NEW ATTACKS ON TAKAGI CRYPTOSYSTEM

Transcription

1 Jounal of Algba umb Thoy: Advancs and Alcatons Volum 8 umb - 7 Pags 5-59 Avalabl at htt://scntfcadvancscon DOI: htt://dxdoog/86/antaa_785 EW ATTACKS O TAKAGI CRYPTOSYSTEM MUHAMMAD REAL KAMEL ARIFFI SADIQ SHEHU and M A ASBULLAH Al-Knd Cytogahy Rsach Laboatoy Insttut fo Mathmatcal Rsach Unvst Puta Malaysa (UPM) Slango Malaysa Datmnt of Mathmatcs Faculty of Scnc Unvst Puta Malaysa (UPM) Slango Malaysa -mal: zal@umdumy sadshhuzz@gmalcom ma_asyaf@umdumy Abstact Ths a ooss th nw attacks on RSA-Takag cytosystm Th fst attack s basd on th uaton X Y ( a b ) fo sutabl ostv Y ntgs a b W show that can b covd among th convgnts of th X contnud factons xanson of and lads to succssful factozaton of th Mathmatcs Subct Classfcaton: Y5 Kywods and hass: m ow modulo factozaton LLL algothm smultanous Dohantn aoxmatons contnud factons Rcvd Jun 7 7 Scntfc Advancs Publshs

2 6 MUHAMMAD REAL KAMEL ARIFFI t al m ow modulus n olynomal tm Th scond and thd attack woks uon ublc kys ( ) whn th xst latons of th sha x y ( a b ) z o of th sha x y ( a b ) z wh th aamts x x y y z a sutably small n tms of th m factos of th modul Alyng th LLL algothm w show that ou statgy nabl us to smultanously facto th ublc ky n olynomal tm Intoducton In cnt yas modulus of th fom hav found many alcatons n cytogahy In [] Bonh t al oosd an ffcnt algothm fo factong modulus of th fom and showd that th algothm uns n olynomal tm whn s lag ( log ) Hnc t s xctd that th factong of th modulus wll b ntactabl whn th bound fo s small Fuoka t al [5] usd th modulus fo n an lctonc cash schm Okamoto and Uchyama [5] usd ky systm wth n dsgnng an lgant ublc Th cytosystm dvlod by Takag ushd n sach n dtmnng th scuty of th modulus In [8] Takag oosd a cytosystm usng modulus basd on th RSA cytosystm H chooss an aoat modulus whch ssts two of th fastst factong algothms namly th numb fld sv and th lltc cuv mthod Alyng th fast dcyton algothm modulo h showd that th dcyton ocss of th oosd cytosystms s fast than th RSA cytosystm usng Chns mand thom known as th Qusuat-Couvu mthod

3 EW ATTACKS O TAKAGI CRYPTOSYSTEM 7 In [7] Saka ovd that usng th lattc ducton tchnus f 95 th dcyton xonnt d thn on can facto th m ow modulus n olynomal tm Asbullah and Affn [] ovd that by takng th tm ( ) as a good aoxmaton of φ ( ) satsfyng th RSA ky uaton d k φ( ) on can yld th factozaton of th m ow modulus (fo mo nfomaton s [] [6] [7]) n olynomal tm Ou fst oosd attack uss th Lgnd thom whch nabls us to fnd th convgnt of th contnud factons that lads to th factozaton of th modulus n olynomal tm Th scond and thd attacks uss lattc bass ducton W a ntstd n th so calld ducd bass of a lattc so as to yld factozaton of th modul K n olynomal tm Th mand of ths a s oganzd as follows In Scton w gv ntoducton to contnud factons lattc bass ducton wth som vous sults In Scton w snt th fst attack and stmaton of th sz of th class of th xonnts fo whch ou attack als In Sctons and 5 w gv th scond and thd attacks W also ovd numcal xaml fo all ou attacks W conclud ths a n Scton 6 Plmnas W stat wth dfntons and motant thoms concnng th contnud factons lattc bass ducton tchnus and som thom fom th vous attacks as wll as som usful lmmas

