Public Auditing Based on Homomorphic Hash Function in

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1 Publc Audtg Bsed o Homomorhc Hsh Fucto Secure Cloud Storge Shufe NIU, Cfe Wg, Xo DU Publc Audtg Bsed o Homomorhc Hsh Fucto Secure Cloud Storge Shufe NIU, Cfe Wg, 3 Xo DU, College of Comuter Scece d Egeerg, Northwest Norml Uversty, Lzhou, Gsu , Ch, E-ml:{sfu76,dux}@wueduc *, Corresodg Author College of Comuter Scece d Egeerg, Northwest Norml Uversty,Lzhou, Gsu , Ch, E-ml:wgcf@wueduc 3, College of Comuter Scece d Egeerg, Northwest Norml Uversty, Lzhou, Gsu , Ch Abstrct Publc verfcto ebles thrd rty udtor (TPA), o the behlf of the dt ower, to verfy the tegrty of cloud storge wth the dt ower s ublc key I ths er, by utlzg homomorhc vector hsh fucto, we roose secure cloud storge scheme suortg rvcy-reservg ublc udtg for multle fles wth dfferet detfer To cheve rvcy reservg ublc udtg, we roose to tegrte the homomorhc ler uthetctor wth rdom vector mskg to hde ler combto of the dt fles Mewhle, the tme t tkes to udt re ot ffected by the umber of fles The securty of ublc udtg scheme reles o the hrdess of the comuttol Dffe- Hellm d dscrete logrthm roblems uder the rdom orcle model Furthermore, through theoretcl lyss d exermetl results, the roosed scheme s demostrted to hve effcet erformce Keywords: Dt Storge, Prvcy-Preservg, Publc Audtblty, Cloud Comutg, Homomorhc Hshg Fucto Itroducto Storg dt the cloud hs become tred Icresg the umber of clets store ther mortt dt remote servers the cloud, wthout levg coy ther locl comuters Sometmes the dt stored the cloud s so mortt tht the clets must esure t s ot lost or corruted For the clets, the tsks of udtg the dt correctess cloud evromet c be formdble d exesve So the clets my resort to deedet thrd rty udtor (TPA) to udt the outsourced dt Becuse TPA c ot retreve the etre storge fle, trdtol crytogrhc rmtves for the urose of dt securty rotecto cot be drectly doted [],[],[3] Ateese et l [] frst roosed rovble dt ossesso (PDP), whch llows clet to verfy the tegrty of her dt stored utrusted server wthout retrevg the etre fle Wg et l [4] costructed ublc udtg mechsm for cloud dt I the scheme, the cotet of rvte dt belogg to ersol clet s ot dsclosed to the thrd rty udtor Wg et l [6] frst roosed mechsm for ublc udtg shred dt the cloud for grou of clets Wth rg sgture-bsed homomorhc uthetctors, the TPA c verfy the tegrty of shred dt but s ot ble to revel the detty of the sger o ech block Che et l [7] troduced mechsm for udtg the correctess of dt wth the mult-server scero, where these dt re ecoded wth etwork codg More recetly, Co et l [8] costructed LT code-bsed secure cloud storge mechsm To overcome ths drwbck, roer roch s to combe the homomorhc uthetctor wth rdom mskg [4][0] Wth rdom mskg, the TPA o loger hs ll the ecessry formto to buld u correct grou of ler equtos d therefore cot derve the ower s dt cotet I ths er, we roose ew rvcy-reservg ublc udtg mechsm for dt storge utrusted cloud I our roch, we utlze homomorhc vector hshg fucto [] to costruct homomorhc uthetctors [],[9],[] so tht the thrd rty udtor s ble to verfy the tegrty of dt wthout retrevg the etre dt To cheve rvcy reservg ublc udtg, we roose to uquely tegrte the homomorhc ler uthetctor wth rdom vector mskg to hde ler combto of the dt fles We hve the followg m cotrbutos: Itertol Jourl of Dgtl Cotet Techology d ts Alctos(JDCTA) Volume7,Number,Jury 03 do:0456/dctvol7ssue87 7

