Randomized Stream Ciphers with Enhanced Security Based on Nonlinear Random Coding

Transcription

1 Jouna of aheacs and Syse Scence 5 (205) do: / / D DAVID PUBLISHING Randozed Sea Cphes wh Enhanced Secuy Based on Nonnea Rando Codng Anon Aekseychuk Segey Gyshakov Insue of Speca Councaon and Infoaon Secuy Naona Technca Unvesy of Ukane "KPI" Kev Ukane Receved: Sepebe / Acceped: Ocobe / Pubshed: Decebe Absac: We popose a faewok fo desgnng andozed sea cphes wh enhanced secuy. The key abue of hs faewok s usng of nonnea becve appngs o keyess hash funcons fo ando codng. We nvesgae he copuaona secuy of he poposed cphes agans chosen-panex-chosen-nazaon-veco aacks and show ha s based on he hadness of sovng soe syses of ando nonnea Booean equaons. We aso povde gudenes fo choosng coponens o desgn andozes fo specfed cphes. Keywods: Syec cypogaphy andozed sea cphe ando codng copuaona secuy chosen-panex-chosen-nazaon-veco aack.. Inoducon In [-5] a genec cass of andozed sea cphes based on on epoyen of dedcaed ando (o hoophonc) codng and eo-coecon codng by nea bnay codes s poposed and suded. One of he goas of desgnng such cphes s o ncease he secuy (whou subsana pefoance educng) of sea cphes cueny used n weess councaon syses pacuay n he GS sandad. Anohe eason s o consuc syec encypon schees whose secuy can be educed o he hadness of soe known aheaca pobe such as he pobe of decodng a ando nea code [6]. Fuhe nvesgaon of he cphes poposed n [3 4] showed [7] ha he copuaona secuy sgnfcany depends on he popees of he coponens and can be consdeaby ess han he desgnes ca. In pacua soe of specfed Coespondng auho: Anon Aekseychuk Doco of Technca Scence Pofesso of Insue of Speca Councaon and Infoaon Secuy of NTUU KPI eseach fed: heoeca cypogaphy E-a: aex-dn@uk.ne. cphes ae vuneabe even o cpheex-ony aacks and he pobe of choosng coponens fo he desgn (accodng o boh secuy and paccay equeens) s non-va and eques fuhe eseach. In hs pape we popose anohe faewok fo desgnng andozed sea cphes wh enhanced secuy. Ths faewok s based on deas fo [8 9] and consss n usng of nonnea becve appngs o keyess hash funcons fo ando codng. I s shown ha he secuy of he poposed cphes n he chosen-panex-chosen-nazaon-veco (IV) aackng scenao s based on he hadness of sovng soe syses of ando nonnea Booean equaons. In addon we povde gudenes fo choosng coponens o desgn andozes fo specfed cphes. 2. Poposed Faewok Fo any naua n denoe by V n he se of a n -densona Booean vecos. The na obecs fo a andozed sea cphe wh paaees N whee < and a key space K ae:

