An improved statistical disclosure attack

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1 In J Grnulr Compung, Rough Ses nd Inellgen Sysems, Vol X, No Y, xxxx An mproved sscl dsclosure c Bn Tng* Deprmen of Compuer Scence, Clforn Se Unversy Domnguez Hlls, Crson, CA, USA Eml: bng@csudhedu *Correspondng uhor Rjv Bg nd Hubo Lu Deprmen of Elecrcl Engneerng nd Compuer Scence, Wch Se Unversy, Wch, KS, USA Eml: rjvbg@wchedu Eml: hxlu@wchedu Absrc: Sscl dsclosure c (SDA) s nown o be n effecve long-erm nersecon c gns mx-bsed nonymsng sysems, n whch n cer observes lrge volume of he ncomng nd ougong rffc of sysem nd correles s senders wh recevers h hey ofen send messges o In hs pper, we furher srenghen he effecveness of hs c We show, by boh n exmple nd proof, h by employng weghed men of he observed relve recever populry, he cer cn deermne more ccurely he se of recevers h user sends messges o, hn by usng he exsng rhmec men-bsed echnque Keywords: sscl dsclosure c; SDA; nonymy; rffc nlyss; secury Reference o hs pper should be mde s follows: Tng, B, Bg, R nd Lu, H (xxxx) An mproved sscl dsclosure c, In J Grnulr Compung, Rough Ses nd Inellgen Sysems, Vol X, No Y, ppxxx xxx Bogrphcl noes: Bn Tng receved hs BS n Physcs from Peng Unversy, Chn n 997, MS n Merls Scence nd Compuer Scence from Sony Broo Unversy n 2000 nd 2002, respecvely, nd PhD n Compuer Scence from Sony Broo Unversy n 2007 He s currenly n Asssn Professor n he Deprmen of Compuer Scence Clforn Se Unversy Domnguez Hlls Hs reserch neress nclude d nonymy nd nonymsng sysems n nformon secury, d preservon n wreless d hoc sensor newors, nd d replcon nd job schedulng n grd/cloud compung Rjv Bg receved hs BS n Compuer Scence from he Brl Insue of Technology nd Scence (BITS), Pln, Ind, nd MS nd PhD n Compuer Scence from he Unversy of Vcor, Cnd He s presenly n Assoce Professor n he Deprmen of Elecrcl Engneerng nd Compuer Scence he Wch Se Unversy, USA Hs curren reserch re s web nonymy, bu n he ps he hs wored n logc progrmmng nd prconssen dbses Copyrgh 20XX Inderscence Enerprses Ld

2 2 B Tng e l Hubo Lu receved hs BS n Compuer Scence nd Technology from he Bejng Foresry Unversy, Chn, nd MS n Compuer Neworng from he Wch Se Unversy, USA He s currenly PhD suden n he deprmen of Elecrcl Engneerng nd Compuer Scence he Wch Se Unversy, USA Hs reserch neress nclude compuer neworng, web nonymy nd prvcy Ths pper s revsed nd expnded verson of pper enled On he sender cover rffc counermesure gns n mproved sscl dsclosure c presened 8h IEEE/IFIP Inernonl Conference on Embedded nd Ubquous Compung (EUC 0), Hong Kong, Chn, 3 December 200 Inroducon A populr echnque o mplemen n nonymy sysem s s mx newor, s proposed by Chum (98) A mx newor s collecon of proxy nodes h rely messges beween senders nd, possbly overlppng, recevers conneced o he newor These nermede proxes re he fundmenl source of he nonymy cheved by such sysems Severl cs by dversres on mx-bsed nonymy sysems, long wh possble counermesures, hve been suded Bc e l (200) nd Rymond (200) conn deled lss of cs Of hese cs, he clss of long-erm nersecon cs s one of he sronges In such n c, pssve globl dversry cn correle senders wh recevers h hey ofen send messges o, by observng messges h ener nd leve he mx over long perod The sscl dsclosure c (SDA), proposed by Dnezs (2003), s member of hs clss of cs, nd s dreced gns sngle sender The c ms o uncover he recevers reled o h sender by eepng rc of, mong ohers, he observed relve populry of he vrous recevers of he sysem Mhewson nd Dngledne (2004) gve n exended verson of hs c by removng some of he resrcons on he number of messges flowng hrough he mx n he orgnl verson of Dnezs (2003) Our pper s n mprovemen on he exended SDA of Mhewson nd Dngledne (2004) Our c s bsed upon he weghed men of he observed relve populry of he recevers over me We show h hs resuls n more ccure concluson hn one obned by he rhmec men mehod of Mhewson nd Dngledne (2004) The res of hs pper s orgnsed s follows In Secon 2, we presen he bsc mx nonymous newor model nd gve n overvew of he SDA echnque (Mhewson nd Dngledne, 2004) Secon 3 presens our mproved echnque for SDA from he cer s perspecve nd gves n exmple We presen forml proof bou he effecveness of he mproved SDA n Secon 4 nd conclude n Secon 5

