An Enegy-Efficient Appoach fo Povenance Tansmission in Wieless Senso Netwoks S. M. Iftekhaul Alam Pudue Univesity alams@pudue.edu Sonia Fahmy Pudue Univesity fahmy@cs.pudue.edu Abstact Assessing the tustwothiness of senso data and tansmittes of this data is citical fo quality assuance. Tust evaluation famewoks utilize data povenance along with the sensed data values to compute the tustwothiness of each data item. Howeve, in a sizeable multi-hop senso netwok, povenance infomation equies a lage and vaiable numbe of bits in each packet, esulting in high enegy dissipation due to the extended peiod of adio communication, and making tust systems unusable. We popose enegy-efficient povenance encoding and constuction schemes, which we efe to as Pobabilistic Povenance Flow (PPF). To the best of ou knowledge, ous is the fist wok to make the Pobabilistic Packet Making (PPM) appoach fo IP taceback feasible fo senso netwoks. We design two bit-efficient povenance encoding schemes along with a complementay vanilla scheme. Depending on the netwok size and bit budget, we select the best method using mathematical appoximations and numeical analysis. Ou TOSSIM simulations demonstate that the encoding schemes of PPF have identical pefomance with a low bit budget ( 32-bit), equiing 33% fewe packets and 3% less enegy than PPM vaiants to constuct povenance. With a two-fold incease in bit budget, PPF with the selected encoding scheme educes the enegy consumption by 6%. Index Tems povenance; tust famewok; pobabilistic packet making; enegy-efficiency; senso netwoks I. INTRODUCTION New mico sensos have enabled wieless senso netwoks (WSNs) to gathe eal-time data fom the physical wold [], [2]. Planet-wide senso netwoks [3], [4], senso netwoks fo lage-scale uban envionments [5], and physical infastuctue systems [6] indicate potential deployments of multihop netwoks consisting of hundeds of senso nodes. In such netwoks, data poduced by the sensos ae collected at the base station and made available to decision makes fo futhe analysis. As the quality of decision making is citically dependent on the quality of tansmitted infomation [5], tustwothiness of infomation and infomation-tansmitting nodes is impotant [7]. In a multi-hop netwok, povenance includes knowledge of the oiginato and pocessing path of data since its geneation. While a few povenance-based tust evaluation famewoks have been poposed [8], [9], they do not investigate enegy dissipation due to povenance tansmission. Povenance of a data item can be epesented as a tee which is embedded as meta-data with the item, and updated along the path used to fowad the item to the base station [9]. In this case, evey intemediate node caies povenance of This wok has been sponsoed in pat by NSF gant CNS-964294. length popotional to the hop count between that node and the oiginato of the data item. In a netwok with a lage diamete (hop count), this inceased meta-data length esults in an extended peiod of adio communication and enegy dissipation at evey intemediate node. We conside a eal deployment of a 46-hop netwok [] in ou simulations, and obseve that aggegated enegy dissipation of the netwok inceases by 27% when a taditional tust famewok is employed. Although lage netwoks can be hieachically oganized [], they still equie a significant numbe of hops [2], with non-negligible enegy usage fo povenance tansmission. Povenance encoding and constuction is simila in natue to the well-known IP taceback poblem [3], [4]. IP taceback aims to detemine the fowading paths of spoofed packets in taditional wied netwoks. Among the many poposed solutions to this poblem, Pobabilistic Packet Making (PPM) can be most easily adapted to WSNs [5]. In ou pevious wok [6], we showed that diect application of PPM to WSNs is infeasible since it embeds a single node identifie in each packet, and hence equies a lage numbe of packets to constuct the fowading path. Instead, we poposed a new appoach, Pobabilistic Povenance Flow (PPF), whee a connected subgaph of the fowading path is pobabilistically incopoated into a packet. In this pape, we extend PPF by designing and analyzing thee new bit-efficient povenance encoding schemes that ensue faste convegence of povenance constuction in an abitaily lage multi-hop netwok. We investigate the selection of encoding scheme and its paametes given a fixed bit budget fo povenance encoding. Ou simulation esults show that PPF with the selected encoding scheme can consume up to 6% less enegy than the taditional appoach, which significantly inceases the netwok life-time. The emainde of this pape is oganized as follows. We fomulate the poblem of enegy-efficient povenance tansmission in Section II. Section III discusses elated wok. Section IV explains thee diffeent encoding schemes unde PPF. We discuss the coesponding appoaches to decode and constuct povenance in Section V. In Section VI, we examine the selection of paametes fo one of the encoding methods. We analyze the bit equiements fo embedding povenance using all encoding schemes in Section VII. Section VIII pesents TOSSIM simulation esults. Finally, Section IX concludes the pape.