4 8 MUHAMMAD REAL KAMEL ARIFFI t al Contnud factons Dfnton (Contnud factons) A contnud facton s an xsson of th fom a a O a m O [ a a K a ] m K wh a s an ntg and a n a ostv ntgs fo n Th a n a calld th atal uotnts of th contnud facton [] Dfnton (Convgnts) Lt x R wth x [ a a K am ] Fo n m th n-th convgnt of th contnud facton xanson of x s [ a a K a ] n Thom (Lgnd) Lt x b a al ostv numb If X and Y a ostv ntgs such that gcd ( X Y ) and Y x X X thn X Y s a convgnt of th contnud facton xanson of x Dfnton (Lattc bass ductons) Lt ntgs and m n b two ostv n L b m R b n lnaly ndndnt vctos A lattc b L sannd by { b L b m } s th st of all ntg lna combnatons of b L b that s m L L( b L b m ) αb α m Th b a calld bass vctos of L and B b L bm s calld a lattc bass fo L Thus th lattc gnatd by a bass B s th st of all ntg lna combnatons of th bass vctos n B

5 EW ATTACKS O TAKAGI CRYPTOSYSTEM 9 Th dmnson (oank) of th a lattc dnotd dm( L ) s ual to th numb of vctos makng u th bass Th dmnson of a lattc s ual to th dmnson of th vcto subsac sannd by B A lattc s sad to b full dmnsonal (o full ank) whn dm( L ) n Thom Lt L b a lattc of dmnson ω wth a bass v K v Th LLL algothm oducs a ducd bass b K bω satsfyng ω b b K b ω( ω) ( ω ) dt Lω fo all ω As an alcaton of th LLL algothm s that t ovds a soluton to th smultanous Dohantn aoxmatons oblm whch s dfnd as follows Lt α K αn b n al numbs and ε b a al numb such that ε A classcal thom of Dchlt assts that th xst ntgs K n and a ostv ntg n ε such that α ε fo n A mthod to fnd smultanous Dohantn aoxmatons to atonal numbs was dscbd by [] In th wok thy consdd a lattc wth al nts Blow a smlasult fo a lattc wth ntg nts Thom (Smultanous Dohantn aoxmatons [8]) Th s a olynomal tm algothm fo gvn atonal numbs α K αn and ε to comut ntgs K n and a ostv ntg such that n( n) max α ε and Lmma Lt Thn b an RSA modulus m ow wth

6 MUHAMMAD REAL KAMEL ARIFFI t al Poof Suos thn multlyng by w gt whch mls that s Also snc thn whch n tun mls Hnc [] Lmma Lt b a m ow modulus wth and b a b sutably small ntgs such that ( ) gcd b a Also lt ( ) b a S wh b a thn S ab Poof St ( ) b a S Thn obsv that (( ) ) ( )( ) b a b a b a S b ab z b a a b ab a b ab ab ab a ab b ab a ab b ab a ab b ab a ( ) ab b a Hnc w obtan ( ) > b a ab S ()

7 EW ATTACKS O TAKAGI CRYPTOSYSTEM Thn w dvd () by w gt S S ab ab ( a b ) ( a b ) ( a b ) a b 6 6 mls that ab S Th Fst Attack on Pm Pow Modul In ths scton w snt a sult basd on contnud factons and show how to facto th m ow modulus f ( ) s a ublc ky satsfyng an uaton X Y ( a b ) wth small aamts X Y and wh a b b a sutably small ostv ntg

8 MUHAMMAD REAL KAMEL ARIFFI t al and Lmma Lt b a m ow modulus wth a b b ntgs such that gcd ( a b) Lt b a ublc ky satsfyng th uaton X Y ( a b ) wth gcd ( X Y ) f X ( a b ) thn Y s among th convgnts of th contnud X facton xanson of Poof Suos that satsfs th uaton X Y ( a b ) wth X ( a b ) and gcd ( X Y ) w gt Thn fom th uaton X Y ( a b ) whn dvdng by X Y X X Y X a b X Assum that f X ( a b ) thn a b X X hold that s X ( a b ) X X( a b ) X( a b ) whch mls X ( a b ) and by Thom w conclud that X Y contnud facton xanson of s among th convgnt of th

9 EW ATTACKS O TAKAGI CRYPTOSYSTEM Thom Lt b a m ow modulus wth Lt a b b ntgs such that gcd ( a b) and lt b a ublc ky satsfyng th uaton X Y ( a b ) wth gcd ( X Y ) f Y X and ( a b ) a b thn fo can b factod n olynomal tm Poof Suos that satsfs an uaton X Y ( a b ) wth gcd ( X Y ) lt X and satsfy th condton n Lmma thn Y s among th convgnt of th contnud facton xanson of X Hnc usng X and Y w dfn S X Y and Lmma shows S that ab It follows that gcd S Th followng algothm s dsgnd to cov th m factos fo m ow modulus n olynomal tm Algothm Inut: Th ublc ky a ( ) satsfyng and Thom Outut: Th two m factos and () Comut th contnud facton xanson of Y () Fo ach convgnt of X S () Comut () gcd S (5) If thn comut S X Y