2 Publc Audtg Bsed o Homomorhc Hsh Fucto Secure Cloud Storge Shufe NIU, Cfe Wg, Xo DU )We cosder m fles wth m dfferet detfers By exlotg homomorhc hsh fucto, uthetctor for fle F ( m, m,, m)(,,, m) s comuted, ot for fles block m Thus for fle F, we roduce oe uthetctor, ot uthetctors,,,, Mewhle, the mout of formto used for verfcto, s well s the comutto cost t the verfcto rocess s deedet from the umber of fles Ths rocesses c sgfctly reduce the commucto, comutto d storge costs )To cheve rvcy-reservg ublc udtg, we roose to uquely tegrte the homomorhc ler uthetctor wth rdom vector mskg w ( w, w,, w ) to hde ler combto of the dt fles 3)The effcecy of the roosed lgorthm wll be ssessed throughtheoretcl lyss d exermetl smulto The rest of ths er s orgzed s follows: I Secto, deftos d relmres re reseted The roosed rvcy-reservg ublc udtg bsed o homomorhc hshg scheme s reseted secto 3 I Secto 4, securty lyss of the roosed scheme s dscussed I Secto 5,the scheme s comlexty s lyzed the sects of comutto costs, furthermore, exermetl results re reseted for the effcecy of the roch ths secto Flly, coclusos d ossble reserch drectos re reseted Secto 6 Deftos d Prelmres We cosder cloud dt storge servce volvg three dfferet ettes, s llustrted Fg : the cloud user, who hs lrge mout of dt fles to be stored the cloud; the cloud server, whch s mged by the cloud servce rovder to rovde dt storge servce d hs sgfct storge sce d comutto resources; the thrd rty udtor (TPA), who hs exertse d cbltes tht cloud users do ot hve d s trusted to ssess the cloud storge servce relblty o behlf of the user uo request Audtg Model We follow smlr defto of revously roosed schemes the cotext of [], [], d dt the frmework for our rvcy-reservg ublc udtg system Defto A ublc udtg scheme bsed o homomorhc hshg cossts of the followg fve lgorthms: SetU, SgGe, Chllege, GeProof d CheckProof k SetU ( ) ( k, sk) : Gve the securty rmeter k, ths fucto geertes the ublc key k d the secret key sk k s ublc to everyoe, whle sk s ket secret by the user SgGe ( k, sk, F) F : Gve k, sk d F, ths fucto comutes verfcto sgture F d mkes t ublcly kow to everyoe Ths sgture wll be used for ublc verfcto of dt tegrty Chllege ( k, F ) chl : Usg ths fucto, the TPA geertes chllege chl to request for the tegrty roof of fles F The TPA seds chl to the server GeProof ( k,, F, chl) P : Usg ths fucto, the server comutes resose P to the F chllege chl The server seds P bck to the TPA CheckProof ( k, F, chl, P) ( success, flure) : The TPA checks the vldty of the resose P If t s vld, the fucto oututs success, otherwse the fucto oututs flure The secret key sk s ot eeded the CheckProof fucto 7

3 Publc Audtg Bsed o Homomorhc Hsh Fucto Secure Cloud Storge Shufe NIU, Cfe Wg, Xo DU Securty Requremets There re two securty requremets for the rvcy-reservg ublc udtg bsed o homomorhc hshg scheme: securty gst the server wth ublc verfblty, d rvcyreservg gst thrd rty udtor We frst gve the defto of securty gst the server wth ublc verfblty I ths defto, we hve two ettes: chlleger tht stds for ether the clet or thrd rty udtor, d dversry tht stds for the utrusted server Defto (Securty gst the Server wth Publc Verfblty) We cosder gme betwee chlleger d dversry tht hs four hses: Setu, Query, Chllege d Forge Setu: The chlleger rus the SetU fucto, d gets the ( k, sk ) The chlleger seds k to the dversry d kees sk secret Query: A dversry dtvely selects some fles F (,,, m) d queres the verfcto sgtures from the chlleger The chlleger comutes verfcto sgture for ech of fles d seds to the dversry Chllege: The chlleger geertes the chl for the fle F (,,, m) d seds ts to the dversry Forge: The dversry comutes resose P to rove the tegrty of the requested fles If CheckProof ( k,, chl, P) success the the dversry hs wo the gme Followg we defe the rvcy-reservg gst TPA udtg, whch s gve Defto 3 I ths defto, we lso hve two ettes: chlleger tht stds for ether the clet or the server, d dversry tht stds for the TPA Defto 3 (Prvcy-Preservg gst Thrd Prty Audtor) We sy rvcy-reservg ublc udtg bsed o homomorhc hshg scheme s rvcy-reservg f there exsts extrcto lgorthm such tht, for every dversry, wheever dversry lyg the gme, oututs dmssble chetg rover P 0 for smled fles F, the extrcto lgorthm recovers F from P 0 3 Prelmres I ths subsecto, we frst troduce bler ms d severl comlexty ssumtos The, we brefly descrbe severl crytogrhc rmtves used ths er Defto 4 Bler M LetG, G T be multlctve cyclc grous of rme order, let g, g be geertors of G A bler m s m e: G G GT wth the followg roertes: ) Comutblty: there exsts effcetly comutble lgorthm for comutg m e b b ) Blerty: for ll uv, G, d b, Z, eu (, v) euv (, ) 3) No-degeercy: eg (, g) Defto 5 Dscrete Logrthm(DL) Problem For Z, gve g, h g G, outut The DL roblem ssumto holds G f o t tme lgorthm hs dvtge t lest solvg the dscrete logrthm roblem G, whch mes t s comuttol fesble to solve the dscrete logrthm roblem G Defto 6 Comuttol Dffe-Hellm(CDH) Problem For Z, gve g, g G, d h G comute h G The co-cdh roblem ssumto holds G d G f o t tme lgorthm hs dvtge t lest solvg the co-cdh roblem G dg 73