A. Netwok Model II. PROBLEM FORMULATION We conside a multi-hop WSN whee changes in topology due to failue o mobility can occu, but ae infequent. We make the following assumptions egading the netwok and taffic: () A Base Station (BS) acts as a cental command authoity and the oot of a outing tee. It has no esouce constaints and cannot be compomised by an attacke. (2) Senso nodes monito thei suoundings and peiodically epot to the base station o thei designated cluste head (if any). (3) Multiple sensos ae used to monito an event. Within a paticula time window, independent obsevations obtained at cluste heads (if any) o the base station fom diffeent sensos ae concened with the same event. (4) A povenance-based tust management method such as [8], [9] is used in the application laye to evaluate and manage tust in an adaptive manne. Moe details can be found in [6]. B. Poblem Statement We conside a netwok of N nodes, whee the maximum length (depth) of any fowading path (tee) is L. Assume that the maximum numbe of bits that can be used to embed povenance infomation in a single packet is B. Based on this bit budget, thee is an intege m, < m L such that at most m consecutive node identities (that is, m consecutive edges) can be embedded into a single packet. We must pefom the following thee opeations: () Povenance Embedding: In a fowading tee G = (V,E) ooted at the base station, each node n i V makes an independent decision whethe to embed its identity into the packet, stating a connected sub-gaph, with pobability p i. We need to design a povenance embedding method to cay a patial path P =< n i,n i2, n im > into a single packet whee n ij V, j m and (n ik,n ik+ ) E, k m. This poblem is a simple extension of the edge sampling appoach in IP taceback [3]. (2) Povenance Constuction: At the base station, we must constuct the entie povenance tee G = (V, E) by exploiting patial path infomation collected fom a numbe of eceived packets, with an uppe bound on the numbe of packets equied to constuct the povenance. (3) Povenance Evolution: Afte topological changes, e.g., due to failues o mobility, we must bound the time that it takes to eflect the changes in the constucted povenance. III. RELATED WORK A few povenance-based tust famewoks have been poposed to date [8], [9]. These famewoks do not conside enegy-efficiency in WSNs. The poblem of povenance tansmission is elated to the IP taceback poblem that detemines the fowading path of spoofed packets [7]. IP taceback methods include hop-by-hop tacing [8], [9], out-of-band ICMP taceback [2], and in-band pobabilistic packet making [3], [4]. Hop-by-hop tacing is not well-suited to WSNs due to its lage stoage equiement. Hot-spot based taceback methods designed fo mobile ad-hoc netwoks [2], [22] stoe packet infomation at the nodes, and taceback is pefomed hop-by-hop to detemine the hot-spot whee the attacke is located. In ou case, povenance infomation is continuously equied at the base station to compute tust scoes of descendant nodes. Hot-spot based methods would incu unnecessay delay in tust scoe calculation. Out-of-band ICMP taceback equies out-of-band communication and inceased bandwidth which limit its usability in esouce-constained WSNs. In this wok, we adapt Pobabilistic Packet Making (PPM) since it does not equie additional stoage o out-of-band communication. PPM assumes static outes which may not hold in ou case. Additionally, PPM equies a significant numbe of packets to constuct the fowading path. Netwok coding vaiants of PPM [23], [24] equie fewe packets to constuct the fowading path. Netwok coding appoaches, howeve, have a high computational complexity and incease the length of the packet, as making coefficients ae tansmitted with the packet. Cheng et al. [25] detemine the optimal making pobability fo each node to educe the numbe of packets equied to constuct the fowading path. IV. PROBABILISTIC PROVENANCE EMBEDDING In this section, we pesent thee povenance embedding schemes as pat of ou Pobabilistic Povenance Flow (PPF) appoach. All thee methods incopoate node identifies into a packet pobabilistically and only diffe in how they encode these identifies. A. Juxtaposition of Ranks In the ank method, instead of embedding the node ID diectly into a packet, ank(id) (defined in Definition below) of the node is embedded, since evey node ID is uniquely identifiable by its ank, and the ank would need fewe bits than the identity. Definition. Conside U = {ID,ID 2, ID N } as the set of N node IDs. Thee is a pemutation of U, σ(u) = {ID a,id a2, ID an }, such that, ID aj < ID aj+, fo j N. Rank of any node ID, ID i U, denoted as ank(id i ), is the position of ID i in σ(u). Assume that the packet meta-data has space to hold identities of up to m nodes. We use a counte of log 2 m bits to tack the numbe of aleady embedded anks in the packet. Initially, the buffe and counte contain zeoes. Evey node n i decides to stat a connected sub-gaph with pobability p i. Once it decides to do so, it ovewites the pevious infomation by doing the following: it zeoes out the entie povenance field and then embeds its ank at the beginning of the buffe and sets the counte to one. If a node decides not to ovewite, it checks fo empty buffe space using the counte field. If thee is space, it adds its ank into the fist available slot in the buffe and incements the counte. Fig. shows an example of this method whee the buffe space can hold at most thee node identities in a single packet.