2 Randozed Sea Cphes wh Enhanced Secuy Based on Nonnea Rando Codng 57 () a appng : V V ; (2) a couave goup opeaon on he se V ; (3) a peuaon ax P of ode ; (4) a keysea geneao ha poduces a sequence f k) f ( )... of -densona Booean vecos 0 ( k deened by a key k K. I s assued ha he funcons f : K V 0... can depend on soe pubc paaees (nazaon vecos). To encyp a panex s 0 s... s whee s V 0... wh a key k K he sende geneaes a sequence of ndependen ando vecos u 0 u... u whee u s unfoy dsbued on he se V and copues he cpheex z 0 z... z as foows: z ( u s ( u )) P f ( k) 0 () whee denoes he bwse XOR opeaon. The egae eceve knowng f (k) can fnd he essage ( z z2 ) z f ( k) whee z V z 2 V and he opeaon s defned by he eaon: x y z y x z x y z V. Afe ha he can ecove s by he foua ( z z (see Fg. ). On he ohe hand he s ) 2 advesay n ode o fnd he key k w be foced o dea wh a couped keysea ( u s ( u )) P f ( k) 0. Le us eak ha he obecs P shoud be chosen unde he equeens fo boh he cypogaphc secuy and he peenaon effcency of ansfoaon (). Takng no accoun he as equeen we can se fo exape a b ( a+ b) od 2 whee abay vecos a b V ae denfed wh he coesponded nubes n he se { } and defne P as he ax of a oaon by a cean nube of bs. The appng shoud be chosen uch oe caefuy because s popees nfuence essenay on he secuy of he consdeed cphe (see deas beow). We popose o use one of he wo genea appoaches: () use as a becve appng on he se V (fo 2 ) wh good cypogaphc popees such as hose used n oden bock cphes; (2) use as a keyess hash funcon (such as Keccak [0]). Takng no accoun he fac ha a secue hash funcon suaes a ando appng (n ou case fo V o V ) suffceny we he as vaan ooks oe pefeabe wh egad o povdng adequae secuy of he andozed cphe. Thus he key abue of he poposed faewok s usng of he above enoned nonnea appngs fo desgnng andozed sea cphes. Ths s he an dffeence fo he faewok descbed n [ 5] whee ony bnay nea ansfoaons pacuay eo-coecon codng of npu essages by nea codes ae used. (Ephasze ha he enoned codng s no used n () a a). Aso noe ha n copason wh andozed bock cphes [8 9] nonnea appngs used n andozes of he poposed sea cphes have sghy dffeen equeens eaed o he specfc aacks pecsey on sea cphes. Ths deenes he dffeences beween desgn cea fo andozes used n andozed bock and sea cphes especvey. In pacua keyess hash funcons can be used n andozed sea cphes wh nonnea ando codng ha dffes he poposed faewok fo he one descbed n [8 9]. 3. Secuy Evauaon of he Poposed Cphes Agans Chosen-panex-chosen-IV Aacks 3. Basc Aack Le s consde one of he os powefu aacks on andozed sea cphes [7] when he advesay has access o he encypon oace wh unknown (chosen unfoy a ando fo he se K ) key k

3 58 Randozed Sea Cphes wh Enhanced Secuy Based on Nonnea Rando Codng u u Fg. Bock daga of poposed andozed sea cphe and can choose on hs own nazaon vecos deenng he funcons f 0... The a of he aack s o ecove fo soe fxed he veco f (k) fo a coecon of essages obaned by encypng es he sae essage s 0 unde he sae IV. In hs case he advesay can deve equaons of he fo ( u ( u )) P f ( k) y whee y y2... y ae known and f (k) u... u ae no. Now consde a oe genea pobe. Le + x y ξ (2)

4 Randozed Sea Cphes wh Enhanced Secuy Based on Nonnea Rando Codng 59 be a syse of ando equaons ove a fne abean goup ( G + ) whee ξ ξ2... ξ ae ndependen ando vaabes wh unfo dsbuon on a se G y x 0 + ξ s he esu of subsuon an unknown eeen x 0 G no -h equaon of he syse. I s equed o ecove hs eeen fo he known vaues y y2... y and. I s obvous ha ecovey of he veco f (k) n he consdeed aackng scenao s educed o he sovng he above fouaed pobe f ( G ) ( V ) x f ( ) + 0 k {( u ( u)) P : u V }. (3) I s aso cea ha he se of a souons of syse (2) s equa o ( y ) and conans a eas one eeen (equa o x 0 ). Fo any x G denoe S ( ξ ) ( x + ξ ) x whee ξ ( ξ ξ2... ξ ). Then S x (ξ ) s he nesecon of ndependen ando ses dsbued as foows: { y : A x + y } P { x + ξ A} A G. To fnd he souon x 0 of syse (2) he advesay can use he foowng os naua agoh. Agoh : exhausve seach ove he vaues of ξ and checkng he condon ξ ( y ) 2 ( s assued ha seachng s execued un he fs success). Le s evauae he e copexy of Agoh. Suppose ha he addon of any wo eeens a b G and he check of condon a fo any a G ake consan e. Le s denoe y { y : y + a } d ( a) a G (4) d ax{ d ( a) : a G \ {0}}. (5) Saeen. Suppose ha d <. Then fo any δ (0) and og( ) δ og( d ) + he souon x 0 of syse (2) can be found wh pobaby a eas δ n e O ( ). Poof. I s suffcen o pove ha he eo pobaby p e of Agoh sasfes he nequay e d p. (6) Suppose ha Agoh akes a sake; hen hee exss an eeen x G \ { } whch beongs o he se S ( ξ ) ( x + ξ. Snce 0 ) ξ2 ξ ae ndependen ando vaabes wh ξ... unfo dsbuon on he se we ge p P { x x e S x 0 ( ξ)} ( { y : y + x } ) x. Now usng (4) (5) and he noaon I (z) z G fo he ndcao of he se we oban ha d ( ) d x) d p e ( d ( x) x x d I ( z) I ( z + a) a 0 z G d I ( z) I ( z + a) z G a 0 d ( ) d. Thus nequay (6) and he saeen ae poved. Cooay. Le condon (3) hods and c ( ) 2 whee c cons. Then he souon