4 Publc Audtg Bsed o Homomorhc Hsh Fucto Secure Cloud Storge Shufe NIU, Cfe Wg, Xo DU Defto 7 Homomorhc hsh fucto [] Let G be cyclc grou of rme order whch the dscrete logrthm roblem s hrd, where s securty rmeter d the ublc rmeters cot descrto of G d rdom geertors g, g,, g G/ The homomorhc hshg o messge v( v, v,, v) Z c be costructed by: v H ( v) def g It s esy to verfy tht the vector hshg fuctos stsfyg followg roertes: )Homomorhsm: For y two messges m,m, d sclrs w, w, t holds tht H( wm w m ) H( m ) H( m ) w w )Collso Resstce: There s o robblstc olyoml-tme dversry cble of forgg ( m, m, m3, w, w) stsfyg both m3 wm wm d w w ( m ) ( m ) ( m ) H H H 3 Theorem [] The homomorhc hshg fuctos s secure ssumg the dscrete logrthm roblem G s hrd 3 The Proosed Scheme I ths secto, we descrbe the roosed udtg scheme Just s metoed Secto, the roosed scheme hs fve fuctos: SetU, SgGe, Chllege, GeProof d CheckProof I our scheme, ech fle vector F (,,, m) s dvded to blocks of equl legths: F ( m, m,, m) To cheve rvcy-reservg ublc udtg, we roose to uquely tegrte the homomorhc ler uthetctor wth rdom vector mskg w ( w, w,, w ) to hde ler combto of the dt fles Let G d G T be multlctve cyclc grous of rme order, d e: GG GT be bler m s troduced relmres Let g be geertor of G, H () s secure hsh fucto: Z Z Our scheme s s follows: SetU ( ) : The cloud user chooses rdom sgg key r ( sk, ssk ), rdom x Z x rdom elemet u G, d comutes v g g, g,, g G/ The secret rmeter s sk ( x; u; ssk) d the ublc rmeters re k ( skvgg ; ; ;, g,, g ) SgGe ( k, skf, ): Gve dt fles F ( m, m,, m) Z, d detfer d of fle F (,,, m), the user comutes: ) H ( d d,, d m ), here d re chose by the user uformly t rdom from Z s the detfer of fle F m d x ) ( F ) ( g u ) for ech F Deote the set of uthetctors by { } m I order to esure the tegrty of fles detfer Oe smle wy to do ths s to comute t d d d u ssg ( d d d u) For smlcty, we ssume the TPA kows the umber of blocks m ssk m The user the seds F log wth the verfcto {, t} to the server d deletes them from locl storge Chllege ( k, F ): The TPA, rdomly chooses, 74