Decision: Ovewite Not Ovewite Not Ovewite Ovewite Not Ovewite Ranks 5 6 5 5 6 5 6 7 9 9 2 Binay counte 7 9 2 Fig.. Povenance encoding using juxtaposition of anks (numbes inscibed in the cicles indicate ank of nodes). B. Pime Multiplication Ou second encoding scheme, the pime method, is based on pime multiplication. In a easonably lage netwok, this method can embed moe node IDs within same numbe of bits (on the aveage), compaed to the ank method. To the best of ou knowledge, this method has not been used in any pio wok. TABLE I BIT REQUIREMENTS FOR MULTIPLICATION OF m NODE IDS, PICKED RANDOMLY FROM THE FIRST f PRIME NUMBERS. f m = 3 m = 4 m = 6 Avg Max Avg Max Avg Max 5 3.8 36 4.76 47 59.82 69 34.25 39 45.35 52 67.2 76 2 37.82 42 49.84 56 73.64 83 5 42.33 47 55.83 62 82.45 93 Definition 2. Let P n be the lagest pime numbe that is less than o equal to the positive intege n. We define the set of usable IDs, Q P,s whee P is a pime numbe and s is a positive intege: Q P,s = {n N 2 n P and n P n s} Definition 3. Fo any positive intege n Q P,s, fo some P and s, we define two functions: pime(n) = P n (the lagest pime numbe that is less than o equal to n). offset(n) = n P n. The pime method is motivated by the idea of using pime numbes as node IDs and encoding a set of IDs though thei multiplication which can be uniquely factoized. Howeve, pime multiplication incus computational and spatial ovehead when the paticipating pime numbes become lage. As shown in Table I, the aveage numbe of bits equied to multiply m pime numbes inceases with the inceasing size of the domain of these numbes. This shows the infeasibility of using pime numbes diectly as node IDs. Thus, we define Q P,s (Definition 2) to ensue that node IDs will not diffe by moe than s fom thei neaest pime numbes whee s is efeed as the speading facto. Then, we encode a sequence of node IDs by multiplying thei neaest pime numbes and summing up the coesponding offset values (Definition 3). Befoe descibing the details of encoding pocess, we descibe the assignment of node IDs using set Q P,s. ) Node ID Assignment: Fo a netwok of N nodes, we pick a set Q P,s = {q,q 2, q z } with the smallest z such that z N. An in-place algoithm is used to poduce a andom pemutation of Q P,s, σ(q P,s ) = {q a,q a2, q az } and membes of σ(q P,s ) ae assigned to all N nodes sequentially. Fo example, in an 8-node netwok, we can pick IDs fo the nodes fom a andom pemutation of Q,7 = {2,3,4,5,6,7,8,9,,}. Fo a given numbe of nodes (N), the bit equiements fo pime multiplication incease the most when s =, which makes Q P,s nothing but a set of pime numbes that ae less than o equal to P ( = Nth pime numbe). With an inceasing value of s, the set Q P,s can contain numbes uppe-bound by elatively smalle value of P ( Nth pime numbe). By tuning s, the lagest element (P ) of Q P,s can be made close to the total numbe of nodes (N). This bings about an inteesting tade-off: eduction of the numbe of bits equied fo pime multiplication vesus incease in the numbe of bits equied fo summation of offsets. We will investigate this tade-off in Section VI. 2) Encoding Pocess: To stoe povenance infomation, we divide the povenance buffe into two pats: poduct and offset. Evey node n i has an ID, say ID i, that is a membe of Q P,s fo some P and s. As with the ank method, once a node n i decides to stat a connected sub-gaph, it cleas the povenance buffe. It then insets pime(id i ) into the poduct pat and offset(id i ) into the offset pat (Definition 3). If a node n j decides not to ovewite, it etieves the values stoed in the poduct and offset pats. It then multiplies the value of the poduct with pime(id j ), adds offset(id j ) to the offset, and stoes the newly calculated values into the espective pats. Fig. 2 shows an example with m = 2. Decision: Ovewite Not Ovewite Not Ovewite Ovewite Not Ovewite 5 Pime multiplication Offset 4 25 8 5 5x3 + 5x3x23 ++2 7 7x29 + Not Ovewite 3 9 7x29x7 Fig. 2. Povenance encoding using pime multiplication (numbes inscibed in the cicles indicate ID of nodes). We no longe need a counte field to tack the numbe of node identities encoded in the povenance buffe because thee is always a unique pime factoization of the poduct pat which gives the numbe of paticipating nodes. C. Rabin s The pime method can typically accommodate moe node identities (m) than the ank method, but pime multiplication esults incease in size as N and m incease. In ode to embed moe node IDs into a single packet without equiing additional bits and excessive computational complexity, we investigate Rabin fingepints [26]. A Rabin fingepint calculates a nea-pefect and spaceefficient unique epesentation of a sequence of bits. Fo a sequence of bits n,n 2, n m, of length m, the Rabin fingepint is given by the following expession, whee α and M ae constant integes: RF(n,n 2, n m ) =(n α m +n 2 α m 2 + +n m ) mod M ++2
