost ters of bt oerto re gve Flly te exteded lgorts s used to solve te roble of vertg RSFMLR crcult trces over Acowledgeets Ts wor ws suorted by te GRRC rogr of Gyeogg Provce [(GRRC SUWON 6-B5) d te Nturl Scece Foudto of C uder Grt No 775 Ter suort s grtefully cowledged Refereces [] P Dvs Crcult Mtrces Wley New Yor (979) [] L Jg X ou Crcult Mtrces Cegdu Tecology Uversty Publsg Coy Cegdu (999) [3] D B Corso GMD G Mz L Mrgr Iverso of crcult trces over MtCo () [4] L Jg B Xu S P Go Algorts for fdg te verses of fctor bloc crcult trces Nuer Mt 5() (6) [5] L Jg B Xu A ew lgort for coutg te verse d geerlzed verse of te scled fctor crcult trx J Cout Mt 6 (8) [6] L Jg SY Lu Effcet lgorts for coutg te l olyol d te verse of level- -crcult trces Bull Kore Mt Soc 4(3) (3) [7] L Jg S Y Lu Level- scled crcult fctor trx over te colex uber feld d te qutero dvso lgebr J Al Mt Cout 4(-) 8 96 (4) [8] S G g L Jg S Y Lu A lcto of te Gröber bss coutto for te l olyols d verses of bloc crcult trces Ler Algebr Al () [9] C Dvd Regulr reresettos of sesle lgebrs serble feld extesos grou crcters geerlzed crcults d geerlzed cyclc codes Ler Algebr Al (995) [] L Jg B Xu Effcet lgort for fdg te verse d grou verse of FLS r- crcult trx J Al Mt Cout 8(-) (5) [] L Jg D H Su Fst lgorts for solvg te verse roble of Ax = b Proceedgs of te Egt Itertol Coferece o Mtrx Teory d Its Alctos C 4 (8) s 9

11 MSETEE 7 IOP Publsg IOP Cof Seres: Ert d Evroetl Scece (7) do :88/755-35/8// [] L Jg Fst lgorts for solvg FLS r-crcult ler systes SCET (X ) 4 44 () [3] J L L Jg N Se Exlct deterts of te Fbocc RFPLR crcult d Lucs RFPLL crcult trx JP J Algebr Nuber Teory Al 8() (3) [4] L Jg J L N Se O te exlct deterts of te RFPLR d RFPLL crcult trces volvg Pell ubers forto teory ICICA Prt II Cou Cout If Sc () [5] A V Ao J E Hocroft J D Ull Te Desg d Alyss of Couter Algorts Addso-Wesley Redg Msscussets (974) [6] D B V Y P Polyol d Mtrx Couttos Fud Algort Br äuser (994) [7] A K Lestr H W Lestr Algorts Nuber Teory I J v Leeuwe edtor Hdboo of Teoretcl Couter Scece Algorts d Colexty A Te MIT Press/Elsever (99) [8] A Scöge V Strsse Scelle Multlto grosse le Cout (97)