Introution to Roust Algorithms Séstin Tixuil UPMC & IUF Motition Approh Fults n ttks our in th ntwork Th ntwork s usr must not noti somthing wrong hppn A smll numr of fulty omponnts Msking pproh to fult/ttk tolrn Prinipl Prolms Tmprtur Prssur CPU CPU CPU Comprtor Throttl Rplit input snsors my not gi th sm t Fulty input snsor or prossor my not fil grfully Th systm might not tolrnt to softwr ugs Tlling Truth from Lis Th Isln of Lirs n Truth-tllrs An isln is popult y two tris Mmrs of on tri onsistntly li Mmrs of th othr tri lwys tll th truth Tri mmrs n rogniz on nothr, ut n xtrnl osrr n t Puzzl 1 You mt mn n sk him if h is truth-tllr, ut fil to hr th nswr You inquir: Di you sy you r truthtllr? H rspons: No, I i not. To whih tri os th mn long? Puzzl II You mt prson on th isln. Wht singl qustion n you sk him/hr to trmin whthr h/sh is lir or truth-tllr?
Puzzl III Puzzl IV Puzzl V You mt two popl A n B on th isln A sys: Both of us r from th lir tri. Whih tri is A from? Wht out B? You mt two popl, C n D on th isln. C sys: Extly on of us is from th lirs tri. Whih tri is D from? You mt two popl E n F on th isln E sys: It is not th s tht oth of us r from truth-tllrs tri. Whih tri is E from? Wht out F? Puzzl VI You mt two popl G n H on th isln G sys: W r from iffrnt tris. H sys: G is from th lirs tri. Whih tris r G n H from? Puzzl VII You mt thr popl A, B, n C You sk A: how mny mong you r truth-tllrs?, ut on t hr th nswr You sk B: Wht i A sy?, hr on. C sys: B is lir. Whih tris r B n C from? Puzzl VII A B B C C C C 0 1 1 2 1 2 2 3 Th Isln of Slti Lirs Inhitnts li onsistntly on Tusys, Thursys, n Sturys. Howr, thy lwys sy th truth on th rmining ys. You sk: Wht is toy? Tomorrow? Rsponss: Stury., Wnsy. Wht is th urrnt y? Th Isln of Rnom Lirs A nw Isln hs thr tris truth-tllrs onsistnt lirs rnomly li or tll th truth How to intify thr rprsntnts of h tri stning in lin with only thr ys/no qustions? Byzntin Gnrls
Sttings Gol Two Gnrls Prox Byzntin gnrls r mping outsi n nmy ity Gnrls n ommunit y sning mssngrs Gnrls must i upon ommon pln of tion Som of th Gnrls n tritors All loyl gnrls i upon th sm pln of tion A smll numr of tritors nnot us th loyl gnrls to opt pln G1 Bsig ity Attk t noon? Ak! Unrlil ommunition mi G2 L1 Th Byzntin Gnrls Prolm G Bsig ity L2 Th (simpl) Byzntin Gnrls Prolm Gnrls l n iisions of th Byzntin rmy Th iisions ommunit i rlil mssngrs Th gnrls must gr on pln ( ttk or rtrt ) n if som of thm r kill y nmy spis Orl Mol A1: Ery mssg tht is snt is lir orrtly A2: Th rir of mssg knows who snt it A3: Th sn of mssg n tt Solution? Rlil Ntworks Rlil Ntworks pln: rry of {A,R}; finlpln: {A,R} Ali:A Ali:A 1: pln[myid] := ChoosAorR() 2: for ll othr G sn(g, myid, pln[myid]) 3: for ll othr G ri(g, pln[g]) 4: finlpln := mjority(pln) Bo:R Chrli:A Bo:R Chrli:A