The fingepint of the concatenation of two sequences X and Y can be computed as follows: RF(X Y) = RF(RF(X) Y) = RF(RF(X) α l )+RF(Y) whee, epesents concatenation and l is the length of Y. ) Encoding s: The patial path tavesed by a packet can be consideed as a bit sequence of IDs of the nodes on that patial path. We aim at tansmitting the fingepint of that bit sequence sequence instead of tansmitting the actual sequence. Evey node uses two constant integes α and M to compute its fingepint. As the packet taveses the path, we could easily compute the contibution of evey node to the fingepint and add it to the contibutions of its pedecesso nodes on that path. Fo example, if the packet taveses the patial path < n,n 2, n m >, node n i, i m has a contibution of ID i α b(m i+) to the fingepint associated with that path. Hee, b is the numbe of bits equied to epesent a single ID. Howeve, the incemental sum of these contibutions equies a lage and vaiable numbe of bits in the packet which is undesiable. Hence, we exploit the concatenation popety of Rabin fingepints, which allows any node n k to compute the fingepint of node IDs fom ID n to ID nk by concatenating its own ID (ID nk ) to the fingepint value of pevious nodes ID n to ID nk. The following equation makes this claim clea: RF(ID k,id k, ID ) = RF(ID k RF(ID k,id k 2, ID )) () Thus, evey node on a path can update the fingepint without equiing any exta bits as the fingepint value is always less than the diviso M. To stoe povenance infomation, we divide the povenance buffe into thee fields: fingepint, intemediate node, and length. As in othe PPF methods, evey noden i decides to stat a connected sub-gaph with pobability p i. Once it decides to do so, it cleas the buffe and insets ID i and into the fingepint and length fields, espectively. If a node n j decides not to ovewite, it etieves the values stoed in the fingepint and length fields. If the length is less than m, it updates the cuent fingepint value (say X) by computing ID j X and incements the cuent value of the length field by one. If the newly computed length is less than o equal to m, ID j is stoed in the intemediate node field that will aid in decoding the povenance as discussed late. 2) Patitioning s: Since we need to exploit pevious knowledge about node odeing to etieve povenance infomation fom the fingepint (as we will discuss in the next section), tansmitting lage patial povenance infomation using fingepints may not always be advantageous. In WSNs, nodes ae vulneable and eo-pone and the outing path may change due to the lossy natue of the wieless medium. This may cause inconsistency between the cuent and the peviously stoed odeing among nodes, making the entie fingepint-based povenance infomation useless. To mitigate this poblem, we divide the fingepint field into ( > ) pats aound the ( ) intemediate nodes so that changes in odeing in one pat do not affect othe pats and changes RF(5) Decision: Ovewite Not Ovewite Ovewite Not Ovewite RF(5) 5 7 9 Intemediate node 5 RF(7, 6, 5) 7 3 RF(9) 9 RF(, 9) 2 Length (a) Embedding povenance with non-patitioned fingepint Decision: Ovewite Not Ovewite Not Ovewite Ovewite 5 Intemediate node 5 7 9 Length RF(7, 6, 5) 7 3 RF(7, 6, 5) 7 RF(7, 8, 9) 5 RF() (b) Embedding povenance with m = 5 and = 2 Fig. 3. Povenance encoding using Rabin fingepints (numbes inscibed in the cicles indicate ID of nodes). in odeing can be eflected. Each pat contains fingepints of at most m+ node IDs whee the length field indicates the total numbe of paticipating node IDs. Fo example, assume we wish to embed infomation about a patial path < n,n 2, n m > into a single packet with m = 7 and = 2. As the packet taveses the netwok, node n 4 becomes the intemediate node and the fist pat of the fingepint contains RF(ID 4,ID 3,ID 2,ID ) and the second pat contains RF(ID 4,ID 5,ID 6,ID 7 ). Both pats of the fingepint ae calculated accoding to equation and the length field indicates the combined length of the both pats. Note that, a non-patitioned fingepint is a special case whee the intemediate node coesponds to the lone fingepint. Fig. 3(a) depicts an example of the nonpatitioned case and Fig. 3(b) depicts the patitioning appoach with m = 5 and = 2. Note that in adjacent fingepints, the ode in which the fingepints ae calculated is opposite. This ensues checking the integity of the patitioned fingepints at no additional cost. V. DECODING AND CONSTRUCTING PROVENANCE When a packet is eceived at the base station, the povenance buffe is examined to etieve the embedded patial povenance (o path) infomation. With the ank embedding appoach, we can easily extact the embedded identities fom the povenance buffe using the length field as the ank of each node ID uses a fixed numbe of bits. Howeve, with both the pime and fingepint embedding method, we assume that infomation about odeing among nodes is known befoehand using a peviously constucted povenance gaph,g pe = (V pe,e pe ) (as discussed in Section V-C). Hee, V pe is the set of node IDs and E pe is the set of edges among those nodes indicating povenance flow. A. Decoding Pocess fo Pime Method We apply a standad pime factoization algoithm ove the poduct pat of the povenance buffe to etieve the set of neaest pime numbes (say, X = {X,X 2, X m }). Then, we pefom a DFS (Depth-Fist-Seach) ove G pe to find all the possible paths consisting of m node IDs whose neaest pime numbes fom any pemutation of X. We need to modify the DFS to compae the neaest pime numbe of a paticula
