A Secure Approach for Caching Contents in Wireless Ad Hoc Networks

Transcription

1 A Secure Approach for Caching Contents in Wireess Ad Hoc Networks Mohsen Karizadeh Kiskani and Haid R. Sadjadpour Abstract Caching ais to store data ocay in soe nodes within the network to be abe to retrieve the contents in shorter tie periods. However, caching in the network did not aways consider secure storage due to the coproise between tie perforance and security). In this paper, a nove decentraized secure coded caching approach is proposed. In this soution, nodes ony transit coded fies to avoid eavesdropper wiretappings and protect the user contents. In this technique rando vectors are used to cobine the contents using XOR operation. We odeed the proposed coded caching schee by a Shannon cipher syste to show that coded caching achieves asyptotic perfect secrecy. The proposed coded caching schee significanty sipifies the routing protoco in cached networks whie it reduces overcaching and achieves a higher throughput capacity copared to uncoded caching in reactive routing. It is shown that with the proposed coded caching schee any content can be retrieved by seecting a rando path whie achieving asyptotic optiu soution. We have aso studied the cache hit probabiity and shown that the coded cache hit probabiity is significanty higher than uncoded caching. A secure caching update agorith is aso presented. Index Ters Secure Caching, Physica Layer Security, Wireess Ad Hoc Networks I. INTRODUCTION Significant advances in wireess and obie technoogies over the past decades, have ade it possibe for obie users to strea high quaity videos and to downoad arge-sized contents. Video streaing appications ike Netfix, Huu and Aazon have becoe increasingy popuar aong obie users. Cose to haf of a video pays were on obie devices ike tabets and sartphones 1 during the fourth quarter of In parae, storage capacity of obie devices has significanty increased without being fuy utiized. Efficient use of this under-utiized storage is speciay iportant for video streaing appications that account for a arge portion of the overa internet traffic. On the other hand, proprietary contents as opposed to usergenerated contents are the ost iportant assets of any content sharing copanies such as Netfix, Huu, and Aazon. These copanies use extree easures to protect their data and therefore, they are hesitant to et the end users cache their contents ocay. One possibe soution is to encrypt each content before caching it ocay. Encryption agoriths reduce the content sharing rate. Further, such agoriths are M. K. Kiskani and H. R. Sadjadpour are with the Departent of Eectrica Engineering, University of Caifornia, Santa Cruz. Eai: {ohsen, haid}@soe.ucsc.edu 1 Index-Q pdf ony coputationay secure and an adversary is abe to break the with tie. For instance, Data Encryption Standard DES) which was once the officia Federa Inforation Processing Standard FIPS) in US is no onger considered secure. In this paper, we propose an inforation theoreticay secure soution for caching contents which cannot be decoded with tie. Physica ayer security in wireess networks has been the subject of any recent research papers. With a focus on physica ayer security in wireess ad hoc networks, we propose a nove decentraized coded caching approach in which rando vectors are used to cobine the contents using XOR operation. Nodes ony transit coded fies and therefore an eavesdropper with a noiseess channe wi not be abe to decode the desired contents. Our technique is quite different fro techniques which expoit wireess channe dynaics to achieve physica ayer security. We deonstrate that coded caching technique is siiar to Shannon cipher [24] probe and it can achieve asyptotic perfect secrecy during fie transissions by using a secure ow bandwidth channe to exchange the decoding gains. The ain contributions of this paper are introduction of decentraized coded caching for wireess ad hoc networks, capacity coputation of coded and uncoded caching for proactive and reactive routing strategies 2, proof of security, coputation of cache hit probabiity, and introduction of update agorith. A shorter version of this paper was presented in [14] that does not incude proofs for Theores 2 and 3 and Lea 4. The cache hit probabiity and cache update agorith were not addressed in [14]. The rest of the paper is organized as foows. In section II, the reated works are described. In section III the network ode, proposed encoding and decoding agoriths and earier resuts on coded caching are presented. Section IV focuses on the network capacity and the security aspect of coded caching is studied in section V. Cache hit probabiity for both coded and uncoded caching approaches is studied in section VI. In section VII, an efficient caching update agorith is proposed. Siuation resuts are provided in section VIII. The paper is concuded in section IX. II. RELATED WORK Many researchers have investigated the probe of caching in recent years. The fundaenta iits of caching over a shared ink is studied in [22]. The authors in [22] suggested to store uncoded contents or uncoded segents of the contents in the caches during cache paceent phase. During content 2 Notice that the notion of proactive or reactive) routing that we study is the extree case when copete or no) network knowedge is avaiabe.