3 An mproved sscl dsclosure c 3 2 Sscl dsclosure c (Dnezs, 2003; Mhewson nd Dngledne, 2004) 2 Mx nonymous newor model A mx newor s collecon of proxy nodes h rely messges beween group of senders nd group of recevers V such newor, he messges sen by ny of he senders cn rrve her recevers nonymously, hus hdng he senders deny from he recevers Some common cs gns such newor re o correle he sysem s ncomng messges wh s ougong ones by observng he enry nd ex sequence numbers or by comprng b perns To hwr such cs, he mx newor collecs cern number of encryped messges n ech round, decryps hem, nd hen rnsms hem smulneously The SDA s rgeed gns sngle sender, clled Alce All oher senders re clled bcground senders The m of hs c s o uncover, over perod of me, he se of recevers h Alce sends messges o, clled Alce s frends The c s crred ou by observng he number of messges sen or receved by ech sender nd recever n ech round Le he ol number of recevers n he sysem be m, nd le nd b be he number of messges sen by Alce nd bcground senders n round, respecvely Fgure shows he messge flow n he mx n round I should be noed h he cer s unble o dfferene Alce s messges from hose of he bcground senders receved by ny recever, nd cn only observe he ol number of messges receved by ech recever n ny round Fgure Messge flow n round Le he m mrx A conn he number of Alce s messges receved by ech of he m recevers n he frs rounds, e, A[, j] s he number of Alce s messges receved by recever n round j Smlrly, le he m mrx B conn he number of bcground senders messges receved by ech of he recevers n he frs rounds Le R A + B, e, R[, j] s he ol number of messges receved by recever n round j Noe gn h he cer cn observe only R, no A or B ndvdully Le α () m A nd β () m B be wo row vecors, whch conn he ol number of messges sen by Alce nd bcground senders n ech round, respecvely We hve α [ ], nd b β [ ] Le r be he column vecor connng he number of messges receved by ech recever n round, h s, r s he h column