node ID with the membes of set X while visiting that node and tack the offset of the ID when a match is found. With this modification, DFS will take O(m( V pe + E pe )) time and poduce all possible sets of node IDs whose neaest pime numbes fom a pemutation of X. Since evey ID does not diffe fom its neaest pime numbe by moe than s, we can find at most s m such sets. Fo each such set, we will sum up the offset values and calculate its diffeence fom the offset value etieved fom the eceived packet. If the diffeence is zeo, we ecod the matched set as the etieved povenance. Othewise, we conclude that topological changes may have occued in the netwok and the peviously stoed povenance gaph may not be up-to-date. Futhe pocessing is necessay to detemine the povenance, such as checking othe combinations of patial paths by consideing nodes that ae o 2-hop away fom the nodes on the ecoded path, o tiggeing the ank ID embedding appoach to ecove the ode. These extensions will be the subject of ou futue wok. B. Decoding Pocess fo Method Upon eception of a packet, we etieve the fingepint(s), associated intemediate node ID(s), and the length field indicating the numbe of paticipating node IDs. Fo evey intemediate node, ID i, i, we pefom a DFS ove G pe = (V pe,e pe ) with following modifications: Set ID i as the oot fo DFS. Seach all the nodes within m+ -hops away fom the node ID i and compute the Rabin fingepint using the concatenation popety. Computing the fingepint takes constant numbe of shift and XOR opeations povided that a pe-computed hash table (of size < 4 kb) contains fequently used values. Afte computing evey fingepint of length m+, we compae them to the etieved fingepints RF i and RF i+. If a match is found, we ecod the matched path oiginating fom the node ID i ; othewise, futhe pocessing is necessay to detemine the exact povenance which we will conside in ou futue wok. Typically, fo evey intemediate node, we ae seaching a smalle potion of the gaph G pe using DFS. In the wost case, we may seach the entie gaph which can be pefomed in O( V pe + E pe ) time. False Positive Rate fo s: Assume that we have at most x = m+ node IDs pe patition whee m is the total numbe of node IDs embedded pe packet. Since decoding each pai of patitions is independent of othes, it suffices to analyze the false positive pobability of fingepinting a path of x node IDs oiginating fom a paticula intemediate node ID. Assume that thee ae n such paths in the povenance gaph. Then, the false positive pobability n2 x.b, whee k is 2 k the numbe of bits used fo fingepinting and b is the length of the bit epesentation of one node ID. If the maximum fan-out of the netwok isf, thennis uppebound by f x which gives, False positive pobability C. Constuction and Evolution of Povenance f2(x ) x.b 2 k (2) = 2(m ) b(m+ )f.2 k. (3) With ou identity embedding methods, povenance constuction is staightfowad once we have decoded patial path infomation fom the eceived packet. Afte collecting sufficient packets with embedded povenance (i.e., when we have at least one ID fom each node), we combine the patial paths to poduce the complete povenance gaph, G = (V,E). Hee, V is the set of nodes and fo some v i V,v j V, (v i,v j ) E iff (ID i,id j ) belongs to some patial povenance encoded in a eceived packet. Howeve, decoding using the pime and fingepint appoaches needs pio knowledge of the ode of nodes. This can be obtained by applying the ank method fist. Afte a configuable peiod of time (geneally geate than povenance convegence time) duing which the povenance is constucted using the ank method, the pime o fingepint embedding method can be employed. In ode to keep node ode infomation up-to-date, nodes utilize the ank appoach evey t embedding seconds. Thus, any topological changes ae eflected in the povenance. Based on the fequency of mobility o failues in the netwok,t embedding can be adjusted. Howeve, a small value of t embedding will educe the benefits of applying the bit-efficient pime and fingepint methods. We ae cuently consideing a eactive appoach to tigge the ank appoach only when necessay. D. Complexity Analysis Decoding using the ank method is staightfowad and takes only O(m) time, whee m is the maximum numbe of nodes embedded pe packet. In case of the pime method, we use the Geneal Numbe Field Sieve (GNFS) algoithm fo pime factoization. The asymptotic unning time fo this algoithm fo a b-bit numbe is O(exp(( 64b 9 ) 3(logb) 2 3)). In an N-node netwok, node IDs equie at most log 2 N bits with the appopiate choice of s and m. Thus, multiplication of m node IDs equies at most m log 2 N bits which makes the complexity of pime factoization O(exp(( 64m log 2 N 9 ) 3(log(m log 2 N )) 2 3)). Then, we need to pefom DFS which takes O(m( V + E )) and s m compaisons which take O(s m ) time. We know N π(n) that s can be appoximated as lnn (as discussed in the next section). Thus, the time equied to decode povenance fom a single packet becomes O(exp(( 64m log 2N ) 3 (log(m log2 N )) 2 3 )+mn 2 +(lnn) m ) 9 which is exponential in tems of mlnn. In case of the fingepint method, we need to seach the povenance gaph and update the fingepint using concatenation while visiting a node on that gaph. This takes O( V + E ) time which is O(N 2 ) in the wost case.