Rlil Ntworks Rlil Ntworks Crshing Ntworks Ali:A Ali:A:A Ali:A Bo:R Chrli:A Bo:R:A Chrli:A Bo:R Chrli:A Crshing Ntworks Crshing Ntworks Crshing Ntworks Ali:A Ali:A Ali:A A A R (R,A,-) Bo:R Chrli:A Bo:R A Chrli:A Bo:R R Chrli:A Crshing Ntworks Ali:A R (R,A,-) Bo:R:R R Chrli:A:A Th Byzntin Gnrls Prolm A gnrl n n-1 liutnnts l n iisions of th Byzntin rmy Th iisions ommunit i mssngrs tht n ptur or ly Th gnrls must gr on pln ( ttk or rtrt ) n if som of thm r tritors tht wnt to prnt grmnt Th Byzntin Gnrls Prolm A ommning gnrl must snt n orr to his n-1 liutnnts gnrls suh tht IC1: ll loyl liutnnts oy th sm orr IC2: if th ommning gnrl is loyl, thn ry loyl liutnnt oys th orr h sns
Orl Mol A1: Ery mssg tht is snt is lir orrtly A2: Th rir of mssg knows who snt it A3: Th sn of mssg n tt 3k+1 nos r nssry (orl mol) Attk Attk Liutnnt 1 Rtrt Liutnnt 2 3k+1 nos r nssry (orl mol) Rtrt Rtrt Liutnnt 1 Attk Liutnnt 2 3k+1 nos r nssry (orl mol) 3k+1 nos r nssry (orl mol) 3k+1 nos r nssry (orl mol) Attk Rtrt Liutnnt 1 Rtrt Liutnnt 2 Rtrt Attk Liutnnt 1 Attk Liutnnt 2 Attk Rtrt Attk Rtrt Liutnnt 1 Rtrt Liutnnt 2 3k+1 nos r suffiint (orl mol) 3k+1 nos r suffiint (orl mol) 3k+1 nos r suffiint (orl mol) Liutnnt 1 Liutnnt 2 Liutnnt 1 () Liutnnt 2 Liutnnt 1 () Liutnnt 2 Liutnnt 3 Liutnnt 3 () Liutnnt 3 (,)
3k+1 nos r suffiint (orl mol) 3k+1 nos r suffiint (orl mol) 3k+1 nos r suffiint (orl mol) Liutnnt 1 () Liutnnt 2 Liutnnt 1 (,) Liutnnt 2 Liutnnt 1 (,) x Liutnnt 2 x Liutnnt 3 (,) Liutnnt 3 (,) Liutnnt 3 (,) 3k+1 nos r suffiint (orl mol) 3k+1 nos r suffiint (orl mol) 3k+1 nos r suffiint (orl mol) Mj(,,x) x z Liutnnt 1 (,,x) Liutnnt 2 Liutnnt 1 (,,x) Liutnnt 2 Liutnnt 1 y Liutnnt 2 Liutnnt 3 (,,x) Liutnnt 3 (,,x) Mj(,,x) Liutnnt 3 3k+1 nos r suffiint (orl mol) Liutnnt 1 Liutnnt 3 Liutnnt 2 3k+1 nos r suffiint (orl mol) Liutnnt 1 Min Liutnnt 3 Min Liutnnt 2 Min Writtn Mol A1-A3: Sm s for A4: A loyl gnrl s signtur nnot forg, n ny ltrtion of th ontnts of his sign mssgs n tt Anyon n rify th uthntiity of gnrl s signtur
k+2 nos r suffiint (writtn mol) k+2 nos r suffiint (writtn mol) k+2 nos r suffiint (writtn mol) Attk:C Attk:C Liutnnt 1 Liutnnt 2 Liutnnt 1 Attk:C:L1 Liutnnt 2 Liutnnt 1 Attk:C:L2 Liutnnt 2 k+2 nos r suffiint (writtn mol) k+2 nos r suffiint (writtn mol) k+2 nos r suffiint (writtn mol) Attk:C Rtrt:C Liutnnt 1 Liutnnt 2 Liutnnt 1 Attk:C:L1 Liutnnt 2 Liutnnt 1 Rtrt:C:L2 Liutnnt 2 k+2 nos r suffiint (writtn mol) Attk:C Rtrt:C:L2 Rtrt:C Attk:C:L1 Liutnnt 1 Liutnnt 2 Aritrry Ntworks Topology Disory Gin synhronous ntwork up to k Byzntin nos h no knows its immit nighors intifirs Gol h no must isor th omplt ntwork topology