5 Publc Audtg Bsed o Homomorhc Hsh Fucto Secure Cloud Storge Shufe NIU, Cfe Wg, Xo DU )wth resect to the mechsm we descrbe the TgGe hse, the TPA verfes the sgture ( ) ssg d d d u v sk, d quts f the verfcto fls Otherwse, the TPA ssk m recovers d, d,, d m, u ) to geerte the chllege messge for the udt chl, the TPA cks rdom c -elemet subset I { s, s,, s c } of set [, ] For ech elemet I, the TPA lso chooses rdom vlue v The messge chl secfes the ostos of the fles requred to be checked 3)The TPA seds chl {(, v)} I to the server GeProof ( k, F, F, chl) : Uo recevg chllege chl {(, v)} I,the server w )chooses rdom vector w ( w, w,, w ), d clcultes R e( g, v) )comutes the ler combto of smled fles secfed chl : T sc μ (,,, ) v F w s 3) clcultes ggregted uthetctor = ( c vm w, c vm w,, c vm w ) s s s T s s s s c v G The seds P = (μ, σ,r) s the resose roof of storge correctess to the TPA CheckProof ( k,, chl, u, R) : Wth the resose P ( μ,, R), the TPA comutes ) H ( d d,, d m ) s ) ID c s vd the checkg the verfcto equto: s u ID Re (, g)? e( g u, v) () The correctess of the bove verfcto equto s elborted s follows: 4 Securty Alyss s w c d v s s c sc w svm svd s c sc svm w svd x ID m x R e(, g) = e( g, v) e( ( g u ) ), g) x e( g, v) e( g u, g ) e( g g u, g ) = e( g u, v) We evlute the securty of the roosed scheme by lyzg ts fulfllmet of the securty requremets descrbed Secto Theorem : Uder the CDH ssumto, the roosed scheme s secure gst the utrusted server Proof We ssume there exsts dversry (Now, the server s treted s dversry) tht ws the chllege cked by d show tht wll be ble to forge determed by the chllege If c brek the ublc udtg scheme, we show how to costruct dversry tht uses order to 75

6 Publc Audtg Bsed o Homomorhc Hsh Fucto Secure Cloud Storge Shufe NIU, Cfe Wg, Xo DU brek CDH A smultes the ublc udtg evromet for s follows: s gve s uts vlues g, g G, h G, ts gol s to outut h Setu: A geertes rdom geertor of G Deote the geertor by g seds mes tht does ot kow the corresodg secret key Hsh Query: Whe receves hsh query for vlue d u v g to Ths, rdom vlues, Z d d m sets u g h, retur to, sce g G, so rdom choose Z m d g ( ) g u m ID, ( ) ( ) g u g Chllege: A geertes chl (, s, s,, s c ) seds t to Forge: comutes resose to rove the tegrty of the requested fle outut u ID (, ) (, ) R e g e g u v w w From the vew of, g g G, g u t g G, where wt, Z By (), c outut w t s c sc s s d comute h ( v ) Theorem 3: Uder the DL ssumtos, from the server s resose P ( μ,, R) TPA cot get y formto bout the clet s dt P ( μ,, R) from the scheme executo Hece, the scheme s rvte gst thrd rty Proof We ssume there exsts dversry (Now, the TPA s treted s dversry) tht ws sc the chllege cked by d show tht wll be ble to recover u vm determed by the s chllege From the vew of TPA, f TPA c recover u, t mes c comute tht: R such ID R e(, g) e( g u, v) () u u ID R e g e g u v (, ) (, ) (3) From (), (3), w w (, ) (, ) e g g e g g We hve w g g w Accordg to Theorem, sce dscrete logrthm roblem G s hrd, dversry c ot sc comute vector w ( w, w,, w ), so c ot recover u vm s 76

7 Publc Audtg Bsed o Homomorhc Hsh Fucto Secure Cloud Storge Shufe NIU, Cfe Wg, Xo DU 5 Performce Alyss I ths secto, we frst evlute the roosed scheme terms of comutto overheds dfferet hses After tht, we reset the exermetl results 5 Comrso lyss We ssume hve m fles wth dfferet detfers our system, ech fle F(,,, m) re dvded to blocks of equl legths Due to TPA sde hs dfferet udtg gols: our scheme, TPA sde c choose c smled fles hvg dfferet detfers for udtg, but [4], TPA sde udt c smled fles, so we wll rmrly ssess the erformce of the roosed udtg schemes o both clet sde d cloud server sde I order to smle our comrso, we covert the exoetto oertos to multlcto oertos, t s ExG Mult G I followg Fgures d Tbles, m deotes the umber of the fles; deotes the umber of fle blocks; c deotes the umber of smled fles; Hsh deotes the hsh oertos; Add deotes the ddto oertos Mult deotes the multlcto oertos; Pr deotes the rg oertos; Tble Clet Sde Our Scheme m( ) Mult Hsh [4] Scheme m ( ) Multm Hsh Tble Server Sde Our Scheme ( c) ( mc) Mult m( c ) Add Pr [4] Scheme mc ( mc ( )) MultmcAddmHshm Pr From Tble d Tble, we c see our lgorthm hve lttle comutto overheds comrg wth Wg s [4] o both clet sde d cloud server sde Furthermore, s show verfcto equto (), the comutto cost t TPA s deedet from the umber of fles O the cotrry, the verfcto sze of [4] s lerly cresg wth the sze of the fles 5 Exermetl Results I the exermet, we frstly mesure the totl comutto tme of our scheme for dfferet choces of rg rmeters Furthermore, we demostrte the effectveess d effcecy of our roosed mechsm the sgture hse, the udtg hse d the geroof hse, resectvely Our mlemetto ws wrtte C usg the Prg-Bsed Crytogrhy Lbrry (lbbc)[3] The m roertes of dfferet tested rmeters re summrzed Tble 3 Tble 3M roertes of tested rg Tye Bse feld(bts) Dlog securty(bts) degree of curve 5 04 e