TABLE II PRIME GAPS BELOW THE NUMBER n. n Obseved mean (µ) Pime Gap µ Obseved stdev gap avg gap avg 5 5.29 3. 5.2.3 5.96 3.55 5.82.2 2 6.6 4.5 6.52. 5 7.48 5.3 7.44. Regading the constuction of entie povenance in an N- node netwok, we epesent the povenance gaph using an adjacency matix. The total numbe of edges of the gaph can be at most O(N 2 ). Since edge insetion equies constant time, the wost case complexity fo constucting the entie povenance is O(N 2 ). We need O(N 2 ) space to hold the povenance gaph apat fom the space equied by the tust famewok. VI. SPREADING FACTOR The pime method equies two paametes s and P that define the set of node IDs, Q P,s. A. Appoximating the Speading Facto Fo a given numbe of nodes, N, we want to detemine the speading facto, s that minimizes the highest value of Q P,s. This value of s depends on the pime gap. Definition 4. A pime gap is the diffeence between two successive pime numbes, p k and p (k+), whee p k is the kth pime numbe. Thus, a pime gap of length n is a un of n consecutive composite numbes between two successive pimes. We use the Pime Numbe Theoem to appoximate the aveage length of pime gaps. The theoem gives an asymptotic fom fo the pime counting function π(n), which counts the numbe of pimes less than some intege n. Accoding to this theoem (poved independently by Hadamad (896) and de la Valle Poussin (896)), π(n) k= k!n (lnn) k+ n lnn + n (lnn) 2 + 2n + (4) (lnn) 3 It has been shown that summation of the fist thee tems in equation 4 is a bette estimate fo π(n) (Debyshie 24, pp. 6-7). Now, we can appoximate the aveage length of pime gaps below n as gap avg (N) n π(n) lnn + (lnn) + 2. 2 (lnn) 3 Table II shows the theoetical mean along with empiical mean and standad deviation of pime gaps fo diffeent values of n. The last column of this table gives the atio between empiical and theoetical mean which justifies the appoximation above. Assume that P n denotes the nth pime numbe. By choosing a speading facto, s, that appoximates to gap avg (N) fo some N, we can obtain a set of numbes TABLE IV AVERAGE CASE BIT REQUIREMENTS FOR VARYING NUMBER OF NODES AND PER-PACKET NODE IDS WITH s. N m = 3 m = 4 m = 5 s avg s avg s avg 5 5 27.82 7 36.6 6 44.35 5 3.25 7 4.53 6 49.9 2 34.5 7 45.4 6 55.8 5 38.99 7 5. 62.4 uppe-bound by some pime numbe P π(n)+ N. We denote this set asq Pπ(N)+,gap avg(n). Due to the high vaiation in pime gaps with espect to gap avg (N) the cadinality of this set becomes less than N. Assume that using the same speading facto (s = gap avg (N)), we find a set of numbes, Q Pπ(N ),s such that Q Pπ(N ),s is the smallest numbe geate than o equal to N. Similaly, by choosing some values lage than gap avg (N) fo speading facto s, we can have a set of numbes Q Pπ(N ),s, whee Q Pπ(N ),s is the smallest numbe geate than o equal to N. Clealy, P π(n ) > P π(n ), which makes the latte set moe favoable in tems of bit equiements (as obseved in Table I). We use the following optimistic choice: s gap avg (N) N π(n) lnn + (lnn) + 2. (5) 2 (lnn) 3 B. Choice of Speading Facto Fo a netwok of size N, we fist calculate gap avg (N) and by setting s = gap avg (N), we pick a set Q Pπ(N ),s such that Q Pπ(N ),s is the smallest numbe geate than o equal to N. Consideing the equiements fo pime multiplication and summation of offset, we estimate the wost case bit equiements pe packet as: m Bwost(s,m) P = log 2 ( (P π(n ) i))+log 2 (m s), i= whee m is the numbe of node IDs embedded pe packet. Then we incease s by one and detemine the coesponding set of node IDs. Afte calculating the wost case bit equiements fo the new set, we compae the newly calculated value with the cuent one. If the newly calculated set outpefoms the cuent one in tems of wost case bit equiements, we set the new set to be the cuent one. We continue until the newly calculated set equies moe bits than the cuent one. Table III shows the compaison among the equied numbe of bits fo diffeent values of s with vaying numbes of nodes and numbe of pe-packet node IDs. Fo a paticula numbe of nodes and pe-packet node IDs, the last column gives the best choice fo s (s ). VII. BIT BUDGET A fixed budget of bits (B budget ) is available fo embedding povenance of at most m nodes within the meta-data of a packet. We give the value ofmfo ou thee encoding methods in this section.