10 MUHAMMAD REAL KAMEL ARIFFI t al Examl Th followng shows an llustaton of ou attack fo X 9 Y 8 a b gvn and as Suos that th ublc ky ( ) satsfy all th condton as statd n th Thom fom th abov algothm w fst comut th contnud facton xanson of Th lst of fst convgnts of th contnud facton xanson of a K Thfo omttng th fst and scond nty and stat wth th convgnt w obtan and S X Y S Hnc gcd S ( ) 7 S Also th convgnt gvs S and 9 wth gcd S

11 EW ATTACKS O TAKAGI CRYPTOSYSTEM 5 8 Thfo w nd to ty fo th nxt convgnt w obtan 9 S X Y and S W comut th gcd S ( ) 77 Fnally wth 77 w comut 8557 whch lads to th factozaton of Estmaton of th numb of s satsfyng X Y ( a b ) W gv an stmaton of th numb of th xonnts whch ou attacks can b ald Lt fo a b b ntgs such that α gcd ( a b) Lt ( a b ) wth α Lmma 5 Lt b a m ow modulus wth Lt a b b ntgs such that gcd ( a b) and suos that s a ublc xonnt satsfyng and two uaton X Y ( a b ) and X Y ( a b ) wth gcd ( X ) fo Y Y X thn X X Y Y ( a b )

12 6 MUHAMMAD REAL KAMEL ARIFFI t al Poof Assum that th xonnt satsfyng th two uaton Y ( a b ) and X Y ( a b ) wth X gcd ( X ) fo Y Y X Thfo ( a b ) uatng th tm ( a b ) w gt X Y X Y () mls X Y X Y ( X X ) ( Y ) Y ( X X ) ( Y Y ) Snc w assum and Y X thn ( X X ) X X thfo ( a b ) ( a b ) wth gcd( ) and X X w obtan X X Y Y Thom 5 Lt b a m ow modulus wth Lt a b b sutably small ntgs such that gcd ( a b) and α ( a b ) Th numb of th xonnts of th fom ( a b ) X ( mod ) wth g cd( X a b ) and X α s at last wh > s abtaly small fo sutably lag

13 EW ATTACKS O TAKAGI CRYPTOSYSTEM 7 Poof Lt a b b sutably small ntgs such that gcd ( a b) α and ( a b ) α and lt X Lt ξ dnot th numb of th xonnts satsfyng ( a b ) X (mod ) wth gcd ( X a b ) and X α ξ X X gcd( X a b ) () Usng th followng sult (s ta [5] Lmma ) wth and m X w gt n a b X ( ) φ a b ( ) ( ) ω a b φ a b ω( a b ξ > X ) a b a b () Thfo ω( a b ) s th numb of sua f dvsos of a b whch s u boundd by th total numb τ ( a b ) of dvsos of a b Hnc usng th dntty that τ ( n) satsfs τ ( n) O( log log n ) (s Hady and Wght [6] Thoms -) It follows that th domnant tm n () s n a b φ( a b ) a b X α and X gvs Substtutng ths wth ξ X ( ) α φ a b φ( ) a b a b α O α φ( a b )

14 8 MUHAMMAD REAL KAMEL ARIFFI t al Also on th oth hand fo n w hav th followng dntty (s Hady and Wght [6] Thom 8) cn φ ( n ) > log log n wh c s a ostv constant Takng n α a b mls that ξ α O α c α log log O ( ) wh α satsfs log log sutably lag and s abtaly small fo Rmak Fom th two dstnct n-bt m ( ) th sultant modulus s ( ) n-bt ntg Thn w can obsv that th numb of xonnts satsfyng ou attack s ( ) n( ) Ths ovs that th a xonntally many xonnts that satsfy ou condtons n th Thom 5 Th Scond Attack on Pm Pow Modul In ths scton fo modul wth th sam sz W suos n ths scnao that th m ow modul satsfyng th uatons x y ( a b ) z W ovd that t s ossbl to facto th modul sutably small f th unknown aamts x y and z a