5 520 Randozed Sea Cphes wh Enhanced Secuy Based on Nonnea Rando Codng x 0 of syse (2) can be found wh pobaby a eas and δ n e. ogδ O 2 + as δ 0 Noe ha n he wos case Agoh eques seachng a ( ) -densona Booean vecos. Hence becoes pacca e.g. as Anohe Vaans of he Aack n a Pacua Case Le s consde an poan pacua case when n syse (2) ( G + ) ( V ) x f ( ) 0 k {( u ( u)) : u V }. (7) Noe ha n hs case paaee (5) concdes wh he quany D ax α V \{0} β V {2 ( ) { z V : ( z α) ( z) β} } (8) whch easues he essance of he appng agans dffeena cypanayss (see [] fo exape). In hs case o sove syse (2) we can appy anohe echnque whch s soees oe effecve han Agoh. Ths echnque s eaed o nea cypanayss and coeaon aacks on andozed sea cphes wh nea ando codng [3 5 7]. Fo any n N a ( a... a n ) b ( b... b n ) V n denoe ab a b a n b n. Le s defne ( ) ( a b) 2 { z V : az b( z) L a V b V ; ax a V b V \{0} { 2 ϕ ( a b)) }. (9) Consde he foowng agoh of ecoveng he souon x 0 of syse (2) unde condon (7). Agoh 2.. Choose neay ndependen vecos ( a b ) whee a V b V \ {0} such ha ( a b ) Fo evey oban fo (2) he syse of equaons ( a u b ( u )) ( a b ) ( a b ) y (0) and ecove he quany ( a b ) usng he aoy ue: f ( a b ) < 2 hen def ( a b ) x 0 0 ( a b ) y < 2 ; f ( a b ) > 2 hen def ( a b ) x 0 0 ( a b ) y > 2 ; 3. Fnd x 0 fo he Obaned Quanes ( a b ) usng Gaussan Enaon Noe ha sep and he ansfoaon of he ax wh he ows ( a b ) on sep 3 of Agoh 2 ae execued ony once (a he sage of pecopuaon). Theefoe he e copexy of hs agoh s deened by he execuon e of sep 2. The poof of he foowng saeen s aos he sae as he poof of Saeen 4 n [7]. Saeen 2. Unde condon (7) he advesay can ecove on sep 2 of Agoh 2 a vaues ( a b ) wh pobaby a eas δ δ (0 ) n O ( og) b opeaons fo 2 2 ax { 2 (( a b )) }n( δ ) abay equaons of syse (2). Noe ha he daa copexy.e. he nube of equaons n syse (0) necessay fo ecoveng one abay quany ( a b ) on sep 2 wh pobaby a eas δ s owe bounded by 2 C 2 (( a b )) whee he vaue C depends