TABLE III WORST CASE BIT REQUIREMENTS FOR VARYING NUMBER OF NODES, PER-PACKET NODE IDS, AND CHOICE OF s. s = gap m N gap avg s = gap avg + s = gap avg + 2 s = gap avg + 3 s = gap avg + 4 avg P max Bits P max Bits P max Bits P max Bits P max Bits s 3 5 5 63 32 587 33 563 33 547 33 52 33 5 3 5 37 36 229 36 63 36 9 36 63 36 5 3 2 6 2683 4 2477 39 2377 39 2269 39 223 39 3 5 7 6779 44 6367 43 679 43 5857 43 5669 44 4 5 5 63 43 587 42 563 42 547 43 52 43 7 4 5 37 47 229 47 63 46 9 47 63 47 7 4 2 6 2683 5 2477 5 2377 5 2269 5 223 5 7 4 5 7 6779 56 6367 57 679 57 5857 57 5669 56 7 5 5 5 63 52 587 5 563 52 547 52 52 52 6 5 5 37 57 229 57 63 57 9 57 63 57 6 5 2 6 2683 62 2477 63 2377 63 2269 62 223 62 6 5 5 7 6779 7 6367 7 679 69 5857 69 5669 69 TABLE V CHOOSING (, m) FOR DIFFERENT BIT BUDGETS IN A 5-NODE NETWORK WITH b = 3 AND ǫ = 5. B budget = 32 B budget = 64 B budget = 28 m x m x m x 2 2 9 9 25 25 2 n.a. n.a. 7 4 2 3 n.a. n.a. n.a. n.a. 6 6 4 n.a. n.a. n.a. n.a. 9 3 Aveage numbe of bits 7 6 5 4 3 2 5 Nodes Nodes 5 Nodes Rank Pime Rank Pime Rank Pime Rank Pime A. Bit Usage fo Rank Method In an N-node netwok, bit equiements fo the ank method ae B R (m) = m log 2 N +log 2 m, whee, the fist tem on the ight hand side indicates the equied numbe of bits to embed m anks, and the second tem accounts fo the counte that tacks the numbe of embedded anks. Thus, we choose the lagest m such that B R (m) B budget. B. Bit Usage fo Pime Method In an N-node netwok, we can pessimistically pick a value of m such that B P wost(m,s ) B budget. This does not guaantee the best usage of available bits since pime multiplication of m node IDs does not always need a fixed numbe of bits (as in the case of ank appoach) andb P wost(m,s ) can hold moe than m node IDs in many cases. Hence, we conside aveage case bit equiements befoe choosing an m fo a paticula bit budget. The aveage bit equiements pe packet ae B P avg(s,m) = m π(n ) ( log 2 π(n ) i=2 (i [π(2 i ) π(2 i )]) +log 2 π(n ) [π(n ) π(2 log 2 π(n ) )] ) +log 2 (m s). Table IV shows the aveage numbe of bits calculated fo diffeent numbes of nodes and pe-packet node IDs with thei coesponding s. We choose an m such that B P avg(s,m ) B budget B P avg(s,m ). Fo example, in a -node netwok with 4 bits available fo povenance embedding, we choose m to be 4 since B P avg(5,3) 4 B P avg(7,4) (Table IV). This choice povides the oppotunity to embed moe node IDs pe packet on the aveage. m=2 m=3 m=4 m=5 Numbe of IDs embedded pe packet Fig. 4. Aveage bit equiements fo pe packet povenance in netwoks of diffeent sizes. (Stacked data along a single column is given in elative values that need to be added to get the absolute values of each textued ba.) C. Bit Usage fo Method In an N-node netwok, the bit equiements fo embedding m IDs pe packet can be expessed as { B F b+log (,m) = 2 m+k, =, (6) ( )b+log 2 m+k, >, whee is the numbe of patitions in the povenance buffe, k is the numbe of bits equied fo each fingepint and b is the numbe of bits equied to epesent one node ID. We need to choose an m and such that B F (,m) B budget. Hee, = denotes the non-patitioned case, whee the entie povenance buffe can be egaded as a single patition. Assuming a false positive pobability of 2 ǫ,ǫ, fom equation 2, k = 2(x )log 2 f +log 2 b+log 2 x+ǫ (7) whee m = (x +). Fist, we conside the case when =, which leads to x = m. Then, combining (6) and (7) we have, 2(m )log 2 f +2log 2 m B budget b log 2 b ǫ. (8) Similaly, consideing >, we have (2(x )log 2 f +log 2 x+b+log 2 b+ǫ)+ log 2 (x +) B budget +b. (9) We detemine the maximum value of m fo diffeent values of using the above two equations. System designes ae left to choose the appopiate pai of(,m) based on the ate of
Numbe of packets 3 25 2 5 5 PPM PPM+NC PPF-Rank PPF-Pime PPF- 5 5 2 25 3 35 Numbe of hops Numbe of packets 8 7 6 5 4 3 2 PPF-Rank PPF-Pime PPF- 5 5 2 25 3 35 Numbe of hops Enegy Consumption (mj).4e+6.2e+6 e+6 8 6 4 2 PPM PPM+NC PPF-32 bits PPF-64 bits 5 5 2 25 3 35 Numbe of hops (a) Povenance constuction with 32-bit budget (b) Povenance constuction with 64-bit budget (c) Aggegate enegy consumption (in mj) Fig. 5. Compaison among diffeent encoding schemes of PPF and PPM vaiants. link failue o changes, and the aveage fan-out of the netwok. In Table V, we conside a 5-node netwok with aveage fan-out f = 4, b = 3 bits, and ǫ = 5 to show the possible choices fo (,m) whee x indicates maximum numbe of node IDs contained in each patition. Note that fanout should be esticted so that excessive enegy consumption at the junction node does not patition the netwok [27]. We also compae the theoetically calculated aveage bit equiements of the fingepint method with the two othe encoding schemes in Fig. 4. Clealy, the fingepint method equies fewe bits than the othe methods, as the value of N and m incease. VIII. TOSSIM RESULTS We conduct simulations using TOSSIM [28] fo netwoks with hop counts anging fom 2 to 3, and numbe of nodes anging fom 3 to 5. Fo enegy analysis, we use POWER- TOSSIMZ [29] which uses the micaz enegy model. We do not conside enegy consumption elated to CPU computations since TOSSIM cannot captue the active CPU time [29]. Howeve, all nodes othe than the base station only pefom encoding opeations which have low computational complexity and ae likely to daw insignificant level of CPU powe. A base