# Crowds: Anonymity for Web Transactions

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11 11 aoymity) if P (I H 1+ ) 1/2. I order to yield probable iocece for the path iitiator, certai coditios must be met i our system. I particular, let p f > 1/2 be the probability of forwardig i the system (see Sectio 4), let c deote the umber of collaborators i the crowd, ad let deote the total umber of crowd members whe the path is formed. The theorem below gives a sufficiet coditio o p f, c, ad to esure probable iocece for the path iitiator. Theorem 5.2. If pf p f (c + 1), the the path iitiator has probable iocece agaist c 1/2 collaborators. Proof. We wat to show that P (I H 1+ ) 1/2 if pf p f 1/2 (c + 1). First ote that ( ) i 1 pf ( c) ( c ) P (H i ) = This is due to the fact that i order for the first collaborator to occupy the ith positio o the path, the path must first wader to i 1 ocollaborators (each time with probability c ), each of which chooses to forward the path with probability p f, ad the to a collaborator (with probability c ). The ext two facts follow immediately from this. P (H 2+ ) = c ( ) k pf ( c) ( c = ) ( ) p f ( c) p f c( c) = 1 2 p f ( c) P (H 1+ ) = c k=1 ( ) k pf ( c) = k=0 ( c ) ( 1 1 pf ( c) pf ( c) ) = c p f ( c) Other probabilities we eed are P (H 1 ) = c, P (I H 1) = 1, ad P (I H 2+ ) = 1 c. The last of these follows from the observatio that if the first collaborator o the path occupies oly the secod or higher positio, the it is immediately preceded o the path by ay ocollaboratig member with equal likelihood. Now, P (I) ca be captured as P (I) = P (H 1 )P (I H 1 ) + P (H 2+ )P (I H 2+ ) = c( p f + cp f + p f ). 2 p f ( c) The, sice I H 1+ we get So, if P (I H 1+ ) = P (I H 1+) P (H 1+ ) pf p f 1/2 (c + 1), the P (I H 1+) 1 2. = P (I) P (H 1+ ) = p f( c 1) As a result of Theorem 5.2, if p f = 3 4, the probable iocece is guarateed as log as 3(c + 1). More geerally, Theorem 5.2 implies a tradeoff betwee the legth of paths (i.e., performace) ad ability to tolerate collaborators. That is, by makig the probability of forwardig high, the fractio of collaborators that ca be tolerated approaches half of the crowd. O the other had, makig the probability

17 17 msecs path legth kbyte images 5 Path Number of 1-kbyte images legth Fig. 5. Respose latecy (msecs) as a fuctio of path legth ad umber of embedded images impacts latecies o paths that use it. Agai, this suggests multiple types of crowds, amely oes cotaiig oly jodos coected via fast liks, ad oes allowig jodos coected via slower liks. 7. SCALE The umbers i Sectio 6 give little isight ito how performace is affected as crowd size grows. We do ot have sufficiet resources to measure the performace of a crowd ivolvig hudreds of computers, each simultaeously issuig requests. However, i this sectio we make some simple aalytic argumets to show that the performace should scale well. The measure of scale that we evaluate is the expected total umber of appearaces that each jodo makes o all paths at ay poit i time. For example, if a jodo occupies two positios o oe path ad oe positio o aother, the it makes a total of three appearaces o these paths. Theorem 7.1 says that the each jodo s expected umber of appearaces o paths is virtually costat as a fuctio of the size of the crowd. This suggests that crowds should be able to grow quite large. Theorem 7.1. I a crowd of size, ( the expected total umber of appearaces 1 that ay jodo makes o all paths is O (1 p f ) ( ). ) Proof. Let be the size of the crowd. To compute the load o a jodo, say J, we begi by computig the distributio of the umber of appearaces made by J

18 18 o each path. Let R i, i > 0, deote the evet that this path reaches J exactly i times (ot coutig the first if J iitiated the path). Also, defie R 0 as follows: ( ) k ( ) 1 P (R 0 ) = (1 p f ) (p f ) k 1 = (1 p f ) 1 1 p f k=0 Ituitively, P (R 0 ) is the probability that the path, oce it has reached J, will ever reach J agai. The, we have P (R 1 ) = 1 ( ) k ( ) ( ) 2 P (R 0) (p f ) k = (1 p f ) 1 1 p f P (R 2 ) = 1 p fp (R 1 ) P (R i ) = 1 p f P (R i 1 ) k=0 k=0 k=0 ( ) k (p f ) k 1 < (1 p f ). ( ) k (p f ) k 1 < (1 p f ) ( ) 2 ( 1 ( ) i ( p f p f 1 ) 3 ) i+1 From this, the expected umber of appearaces that J makes o a path formed by aother jodo is bouded from above by: ( ) [ 1 pf ( ) ] k ( ) ( ) 1 1 pf p f ( 1) k = 1 1 p f p f ( 1) 1 1 p k=0 f (1 + p f ( 1)) 2 p f ( 1) < (1 + p f ( 1)) 2 = < 1 (1 p f )( 1) + 1 ((1 p f )( 1)) 2 2 (1 p f ) 2 ( 1) Therefore, the expected umber of appearaces that J makes o all paths is bouded from above by: ( 2 (1 p f ) 2 ( 1) = ) (1 p f ) CROWD MEMBERSHIP The membership maiteace procedures of a crowd are those procedures that determie who ca joi the crowd ad whe they ca joi, ad that iform members of the crowd membership. We discuss mechaisms for maitaiig crowd membership i Sectio 8.1, ad policies regardig who ca joi a crowd i Sectio Mechaism There are may schemes that could be adopted to maage membership of the crowd. Existig group membership protocols, tolerat either of beig (e.g., [Cristia 1991;

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