4 4 B Tng e l of R Le o be he vecor connng he observed relve populry of ech recever n round, whch s defned s he frcon of he messges receved by ech recever n round, e, o r / ( + b) Le O be he vecor connng he cumulve observed relve populry of ech recever fer rounds, h s, O s he verge of he vecors { o }: O o SDA explos he fc h for lrge vlues of, he bove cul recever populry pproxmes he expeced one, whch s formuled below Le m be he verge number of messges sen by Alce n ech round, e, m ( ) Smlrly, le n ( ) + b be he verge number of ol messges sen n ech round Le v be he vecor h conns he relve degrees of frendshp wh Alce of ll he recevers, h s, he frcon of Alce s messges receved by ech recever n ll he rounds The gol of SDA s o deermne v Le u be he vecor h conns he observed recever populry wh bcground senders of ll he recevers, e, u s he verge of he vecors n he followng se: o, 0 { } In oher words, u s obned by observng he bcground senders behvour n rounds n whch Alce does no send ny messges We ssume h here re enough rounds n whch Alce does no prcpe, whch fcles he compuon of u Ths ssumpon s no n unresonble one, becuse mos senders conneced o n nonymous sysem, such s users who browse he web, re onlne only some of he me nd offlne mos of he me If Alce s n ordnry user, u cn be obned esly durng rounds h she s offlne Recll h, he gol of SDA s o deermne v, whch conns nformon bou he relve degrees of frendshp wh Alce of ll recevers The expeced recever populry cn be expressed s: mv ( n m) + u n The bove expresson s bsed upon he premse h of he n verge number of messges sen n round, m messges from Alce should rech recevers ccordng o her degrees of frendshp wh Alce n v, nd he remnng n m messges from bcground senders should rech hem ccordng o her degrees of frendshp, wh ll he bcground senders s whole, n u Accordng o he Lw of Lrge Numbers (Grmme nd Srzer, 992), when s lrge enough, he bove expeced populry pproxmes O, he observed one, e, mv + ( n m) u O n By rerrngng, we ge

5 An mproved sscl dsclosure c 5 no ( n m) u v m All vlues on he rgh sde of he bove equon cn be obned by observng he mx over me, hus mng possble resonble esme of he recevers degrees of frendshp wh Alce 3 An mproved SDA usng weghed men In he bsc SDA proposed by Dnezs (2003), he mx oupus fxed number of messges n ech round, excly one of whch s sen by Alce Compung O s n rhmec men of he o vecors s ll h s needed for h model Also, n h model u corresponds o unform dsrbuon over ll recevers, whch s fxed nd does no need o be compued The model presened n Secon 2 s n exenson developed by Mhewson nd Dngledne (2004), n h he number of messges rnsmed by he mx n ech round cn vry, Alce s llowed o send ny number of messges n ech round, nd u need no be unform However, he SDA of Mhewson nd Dngledne (2004) connues o employ he sme rhmec men mehod for compung O, whch cn be mde more ccure by nsed employng weghed men bsed upon he ol number of messges oupu by he mx As n exmple, suppose A nd B re he only recevers n he sysem If n Round, A receves messge nd B receves 3 messges, hen o 025, 075 Now, f n Round 2, A receves 300 messges nd B receves 00 messges, hen o 2 075, 025 An rhmec men of hese wo vecors gves O 05, 05 On he oher hnd, men weghed by he ol number of messges n ech round would resul n O, 0745, 0255, whch beer reflecs he poron of he ol number of messges receved by he wo recevers so fr As he nuon behnd O s he cumulve observed relve populry of recevers so fr, s compuon bsed upon weghed verge s more n lne wh h nuon In oher words, O [] should be clculed s he frcon of ol messges receved by recever n ll he rounds We hus propose he followng defnon of O : ( + b) o[] O [], ( + b ) whch cn be smplfed o be r O [ ], for ny recever ( + b )

6 6 B Tng e l In order o beer sudy he effecveness of SDA by he bove weghed men mehod, we exend he exmple o ol of seven rounds, s shown n Tble The ble shows he number of messges sen by Alce nd he bcground senders o he wo recevers, A nd B, n ech of he seven rounds Whle such nformon s no vlble o he cer, we use o compre he effecveness of he c ccordng o he old nd new defnons of O Tble Messges sen o recevers A nd B Round number Alce o A Alce o B Bcground o A Bcground o B ,72, , ,080, Tol ,469 3,46 The vlues m nd n cn be deermned from Tble o be 84 nd,074, respecvely From round 5, n whch Alce does no send ny messges, u s esmed o be 0502, 0498 By usng hese vlues of mn,, nd u n equon (3), long wh he vlue of O s he rhmec men of he o vecors, we ge v 044, 056 The bove vlue of v s msledng s suggess B beng more lely o be Alce s frend hn A On he oher hnd, our new defnon of O s weghed men resuls n v 08, 09, whch s much closer o s cul vlue , from hese seven rounds, whch s 083, 07 4 Proof of he effecveness of he mproved SDA usng weghed men We begn wh he followng defnons: r A : he vecor connng he number of Alce s messges receved by ech recever n round r B : he vecor connng he number of bcground senders messges receved by ech recever n round v []: he recever s cul relve degree of frendshp o Alce fer rounds: rel A r [] rel v