15 EW ATTACKS O TAKAGI CRYPTOSYSTEM 9 Thom 6 Fo lt b modul Lt mn Lt K b ublc xonnts Dfn δ ( ) α( ) ( ) wh α Lt a b b sutably small α ntgs such that a b If th xst an ntg δ x and δ ntgs y and z such that ( x y a b ) z fo K thn on can facto th modul K n olynomal tm Poof Fo and lt b modul Lt mn and suos that y and a b thn th uaton x y ( a b ) z can b wttn as δ α ( a b ) z x y (5) δ Lt mn and suos that y z α a thn and b ( a b ) z ( a b ) z α α α

16 MUHAMMAD REAL KAMEL ARIFFI t al 5 Substtut n to (5) to gt α y x Hnc to shows th xstnc of th ntg x w lt α ε wth ( ) ( ) ( ) α δ thn w hav ε δα δ Thfo snc ( ) fo w gt ( ) ε δ It follows that f δ x thn ( ) x ε Summazng fo K w hav ( ) x y x ε ε Hnc t satsfy th condtons of Thom and w can obtan x and y fo K xt usng th uaton ( ) z b a y x Snc z Thn Lmma mls that S ab z wth y x S fo K w comut gcd S Whch lads to factozaton of modul K

17 EW ATTACKS O TAKAGI CRYPTOSYSTEM 5 Examl As an llustaton to ou attack on m ow modul w consd th followng th m ow and th ublc xonnts: Thn mn( ) Snc and a b wth α w gt δ ( ) α( ) ( ) 5 α and ε 5587 Usng Thom wth n w obtand C ( n )( n) n [ ε n ] Consd th lattc L sannd by th matx M [ C ] [ C ] [ C ] C C C Thfo alyng th LLL algothm to L w obtan th ducd bass wth followng matx:

18 5 MUHAMMAD REAL KAMEL ARIFFI t al K xt w comut K M Thn fom th fst ow w obtand x 7 y 67 y 558 y 7 Hnc usng x and y fo dfn S x y w gt S S S And Lmma mls that z ab S fo whch gvs S S S

19 EW ATTACKS O TAKAGI CRYPTOSYSTEM 5 Thfo fo w comut gcd S that s Fnally fo w fnd hnc whch lads to th factozaton of th modul and 5 Th Thd Attack on Pm Pow Modul W snt an attack on th m ow modul Fo and w consd th scnao whn th modul satsfy uatons of th fom x y ( a b ) z fo K wth sutably small unknown aamts x y and z Alyng th LLL algothm w show that ou aoach nabl us to facto th m ow modul n olynomal tm Thom 7 Fo and lt b modul wth th sam sz Lt K b ublc xonnts wth mn β Lt β δ ( β α ) ( β α ) ( ) wh α Lt a b b sutably ntgs such that a b α If th xst an ntg δ y and ntgs δ x such that x y ( a b ) z fo K thn on can facto th modul K n olynomal tm

20 5 MUHAMMAD REAL KAMEL ARIFFI t al Poof Fo and lt b modul Thn th uaton x y ( a b ) z can b wttn as ( a b ) z y x (6) Lt max and suos that and a b thn α y δ β z mn ( a b ) z ( a b ) z β β α α β α β Pluggng n to (6) to gt y x α β Hnc to shows th xstnc of th ntg y and ntgs x w lt β α ε wth δ ( β α ) ( β α ) w gt ( ) δ δ ε αβ

21 EW ATTACKS O TAKAGI CRYPTOSYSTEM 55 ( ) Thfo snc fo w gt It follows that f y thn K w hav δ ( ) y ε ( ) δ ε Summazng fo y x ( ) ε y ε Hnc t satsfy th condtons of Thom and w can obtan y and x fo K xt fom th uaton x y ( a b ) z Snc z S Thn Lmma mls that z ab wth S x y fo K w comut gcd S Whch lads to factozaton of modul K Examl As an llustaton to ou attack on m ow modul w consd th followng th m ow and th ublc xonnts:

22 56 MUHAMMAD REAL KAMEL ARIFFI t al Thn max( ) β Also mn ( ) wth β 98 Snc and a b wth α w gt ( β α ) ( β α ) δ 6 and ( ) αβ ε Usng Thom wth n w obtand C ( n ) ( n) n [ n ε ] 9559 Consd th lattc L sannd by th matx M [ C ] [ C ] [ C ] C C C Thfo alyng th LLL algothm to L w obtan th ducd bass wth followng matx: K xt w comut K M