station with no esouce constaints can pefom the decoding opeations, e.g., pime factoization, fo a modeate numbe of nodes in easonable time. All expeiments ae pefomed using a tansmission ate of 25 kbps, the default tansmission ate of the micaz mote, whee evey data-geneating senso sends data towads the base station evey 2 s. The pobability fo embedding a node ID is p = 25. All esults ae aveaged ove uns, and we find the standad deviation to be extemely small. We conside a 5-node netwok to compae the pefomance of the encoding schemes of PPF with two vaiants of pobabilistic packet making: PPM [3], [4] and PPM with Netwok Coding [23], [24], as they ae the closest to ou appoach (though they wee designed fo wied IP netwoks). We place the same constaint on usable bits (32 bits) fo povenance embedding pe packet on all schemes. Fig. 5(a) shows the numbe of packets equied to constuct povenance fo inceasing numbes of hops fom a single souce to the base station. The esults eveal that all thee schemes of PPF have identical pefomance in this case since they can embed only 3 node IDs on aveage pe packet using 32 bits. Howeve, they equie at least 33% fewe packets than both PPM vaiants. The oiginal PPM scheme equies a lage numbe of packets since it embeds a single node ID pe packet. Netwok coding-based PPM (PPM+NC) embeds a linea combination of 3 node IDs in a packet. Howeve, in ode to constuct a fowading path of length d hops, PPM+NC conveges upon eception of d unique linea combinations of node IDs, wheeas PPF only equies d + diffeent node IDs. We pefom the same expeiment in a 5-node netwok with a 64-bit budget in Fig. 5(b). Since TOSSIM does not scale to 5 nodes, we andomly assign node IDs fom a set of 5 numbes and take the aveage ove expeimental esults of seveal TOSSIM uns. We find that the pime method equies fewe packets than the ank method, while the fingepint method outpefoms both in this case. The eason is that with a 64-bit budget, the fingepint method ( = 2) embeds 7 node IDs pe packet with a low false positive ate (<.), wheeas the pime and ank methods embed 5 and 4 node IDs on aveage, espectively. Fig. 5(c) compaes the aggegate enegy consumption fo the two PPM vaiants and the two PPF methods that equie the lowest numbe of packets fo the two bit budgets (i.e., pime method with 32-bit budget and fingepint method with 64-bit budget). PPF with 32-bit budget consumes at least 3% less enegy than the PPM vaiants. PPF with 64-bit budget educes enegy consumption by moe than 6% compaed to its 32-bit countepat. Pecentage of eo.8.6.4.2 (=, m=9) (=2, m=7) (=3, m=7).5..5.2.25 Failue ate (a) Decoding eo of the fingepint method (B budget = 64 bits) Numbe of packets 2 8 6 4 2 (=, m=9) (=2, m=7) (=3, m=7).5..5.2.25 Failue ate (b) Povenance constuction using fingepint method (B budget = 64 bits) Fig. 6. Povenance and associated decoding eo with vaying ate of link changes. To obseve the effect of patitioning fingepints, we study the decoding eo of the fingepint method with espect to the ate of link changes. Hee, decoding eo denotes pecentage of node IDs that cannot be decoded due to link changes o false positive ates, whee ate of link changes indicates the aveage numbe of link changes pe unit time. We atificially intoduce link failues and associated path changes which ae andomly distibuted ove a time window of 2 s along a 3-hop path. Fig. 6(a) shows that as link changes incease, the fingepint method with two patitions ( = 2) has a decoding eo lowe
than the non-patitioned case because of its low sensitivity to topological changes. Howeve, the fingepint method with = 3 suffes fom a false positive ate of about.6 with 64- bit budget in this paticula expeiment, and pefoms wose than the = 2 case in the pesence of low ate of link changes (when decoding eo due to link changes is small). With a high ate of link changes, the case of = 3 shows a small impovement ove the = 2 case, but a elatively highe false positive ate in a dense netwok will nullify that impovement (as indicated in Table V). Fig. 6(b) shows the effect of the decoding eo in constucting povenance whee the fingepint method with = 2 conveges with a fewe numbe of packets even in the pesence of a high ate of link changes and suffes fom a negligible false positive ate. Tust scoe.8.6.4.2 2 3 4 5 Fig. 7. Taditional Tust Model Tust Model with PPF Numbe of iteations Tust scoes in a -hop netwok. We also integate PPF with a povenance-based tust famewok to iteatively compute tust scoes. Fig. 7 shows that the tust scoe calculated using PPF evolves coectly as soon as the entie povenance is constucted at the base station. PPF accuacy in tust scoe calculation is simila to the taditional appoach that includes evey node ID on the fowading path in the povenance. IX. CONCLUSIONS We have pesented an enegy-efficient povenance tansmission and constuction appoach fo lage-scale multi-hop wieless senso netwoks, based on the idea of pobabilistic incopoation of node identities. We adapt the pobabilistic packet making (PPM) appoach fo IP taceback, and popose thee povenance encoding methods with a space constaint on the size of povenance data in each packet. We analyze the suitability of the methods based on the netwok size and bit budget via mathematical appoximations and numeical methods. 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