2 2 deivery phase, they proposed to broadcast coded fies to the users which resuted in significant uticasting gain. Such a gain is achievabe by taking advantage of content overap at various caches in the network, created by a centra coordinating server. This work was ater extended in [23] to scenarios with decentraized uncoded cache paceent and there are any other reated works that foowed the sae set of assuptions. Whie references [22], [23] study the probe of caching in broadcast channes, we study the probe of caching in wireess ad hoc networks using utihop counications. The authors in [9] studied the probe of caching in utihop networks. They assued that during the cache paceent phase, uncoded contents are stored in the caches independenty in a decentraized fashion. They studied the throughput capacity of wireess ad hoc networks. In this paper, we introduce decentraized coded caching for wireess ad hoc networks and copare our resuts with uncoded caching resuts in [9]. In decentraized uncoded caching schee that we have considered in this paper, we assued that different contents are seected unifory at rando and paced in the caches during the cache paceent phase. We can therefore ode the uncoded caching strategy as a coupon coector probe with group drawings as studied in [27] and [10]. Siiar to the cassica coupon coector probe, it is proved in [10] that unifor seection of cached contents resuts in the iniu possibe vaue for the average nuber of required hops. Hence, in this paper we study the case of unifor cache paceent which is the best possibe scenario in ters of network capacity. Our proposed decentraized coded caching schee utiizes rando unifor LT codes [20], [21], [26] to encode the contents and store the during cache paceent phase. In rando binary unifor LT codes [21], a rando code of ength is chosen fro F 2 such that any of its eeents is either equa to zero or one. Such a rando code can be represented by a rando vector of ength in F 2. Then contents with indices corresponding to one in the rando vector are added together and the encoded fies are cached in the nodes. Physica ayer security has attracted any researchers in recent years. A survey of recent progress in this fied can be found in [1], [25]. Security aspects of network coding is studied in references ike [2], [19], [28]. In this paper, we wi study the security of LT coding-based caching technique. We specificay investigate the ast hop counication security which is the ost vunerabe transission ink. We prove that for a certain regie of caching sizes in the network, asyptotic perfect secrecy is achievabe for the ast hop 3. Further, a secure caching update agorith is proposed. Our proposed decentraized caching schee can be used to create a distributed network coding based storage syste. The concept of network coding for distributed storage was originay studied in [4], [6] where fies are divided into packets and nodes need to coect a the packets to be abe to reconstruct their desired contents. In our proposed distributed 3 Security investigation for other hops reains as future work. c 1 sn) c 2 c 1 sn) anchor node requesting nodes s g n) Fig. 1: Each oca group with side ength s g n) contains any square-ets of side ength c 1 sn). Each square-et has one randoy seected anchor node. caching schee, the requesting node needs to decode ony one content. The authors in [4], [6] study the repair probe which is the probe of recovering data when soe nodes fai. Further, whie they address the repair probe using MDS codes, our proposed coded caching technique is based on LT codes [20]. This paper focuses on the scaing capacity and security of a network with decentraized coded caching which is not studied in [4], [6]. LT coding based storage is studied in references ike [3], [29]. Reference [3] investigates the repair probe of LT codes in coud services and [29] proposes new types of LT codes for storage systes. Use of fountain codes and Raptor codes [26] has aso been studied in [5], [17]. None of these references have studied capacity, cache hit probabiities, cache update agoriths and security of LT code based storage in wireess ad hoc networks. A. Network Mode III. PRELIMINARIES We consider a dense wireess ad hoc network in which n nodes are unifory distributed over a square of unit area as depicted in Figure 1. These nodes use utihop counications to request one of contents fro a set abeed as F = {F 1, F 2,..., F }. The requested content is denoted by F r and the iniu nuber of nodes to decode F r by N r. The iniu nuber of nodes to decode any requested content is denoted by N. We aso assue that each node can cache M fies of equa size each containing Q bits. In practice, the contents are divided into equa-sized chunks and the chunks are cached. Such an assuption is coon in any references incuding [13], [22], [23]. The unit square area in Figure 1 is divided into any square-ets each with a side ength of c 1 sn) where sn) = ogn)/n. It is shown [18] that each square-et contains Θogn)) nodes with probabiity 1. To avoid utipe access

