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1 Supplementary Material (fr Parasitic Cmputing, by Albert-László Barabási, et al.) 1 Encding 3-SAT prblem using TCP checksum In the checksum, ne can use up t three variables withut verflw, as. Thus a 3- SAT prblem can be encded in a way similar t the 2-SAT implementatin described in the manuscript, assuming that there are apprpriate peratrs whse lgical tables match the checksum. There are 3 different peratins in additin t the NOT peratr fr each variable that can be perfrmed by the TCP checksum. (Nte: that the slutin t the 2-SAT prblem uses AND and XOR between variables and NOT peratr.) Operatin I: ALL TRUE (AND) crrespnding t "!# %$'& Operatin II: TWO TRUE crrespnding t (*) +-,./ :8<;=,>71)/,>?@. 8:AB,>7C8<467CD8 %E % F GHGI"!J $K& () +0 LM. N4POQ0R :8<;=,>71)/,>?@. 8:AB,>7C8<467CD8 6S "!UT S V"!UT S "! 6S T S T S "!W XT W!W TV"!U Operatin III: ONE TRUE crrespnding t - %F ZY [Y/\! $'& () +]0 L@. N0 L8< :8Q;I,>71)/,>?M. 8:A') AZ4679D8 X S S "!UT 6S S "!UT 6S S ^V"! XT T "!U 6S T S W!W 6S T S "!W 6S KT S Z! Fr peratins II and III, (x:y:z) and (x.y.z), the first definitin lks simple, hwever, because we are using Cnjunctive Nrmal Frm (cnnected with AND( ) between clauses) it is mre apprpriate t use the secnd definitin. Althugh a 2-SAT prblem is nt NP-cmplete, a mixed prblem, where any number (greater than zer) f 3-SAT clauses is cmbined with 2-SAT clauses, is NP-cmplete. T illustrate the use f these peratrs, cnsider the fllwing sentence with eight variables Z_àb "`:Y/Y/Y/àdc : #_Y d "Y beh!f 6S Z_<GHbghG S dï!w 6S d ' S beb dj!u bg"y S dj Y Zk!f b "Y di"y S #k>!w deg S dihghdc!w b GHdjhGHbc!U 6S #_ly be"y dj! The nly slutin is `:F `:F `: `:F `: `:F `m!. The checksum that wuld frce TCP t drp everything but the crrect slutin has the frm As a test we have implemented their 3-SAT prblems as well as described in n Cmputing with TCP Our implementatin f parasitic cmputing explits a reliability mechanism in the transmissin cntrl prtcl (TCP). The sender f a TCP segment cmputes a checksum ver the entire segment, which prvides 1

2 F Vers Service Packet Length Identificatin Fragment ffset Time t live Prtcl IP header checksum IP surce address IP destinatin address TCP surce prt TCP destinatin prt Sequence number Acknwledgment number TCP hdr Len IP hdr Len IP Flags Unused TCP Flags Windw size TCP checksum Urgent pinter a b IP Header TCP Header Data Figure 1: TCP/IP segment used fr parasitic cmputing reliability against bit errrs that might ccur in transprt. It inserts the cmplement f the checksum in the message. The receiver als cmputes a checksum in rder t verify data integrity. The receiver drps an entire segment if its checksum des nt add up, assuming that the message was crrupted in transit. Because TCP is reliable, a sender will retransmit each segment until it is acknwledged by the receiver [1, 2]. We explit the TCP checksum t answer the fllwing questin: Is p equal t q? The checksum value in a TCP packet is determined by q. The data cntains 16-bit wrds and p. TCP pr cmputes the checksum, if q, then TCP rejects the segment. Therefre, a message is a valid TCP pb segment if and nly if q. p< Suppse ne needs t find tw numbers which add up t a certain value, q. One culd generate guesses fr and p, add them, and test if the sum is equal t q. The checksum mechanism f TCP can be used t slve this prblem. First, cmpute a checksum fr the answer, q, as the data part f the the TCP segment. Next, send a TCP segment where the data part cntains candidate addends ts and pvu, as shwn belw. TCP header q s... w\x w\x... data p u Dummy packet fr cmputing checksum Slutin packet Finally, cntinue t send TCP segments until ne is received meaning that the checksum was verified. Fr p u r any s q, the TCP segment is drpped because the checksum verificatin fails. Cnsequently, nly ypz the crrect slutin (where q ) is a well-frmed TCP segment. The checksum in the TCP header is w{x nt q because the checksum cmputatin is cmputed ver the entire header and it is cmplemented. If the header fields are the same, the checksum cmputed fr a packet with and p as data is the same as that cmputed fr a packet with q and F (pad t keep length the same). Figure 1 shws a schematic diagram f the specially cnstructed TCP segment that is used in ur explit. The first 20 bytes are the IP header, the next 20 are the TCP header, and the last 4 bytes are the data. The 2