8 Publc Audtg Bsed o Homomorhc Hsh Fucto Secure Cloud Storge Shufe NIU, Cfe Wg, Xo DU The comuttos re ru o PC wth 9 GHz CPU frequecy d 4 GB of RAM, usg Lux oertg system We utlze two elltc curve, oe s Tye wth bse feld sze of 5 bts d the embeddg degree, the other s Tye wth bse feld sze of 04 bts d the embeddg degree The securty level s chose to be 5 d 04, resectvely Tble 4 shows the totl comutto tme of our scheme for dfferet choces of rg rmeters We c see tht the comutto cost hevly deeds o the selected Tye of rg, d for ech rg rmeters, the comutto cost creses wth the cresg umber of udtg fles Tble 4The totl comutto tme (s) of the roosed scheme uder dfferet rg rmeters m=600 c=00 c=300 c=400 c=500 c=600 Tye Tye e Furthermore, we demostrte the effectveess d effcecy of our roosed mechsm sgture hse, udtg d the geroof hses, resectvely 5 Performce of the Sgture hse For ech sesor odes, comutto costs for the sgture mly rely o the messge sze I ths exermet, we set the umber of fles m 50, the dmeso of messges set to 5,0,50,00, wheres the rg rmeter s fxed t Tye d Tye e As show Fgure, f lrger messge sze s used, the totl comutto cost wll crese becuse of the cresg umber of exoetto oertos Sgture tme (ms) Tye Tye e The umber of fle blocks Fgure Comutto costs of the clet s sgture wth m = 50 d = 5, 0, 50, 00 5 Performce of the Geroof hse For the Geroof hse, the effcecy of comutto mly deeds o both the umber of fles d the messge sze Thus, ths exermet, we set the legth of messge t 300 bytes, wheres, the umber of fles m s set to 50, 00, 00 As llustrted Fgure 3, the comutto cost t the cloud server hevly deeds o the selected Tye of rg, d for ech rg rmeters, the comutto cost creses wth the cresg umber of fles 78

9 Publc Audtg Bsed o Homomorhc Hsh Fucto Secure Cloud Storge Shufe NIU, Cfe Wg, Xo DU 500 Geroof tme(ms) Tye Tye e The umber of udtg fles Fgure 3Comutto costs of the server s geroof wth m = 50, 00, Audtg tme(ms) Tye Tye e The umber of fles Fgure 4 Comutto costs of the TPA s udtg wth m = 00, 00, 300, 400, Performce of the Checkroof hse From Fgure 4, we c see tht comutto cost t the Checkroof hse s deedet from the umber of fles Our exermetl results ths secto show tht our scheme hs better erformce whe udtg lrge umber of fles 6 Summry I ths er,we roose ew ublc udtg scheme bsed o homomorhc hshg secure cloud storge The roch s roved to be secure gst utrusted server It s lso rvte gst thrd rty Both theoretcl lyss d exermetl results demostrte tht the roosed scheme hs very good effcecy the sects of comutto costs Curretly we re stll workg o extedg the scheme to suort dt dymcs 7 Ackowledgemets The uthors wsh to thk the oymous referees for ther tece redg ths muscrt d ther vluble commets d suggestos The work ws suorted by the Ntol Nturl Scece Foudto of Ch uder grt 60395, Refereces [] A Juels d Jr B S Klsk,"Pors: roofs of retrevblty for lrge fles", ACM cof o Comuter d Commuctos Securty, , 007 [] G Ateese, R Burs, R Curtmol, J Herrg, L Ksser, Z Peterso, d D Sog, "Provble dt ossesso t utrusted stores", ACM cof o Comuter d Commuctos Securty, ,

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