3 3 interference, a Protoco Mode [30] is considered for successfu counication between nodes. A Tie Division Mutipe Access TDMA) schee is assued for the transission between the square-ets. With the assuption of Protoco Mode, if each square-et has a side ength of c 1 sn) for a constant c 1, then the square-ets with a distance of c 2 = 2+ c 1 square-ets apart can transit siutaneousy without significant interference [18] for a constant vaue. For our proactive routing anaysis, we assue that one of the nodes in each square-et is randoy chosen which is caed anchor node. This node coects a the inforation in the square-et and cobines the and reays the coded inforation to the next hop toward the requesting node. It ) is known that a iniu transission range of Θ ogn) n ensures network connectivity [11] in such a dense network. For the case of uncoded caching with proactive routing, [9] proposes a soution in which groups of neighboring squareets wi cooperate together to for a oca group. It is proved [9] that for a square oca group with side ength s g n) = Θ ) nm, any requested content is avaiabe in at east one node in the oca group. A routing protoco [18] within the oca group connects the source to destination through a series of horizonta and vertica square-ets. The paper assues that a the nodes inside a oca group have a goba knowedge of cached contents within each oca group. Such an assuption resuts in significant overhead which requires aocating part of network resources to the exchange of this inforation. Further, exchange of this inforation poses significant security threat and aows an eavesdropper to find out which contents are cached in the oca group. In this paper, we propose a decentraized coded caching approach based on rando vectors. In this technique, the contents are randoy and independenty cobined and stored in caches. When a node requests a content, a unique inear cobination of coded fies can reconstruct the requested content. Therefore, each node first cobines its encoded fies and then transits it to the anchor node yeow circes in Figure 1). The anchor node aso adds soe inforation fro its cache and forwards the newy updated fie to the next anchor node coser to the requesting node as shown in the ower eft oca group in Figure 1. This process continues unti the requesting node receives a the required coded fies for decoding as shown in Figure 2. For reactive routing approach, a rando direction in the network is seected. Using the cached inforation in this rando direction, the desired content can be obtained as shown in the upper eft corner of Figure 1. Interestingy, we prove seecting a rando direction is optia in ters of nuber of hops traversed to retrieve the content. In both cases, perfect counication secrecy is achievabe. B. Decentraized Coded Cache Paceent In our proposed coded cache paceent approach, a randoy encoded fie r i j is created and paced in the jth cache ocation of node i. This randoy encoded fie is a bit-wise N 1 N 2 N 0 N 3 N 4 N 5 N 10 N 9 N 8 N 6 N 7 Fig. 2: When a node N 0 requests a fie, it starts gathering the coded fies fro a the nodes in its oca group to construct the requested content. suation of rando contents fro F. In other words, r i j = =1 a i,j F = v i jf, 1) where F = [F 1 F 2... F ] T represents the contents vector and vj i = [a i,j 1 a i,j 2... a i,j ] T F 2 is a rando row vector with each eeent equa to 0 or 1 and the suation is carried over Gaois Fied GF2). This process is repeated independenty for a cache ocations of a nodes 4. Notice that based on this construction, each vector vj i is unifory seected fro the set of a vectors in F 2. Notice that this specific choice of fountain codes is known as Rando Linear Fountain RLF) codes [21] or Rando Unifor LT codes [26]. C. Content Reconstruction In order to decode any of the contents, nodes need to find ineary independent vectors vj i to span the -diensiona space. For each requested content F r, a unique cobination of these encoded fies wi generate F r. In both routing scenarios, the coputation of appropriate gains for the cobination of coded fies is carried by the requesting node and this inforation is reayed to the neighboring nodes. As depicted in Figure 2, node N i in the routing path can contribute up to M ineary independent row vectors v1, i v2, i..., vm i to span the entire space. Node N i appies gain b i j {0, 1} to its jth cached fie r i j and then adds in binary fied) M j=1 bi j ri j to the fie it has received fro previous hop and passes the newy constructed fie to the next hop toward requesting node N 0. After a tota of N r transissions, the requesting node receives N r M i=1 j=1 bi j vi j )F. It wi then appies decoding coefficients to its own cached fies and adds it to the received fie to reconstruct the desired content. Note that each reay node adds soe encoded fies to the received fie and reays it forward. The coefficients b i j {0, 1} are seected such that the inear cobination of encoded fies produce the desired requested content. D. Prior Resuts The coded caching was originay introduced in [12], [15] for ceuar networks. The foowing ea was proved in [12], [15]. 4 In practice, each node chooses M ineary independent rando vectors during cache paceent phase. However, to sipify the anaysis, we drop the independence assuption for different cache ocations of a node. Therefore, anaytica capacity resuts found in this paper are pessiistic.