3 } Initiatr Nde Acceptr Nde Time SYN SYN+ACK ACK Cnnectin Established Figure 2: Establishing a TCP cnnectin. values f and p and the checksum (w{x ) are shwn in blue. TCP is a cnnectin-riented prtcl meaning that a cnnectin must first be established befre data is sent. A cnnectin is established with three messages, see Figure 2. First, the initiatr sends a SYN (the TCP SYN flag is set) t a remte nde. If the remte nde is willing t accept a new cnnectin, it respnds with a SYN/ACK (bth the SYN and ACK flags are set) message. Finally, the initiatr sends an ACK message [1, 2]. In these messages ndes exchange initial sequence numbers. The test TCP segments, cnstructed as abve, are sent fllwing the establishment f a cnnectin. Because we are cnnecting t an HTTP server, the respnse received depends n the cnfiguratin f the server. Hwever, all we care abut is whether there is a respnse r nt. The crrect slutin results in a valid TCP segment, but it is nt a valid HTTP request. Therefre, the respnse will be smething like that in Figure 3. 3 Algrithm An 8-variable 2-SAT prblem prblem is shwn belw. _Z!U e g!w _Z S g!w ' i!u e j!f g k!w j' c!u ' g! The slutin vectr, }, has 8 elements that can range ver 0 and 1. Thus there are E c ~ `:F `: `: `: `: `: `:F ƒ. The nly crrect slutin t (1) is: } specially cnstructed TCP/IP packet, as shwn in Figure 1. The high-level algrithm is: S = create TCP segment S.checksum = checksum freach } s@ S.data = pad with zeres( } s ) send S receive answer if answer = true write } s is a slutin (1) pssible slutins fr. Our algrithm tests each slutin using a First, we create a TCP segment that cntains all the standard header infrmatin required by the prtcl. Next the checksum field is set. The slutin determines the checksum, as discussed in Figure 3 in the text; 3

4 q q <html> <head> <Title> Ntre Dame Cmputer Science and Engineering </Title> </head> <bdy bgclr=white> <center> <img align=middle src= alt="[ U_N_I_V_E_R_S_I_T_Y f N_O_T_R_E D_A_M_E ]"> <h1></h1><h2><p> That feature is nt implelemented n this server (501): <br> /index.html < <H2> If yu feel this message is in errr, please cntact <br> <em><a href="mailt:www@ </em></h2> <H4> Please include the full URL that yu are trying t access, <br>r we may nt be able t prvide yu with any assistance.<br><p> <img align=bttm src= <p> <a href= <img align=tp src= alt="[back TO ND CSE HOME PAGE]"></a> </center> </bdy></html> Figure 3: Respnse frm HTTP server therefre, there is a single checksum fr all tests. In the main lp f the algrithm, a test slutin is placed int the data field f the segment. Then the packet is sent, and we wait fr an answer. A crrect slutin induces a respnse frm the remte nde. Therefre, a respnse means a crrect slutin. An incrrect slutin is deduced by nt receiving a respnse. This is dne by timing ut: if a respnse is nt received within a certain amunt f time it is presumed a negative answer. This is discussed in greater detail in n 2 and n 4. 4 Implementatins In ur implementatin a single master nde cntrls the executin f the algrithm. There are several ways t implement the basic algrithm discussed abve. The tw majr chices are (a) cncurrency and (b) cnnectin reuse. Regarding (a), the master nde can have many cmputatins ccurring in the web cncurrently. Each cncurrent cmputatin requires a separate TCP cnnectin t a HTTP hst. Regarding (b), befre a TCP cnnectin can be used, it must be established. Once established, TCP segments can be sent t the remte hst. When multiple guesses are sent in ne cnnectin, it is impssible p:_` _K pa ` ^ t knw t which guess a crrect slutin refers t. Fr example, suppse guess and are sent ne after the ther in a single cnnectin. Further suppse that nly ne slutin is crrect. We expect t get ne respnse back. But we cannt tell t which slutin the respnse refers. The implementatin used in this paper is a prttype that is nt designed fr efficiency f executin. In ur prttype implement there is n cncurrency and each cnnectin is used fr exactly ne cmputatin. 4