6 Randozed Sea Cphes wh Enhanced Secuy Based on Nonnea Rando Codng 52 ony fo δ. Theefoe Agoh 2 becoes pacca f he vaue of (9) s suffceny sa (e.g. L 2 32 ). In concuson consde anohe possbe appoach (based on deas fo agebac cypanayss) fo sovng syse (2) unde condon (7). Denoe x x x ) ξ u ( u )) ( 2 ( y ( α β ) whee x u α V x 2 β V. Then (2) s equvaen o he syse of equaons: x u α x ( u ) β 2 whch can be wen as foows: ( z ( α α )) ( z β β 2 () ) x α z 2 ( z) β u x α α z. The foowng saeen s obvous. Saeen 3. Unde condon (7) he copuaona secuy of he consdeed cphe s uppe bounded by he e copexy of sovng he syse of equaons () fo abay (known) vecos α β. Thee a o of faes of Booean appngs wh sa vaues of paaees (8) and (9) (see [ 2] fo exape). Bu no a of he guaanee hgh copexy of sovng syses of he fo (). As an exape consde he appng 2 2 ( x ) x x GF ( 2 ) whee 2 s even wdey used n oden bock cphes. I s known ha 2 2 D L 2 bu each sepaae equaon of syse () has a os fou souons whch can be found n ea e [3]. Thus havng a sa (naey og + δ + og δ + + ) nube of og( ) 2 D equaons n syse (2) we can fnd n ea e s (unque wh pobaby a eas δ δ (0) ) souon x 0 by sovng syse (). A he sae e he pobe of sovng a syse of he fo () fo abay appng : V V s copuaonay had. oeove no effcen agohs of sovng such syses fo any oden copuaonay secue hash funcons ae known so fa. I sees vey key ha he exsence of such agohs can be an undesabe popey whch w aow us o dsnc a hash funcon fo he uy ando appng. 4. Concuson In conas o befoe known appoaches [ 5 8 9] he descbed faewok gves oe possbes fo desgnng copuaonay secue andozed sea cphes. Ths s acheved by enagng he cass of ansfoaons used n he consucon of a andoze. As he opeaon n () he addon oduo 2 o he bwse Booean addon of bnay vecos can be used besdes n he as case he ax P can be chosen as he deny ax of ode. Unde condon (7) he copuaona secuy of he poposed andozed sea cphes n he chosen-panex-chosen-iv aackng scenao s deened by he foowng popees of he appng : V V : (a) age vaue of o ess exhausve seach (Agoh ); (b) sa vaue of (9) o ess he nea-ype aack (Agoh 2); (c) hgh e copexy of sovng syses of equaons (). In ode o ncease he paccay of he encypon schee s aso desabe o choose he quany suffceny age n copason wh. Fo exape we can pu 28 ha povdes encypon ae 2 ndependeny fo he choce of. In ode o ess he consdeed above (as we as ohe possbe) aacks s desabe ha appng

7 522 Randozed Sea Cphes wh Enhanced Secuy Based on Nonnea Rando Codng shoud have popees sa o hose of ando equpobabe appng fo V o V. Fo hs pon of vew s naua o choose as one of oden keyess hash funcons. We conecue ha any genea aack on he andozed sea cphe can by effceny ansfoed n an appopae aack on he undened hash funcon bu cueny we don have an accuae poof of hs saeen. Refeences [].J. haevć H. Ia A sea cpheng appoach based on weap channe codng 8 h Cena Euopean Confeence of Cypogaphy 2008 Gaz Ausa Juy 2-4 E-Poc. (3 p.) [2].J. haevć H. Ia An appoach fo sea cphe desgn based on on copung ove ando and sece daa Copung 2009 Vo pp [3].J. haevć H. Ia An nfoaon-heoec and copuaona copexy secuy anayss of a andozed sea cphe ode 4h Wesen Euopean Wokshop on Reseach n Cypoogy WeWoRC 20 Wea Geany Juy Conf. Recod 20 pp [4].J. haevć H. Ia Epoyen of hoophonc codng fo poveen of cean encypon appoaches based on he LPN pobe Syec Key Encypon Wokshop SKEW 20 Copenhagen Denak Feb. 6-7 E-Poc. (7 p.) 20. [5].J. haevć F. Ogge H. Ia Hoophonc codng desgn fo councaon syses epoyng he encodng-encypon paadg axv: v [cs.cr] 29 Dec 200. [6] E.R. Beekap R.J. cece H. van Tbog On he nheen nacaby of cean codng pobes IEEE Tans. on Info. Theoy 978 Vo. 24 No. 3 pp [7] A.N. Aekseychuk S.V. Gyshakov On he copuaona secuy of andozed sea cphes poposed by haevć and Ia Zakhs Info. 204 No. 4 pp [8] A.N. Aekseychuk Anayca esaes of heoeca secuy of andozed bock cphes agans dffeena cypanayss Zakhs Info No. 3 pp (n Russan). [9] A.N. Aekseychuk Suffcen condons fo andozed bock cphe-syses o be secue agans couave daga cypanayss Daa Recodng Soage and Pocessng 2007 Vo. 9 No. 2 pp (n Russan). [0] ECRYPT II: Fna hash funcon saus epo hp:// 3 Jan [] A. Caneau Cypogaphc funcons and desgn cea fo bock cphes INDOCRYPT 200 LNCS 2247 Spnge Veag 200 pp. -6. [2] C. Cae Vecoa Booean funcons fo cypogaphy n Booean odes and ehods n aheacs Copue Scence and Engeneeng Cabdge Unvesy Pess 200 pp [3] K. Nybeg Dffeenay unfo appngs fo cypogaphy Advances n Cypoogy EUROCRYPT 93 LNCS 765 Spnge Veag 994 pp