Wk Topology Disory Trmintion ithr ll non-fulty prosss trmin th systm topology or t lst on tts fult Sfty for h non-fulty pross, th trmin topology is sust of tul Vliity fult tt only if it in xists Wk Topology Disory Wk Topology Disory Wk Topology Disory Bouns nnot trmin prsn of g if two jnt nos r fulty nnot (ompltly) sol if ntwork is lss thn k+1 onnt Strong Topology Disory Trmintion ll non-fulty prosss trmin th systm topology Sfty for h non-fulty pross th trmin topology is sust of tul Strong Topology Disory Strong Topology Disory Strong Topology Disory Strong Topology Disory Bouns nnot trmin prsn of g if on nighor is fulty nnot sol if ntwork is lss thn 2k+1 onnt
Solutions Prliminris Min i Mngr s thorm: if grph is k onnt thn for ny two rtis thr xists k intrnlly no-isjoint pths onnting thm singl (non-sour) no nnot ompromis info if it trls or two no-isjoint pths Solutions Prliminris Common Fturs ry solution ssntilly inols flooing h no s nighor info to th othr nos solutions iffr on how th nos forwr nighorhoo info ri from othr nos A Ni Solution Stor trl pth in mssg, forwr mssg tht ontins simpl pth to ll outgoing links Sols strong (n wk) topology isory prolms A Ni Solution h g f A Ni Solution Stor trl pth in mssg, forwr mssg tht ontins simpl pth to ll outgoing links Sols strong (n wk) topology isory prolms rquirs xponntil numr of mssgs Dttor Bsi sign propgt nighor info mssg for h pross xtly on (first tim) if ri iffrnt info for sm pross, signl fult sin ntwork is k+1 onnt, info out non-fulty nos rhs ry no Dttor Dttor Dttor Hnling fk nos fulty pross my sn info out nonxistnt (fk) nos thus ompromising sfty n trmintion only fulty nos n onnt to fk nos? (isor ntwork is lss tht k+1 onnt) Hnling fk nos fulty pross my sn info out nonxistnt (fk) nos thus ompromising sfty n trmintion whn th ntwork is not ompltly isor yt, it my lso lss thn k+1 onnt, prolms with liity g h f
Dttor Nighorhoo losur onnt ll nos whos nighor informtion is not ri th onntiity of this grph is no lss thn th tul topology if th onntiity of this grph flls low k+1, signl fult,, Dttor, Dttor Dttor,, Dttor Hnling Fults??? Hnling Fults Hnling Fults Dttor Dfinition Solution is jnt-g omplt if nonfulty nos isor ll non-fulty nos n thir jnt gs
Dttor Explorr Explorr Thorm Dttor is n jnt-g omplt solution to th wk topology isory prolm if th onntiity of th systm xs th mximum numr of fults Min i ollt no s nighor informtion suh tht th info gos long mor thn twi s mny no isjoint pths s mx numr of fulty nos Confirm nighor informtion k+1 isjoint pths from sour non-intrsting pths from k+1 onfirm nighors Explorr <,> Explorr Explorr :<,> :<,>,:<,> :<,> Dfinition Solution is two-jnt g omplt if non-fulty nos isor ll non-fulty nos n gs jnt to two nonfulty nos Thorm (gnrliz) Explorr is two-jntg omplt solution to th strong topology isory prolm in s th grph onntiity is mor thn twi th numr of fults Composing Dttor n Explorr Osrtion Dttor uss lss mssgs whn thr r no fults I run Dttor, if no isors fult, inok Explorr rquirs 2k+1 onnt topologis