5 4.1 Reliable Cmmunicatins Any message can get lst. In a reliable system, the sender f a message saves a cpy f the message and waits fr an acknwledgement f the message. If after sme time, the sender has nt received an acknwledgement, it will re-send the message (frm the cpy). The sender will cntinue t d this until an acknwledgement is received. In general, there is n upper bund n hw lng a message might take t be delivered. Cnsequently, in a distributed system, it is nt pssible t distinguish between a lst message and a delayed message. Therefre, a message is assumed lst after sme time-ut perid. A time-ut value that is t small declares t many delayed messages as lst. On the ther hand, a value that is t large unnecessarily slws dwn the system. Our explit circumvents the reliability mechanism in TCP. Furthermre, because an invalid slutin fails the checksum, it is as if it never arrived. Therefre, the receiver will nt send an acknwledgement f the message. There are tw undesirable utcmes that culd ccur: A false negative ccurs when a packet fr a valid slutin is drpped due t a data crruptin r cngestin, and A false psitive ccurs when a bit errr changes an invalid result int an valid result. The latter is very rare statistically and all but impssible in practice. Althugh the frmer is als unusual, it is frequent enugh that it shuld be cnsidered further. First, let s cnsider the errrs that are caught by the TCP checksum. Every transmissin link (hardware devices such as ethernet) cmputes a checksum n its packets. The TCP checksum catches errrs that pass the link checksum, but still have sme data crruptin. Because the data was nt damaged in transmissin (where it wuld have been caught by the transmissin link checksum), it must have ccurred in an intermediate system (ruter, bridge, gateway, etc.) r at an end pint (sender r receiver) [3]. Such errrs ccur very infrequently. Research shws that the TCP checksum fails abut 1 in E _ ˆ messages [4]. The prbability f receiving a false psitive is the prbability f a errr times the prbability that it changes an invalid slutin int a valid slutin. The prbability f the latter event is infinitesimal. Secnd, an IP packet might be drpped due t data crruptin r cngestin. The rdinary TCP reliability mechanism handles this, but it is disabled in ur prttype. Our test shw false negatives ccur between 1 in abut 100 and less than 1 in 17,000. The errr rate is strngly crrelated with the distance (number f hps) between end pints. 4.2 Dealing with an Unreliable System This sectin describes tw appraches t using this unreliable system. First, ne culd ask every questin multiple times. The prbability f false negatives is almst certainly uncrrelated. Therefre, if Š is the the likelihd prbability f a false negative, then Š{ is the prbability f Œ false negative. Because ŠV f a false negative all but disappears fr small values f Œ. Secnd, ne culd ask a questin, Ž S, and its cmplement, Ž. Absent any errrs, ne will get exactly ne respnse. If n respnse is received, ne must assume that a prblem ccurred. Then, the questins shuld be asked again. This slutin results in a reliable system, but requires the every questin als have a cmplement. 5

6 ! F F u q E 4.3 Implementing the 3-SAT Prblem The 3-SAT prblem has three variables per clause. This easily can be supprted with the prttype. The algrithm described in n 3 des nt have t be mdified. The nly change is t hw the packet is cnstructed. Each candidate slutin cntains three 16-bit wrds, which are added tgether and cmpared t answer the pz questin: Is q equal t? The sender cmputes the checksum ver three 16-bit wrds and the header, as shwn belw. TCP header s... w\ w\... data p u Dummy packet fr cmputing checksum Slutin packet The data part f the packet cntains three data wrds. Data wrds are cnstructed with zer padding, as dne in the 2-SAT prblem. On the receiver side, a checksum in cmputed ver the TCP packet just as befre. A respnse is sent p nly if q des indeed equal. The nly difference is between this and the 2-SAT prblem, is that the packet is 2 bytes lnger. 5 Scalability Definitin f 3-SAT prblem: 3-SAT is a special case f the satisfiability prblem whereby all frmulae are in a special frm. A literal is a Blean variable r a negated Blean variable, as in r S. A clause is several literals cnnected with ORs, as in _ T S T S e!. A Blean frmula is in cnjunctive nrmal frm, called a CNF-frmula, if all clauses are cnnected with ANDs, as in Z_ZT S b 'T S de't^bg"!w de't S dibt bjh!f dg't^bi't S dj!6y It is a 3CNF-frmula if all the clauses have exactly three literals, as in Z_ZT S d BT S bë!w de't S dibt dj!u be't S bj Tdg"!6Y 6 The 3-SAT prblem is determining the satisfiability f a 3CNF-frmula. It is pssible t reduce any %! - SAT prblem int a 3-SAT prblem. First, if a frmula has mixed ANDs and ORs, by the distributive laws, X T "! ^[!ft V\!6` XT "! T^[!f TV\!6` we can replace an OR f ANDs with an equivalent AND f ORs. Ding s may significantly increase the size f each sub-frmula, but nly by cnstant factr. By this prcess, we have written the frmula in CNF. All that remains is t make all clauses cntain three literals. In each clause that currently has ne r tw literals, replicate ne literal until the ttal number is three. In each clause that has mre than three literals, split it int several clauses and add additinal variables t preserve the satisfiability r nn-satisfiability f the riginal. _[T ZT e#t g! Fr example, the fllwing clause _[T #T "!I 6S T e#t g! is replaced with where z is a new variable. It is straightfrward t extend this int the clause cntaining Œ literals. Fr example, _ft <TXY/Y/YHT can be rewritten with the Œ _ft <TXt_!d 6S t_ft e<tx9![ 6S 9 T g<t clauses 9e"!fVY/Y/Y" 6S l š e T prblem. š _ T! Therefre, it is always pssible t reduce 6! -SAT prblem int 3-SAT 6

7 References [1] Stevens, W. R TCP/IP Illustrated. Addisn-Wesley, Reading, Massachusetts. [2] Fruzan, B. A TCP/IP Prtcl Suite. McGraw-Hill, Bstn, Massachusetts. [3] Jnathan Stne and Craig Partridge. When the CRC and TCP checksum disagree. In SIGCOMM 2000, September [4] Jnathan Stne, Michael Greenwald, Craig Partridge, and Jim Hughes. Perfrmance f checksums and CRCs ver real data. IEEE Trans. n Netwrks, Octber

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