23 EW ATTACKS O TAKAGI CRYPTOSYSTEM 57 Thn fom th fst ow w obtand y 75 x 9 x x Hnc usng x and y fo dfn S x y w gt S S S And Lmma mls that z ab S fo whch gvs S S S Thfo fo w comut gcd S that s Fnally fo w fnd hnc whch lads to th factozaton of th modul and

24 58 MUHAMMAD REAL KAMEL ARIFFI t al 6 Concluson W oosd th fst attack basd on th uaton X Y ( a b ) fo sutabl ostv ntgs a b Usng contnud facton w show that X Y can b covd among th convgnts of th contnud factons xanson of Futhmo w show that th st of such wak xonnts s latvly lag namly that th numb s at last ε wh ε s abtaly small fo sutably lag Hnc on can facto th m ow modulus n olynomal tm Fo w thn snt scond and thd attacks on th m ow modul fo K Th attacks wok whn ublc kys ( ) a such that th xst latons of th sha x y ( a b ) z o of th sha x y ( a b ) z wh th aamts x x y y z a sutably small n tms of th m factos of th modul Basd on LLL algothm w show that ou aoach nabl us to smultanously facto th m ow modul n olynomal tm Rfncs [] M R K Affn and S Shhu w attacks on m ow RSA modulus Asan Jounal of Mathmatcs and Comut Rsach (6) 77-9 [] M A Asbullah and M R K Affn w attacks on RSA wth modulus usng contnud factons Jounal of Physcs Confnc Ss Volum 6 o IOP Publshng 5 [] D Bonh G Duf and Howgav-Gaham Factong fo lag Advancs n Cytology CRYPTO 99 Lctu ots n Comut Scnc 59 (999) 6-7

25 EW ATTACKS O TAKAGI CRYPTOSYSTEM 59 [] J Blom and A May A gnalzd Wn attack on RSA In Publc Ky Cytogahy - PKC Lctu ots n Comut Scnc 97 () - [5] A Fuoka T Okamoto and S Myaguch ESIG: An ffcnt dgtal sgnatu mlmntaton fo smat cads Advancs n Cytology EURO-CRYPT 9 Sng-Vlag (99) 6-57 [6] G H Hady and E M Wght An Intoducton to th Thoy of umbs Oxfod Unvsty Pss London 975 [7] J Hnk On th Scuty of Som Vaants of RSA PhD Thss Watloo Ontao Canada 7 [8] Howgav-Gaham and J P Sft Extndng wns attack n th snc of many dcytng xonnts In Scu twokng-cqre (Scu) 99 7 (999) 5-66 [9] A K Lnsta H W Lnsta and L Lovasz Factong olynomals wth atonal coffcnts Mathmatsch Annaln 6 (98) 5-5 [] A May w RSA Vulnablts Usng Lattc Rducton Mthods PhD Thss Unvsty of Padbon [] A ta Dohantn and lattc cytanalyss of th RSA cytosystm Atfcal Intllgnc Evolutonay Comutng and Mtahustcs Sng Bln Hdlbg () 9-68 [] A ta Cytanalyss of RSA usng th ato of th ms Pogss n Cytology- AFRICACRYPT 9 Sng Bln Hdlbg (9) 98-5 [] A ta M R K Affn D I ass and H M Bahg w attacks on th RSA cytosystm Pogss n Cytology-AFRICACRYPT Sng Intnatonal Publshng () [] A ta A w Vulnabl Class of Exonnts n RSA [5] T Okamoto and S Uchyama A nw ublc-ky cytosystm as scu as factong Advancs n Cytology-EUROCRYPT 98 Sng-Vlag (998) 8-8 [6] R Rvst A Sham and L Adlman A mthod fo obtanng dgtal sgnatus and ublc-ky cytosystms Communcatons of th ACM () (978) -6 [7] S Saka Small sct xonnt attack on RSA vaant wth modulus dsgns Cods and Cytogahy 7() () 8-9 [8] T Takag Fast RSA-ty cytosystm modulo k Cyto 98 Sng (998) 8-6 In Advancs n Cytology- [9] M Wn Cytanalyss of shot RSA sct xonnts IEEE Tansactons on Infomaton Thoy 6 (99) g