7 An mproved sscl dsclosure c 7 Ow[] nd O[]: he cumulve observed relve populry for recever by weghed men nd rhmec men, respecvely From Secons 2 nd 3, we hve: r Ow, () ( + b ) r o + b O (2) uw[] nd u[ ]: he cumulve observed relve populry wh he bcground senders, n rounds when Alce does no send messges, by weghed men nd rhmec men respecvely vw[] nd v[]: he recever s relve degree of frendshp o Alce fer rounds, obned by weghed men nd rhmec men respecvely: [] ( ) now n m uw[] vw, (3) m no ( n m) u[] v (4) m By subsung () no (3) nd (2) no (4), we ge r b uw[] vw, (5) r ( + b) b u[] + b v To show h by employng weghed men of he observed relve recever populry, he cer cn deermne more ccurely he se of recevers h user sends messges o hn usng exsng rhmec men-bsed one, we show h hs s he cse for ech recever Theorem : For ny recever, vrel[ ] vw[ ] vrel[ ] v[ ] 2 2 Proof: We need o show h ( vrel[ ] vw[ ]) ( vrel[ ] v[ ]), e, 2 vrel[ ] ( v[ ] vw[ ] ) ( v[ ] vw[ ] ) ( v[ ] + vw[ ] ) (7) We frs show h v vw > 0, nd hen we wll only need o prove 2 vrel[ ] v[ ] + vw[ ] From (6) nd (5), (6)

8 8 B Tng e l v v w r ( + b) b u ( r b uw[] ) + b r We hve h + b w ( + b ) r + b ( u u ) r r ( + b) ( + b) + ( + b) + b + b r2 r + + ( + b) + b + b 2 2 > r + r + + r r [] 2 Furhermore, snce he observed recever populry wh bcground senders does no depend on wheher he cer uses he weghed men or rhmec men, we cn ssume h uw u[] Therefore, we hve h v vw > 0, nd from (7), we only need o prove v + v 2 v [], shown s below: v + v > w rel r ( + b) b u + ( r b uw[] ) ( r + r ) b ( uw + u[] ) w + b w B r r [] A r [] 2 v [ ] rel r b u

9 An mproved sscl dsclosure c 9 5 Conclusons We hve presened n mproved SDA by employng weghed men of he cer s observons nd proved h hs mes he c more ccure hn h of Mhewson nd Dngledne (2004), whch employs rhmec men of he cer s observons Acnowledgemens The reserch descrbed n hs pper ws prlly suppored by he Uned Ses Nvy Engneerng Logscs Offce conrc no N C-3077 References Bc, A, Möller, U nd Sglc, A (200) Trffc nlyss cs nd rdeoffs n nonymy provdng sysems, n Proceedngs of he 4h Inernonl Worshop on Informon Hdng, pp Chum, D (98) Unrceble elecronc ml, reurn ddresses, nd dgl pseudonyms, Communcons of he ACM, Vol 24, No 2, pp84 88 Dnezs, G (2003) Sscl dsclosure cs: rffc confrmon n open envronmens, n Proceedngs of Secury nd Prvcy n he Age of Uncerny (SEC 2003), pp Grmme, GR nd Srzer, DR (992) Probbly nd Rndom Processes, 2nd ed, Clrendon Press, Oxford Mhewson, N nd Dngledne, R (2004) Prccl rffc nlyss: exendng nd ressng sscl dsclosure, n Proceedngs of he 4h Prvcy Enhncng Technologes Worshop, pp7 34 Rymond, J-F (200) Trffc nlyss: proocols, cs, desgn ssues, nd open problems, n Proceedngs of Inernonl Worshop on Desgn Issues n Anonymy nd Unobservbly, pp0 29

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