4 4 Lea 1. Let vector vj i F 2 have equiprobabe eeents. The average nuber of vectors vj i to span the -diensiona space equas to 1 E unifor = + 2 i 1 = + c 3, 2) i=1 where c 3 asyptoticay approaches the Erdős Borwein constant ). Lea 1 shows that the average nuber of cached fies to reconstruct any content is equa to + c 3 which is very cose to for arge vaues of. This shows that rando unifor vectors perfor cose to optia in ters of iniizing the nuber of cache ocations to retrieve contents. Based on Lea 1, we have the foowing coroary. Coroary 1. If rando unifor vectors are used to create encoded fies and then these fies are independenty cached in node caches, then on average E[N] = + c 3 )/M = Θ/M) nodes are required to decode any requested content. Theore 1. In the proposed coded caching schee, seecting a rando direction is asyptoticay optia in ters of iniizing the nuber of hops required to retrieve a the contents. Proof. In the proposed coded caching schee, rando vectors in F 2 are used for encoding the contents. Lea 1 shows that on average + c 3 rando vectors are needed to span the -diensiona space of F 2. Therefore, with the proposed decentraized coded caching schee, on average E[N] = + c 3 )/M nodes are required to decode any requested content regardess of which routing direction is chosen for content retrieva. On the other hand, to be abe to retrieve a the contents, at east cache ocations are necessary. This eans that at east /M nodes are required for content retrieva in any caching schee. Since + c 3 is very cose to for arge vaues of, then Θ + c 3 )/M ) = Θ /M ) = Θ/M), this proves that seecting any rando direction is asyptoticay optia in ters of the iniu required nuber of hops for content retrieva. Notice that Theore 1 is intuitivey vaid since we have used copetey rando vectors and this eans that a of the contents are equay distributed in a directions. IV. CAPACITY This section is dedicated to coputation of network throughput capacity of decentraized coded and uncoded caching schees for proactive and reactive routing techniques. First, we define the precise notions of achievabe throughput and network throughput capacity as foows. Definition 1. A network throughput of λn) contents per second for each node is achievabe if there is a schee for scheduing transissions in the network, such that every content request by each node can be served by a rate of λn) contents per second. Whether a particuar network throughput is achievabe depends on the specific cache paceent and node ocations in the network. Since the ocation of the nodes and the cache paceent in nodes is rando, we wi define the network capacity as the axiu asyptotic network throughput achievabe with probabiity 1. Definition 2. We say that the throughput capacity of the network is ower bounded by Ωgn)) contents per second if a deterinistic constant c 5 > 0 exists such that i P[λn) = c 5gn) is achievabe ] = 1. 3) n We say thet the network throughput capacity is upper bounded by Ogn)) contents per second if a deterinistic constant c 6 < + exists such that i inf n P[λn) = c 6gn) is achievabe ] < 1. 4) We say that the network throughput capacity is of order Θgn)) contents per second if it is ower bounded by Ωgn)) and upper bounded by Ogn)). In this paper, we study the throughput capacity after the cache paceent phase and during the content deivery phase. In the foowing, we wi describe the necessary size of the oca group to decode a the contents in proactive routing approach. Reark 1. Coroary 1 suggests that for decentraized coded caching on average Θ/M) nodes are needed to decode any desired content. Since any oca group in proactive routing on average has Θns g n) 2 ) nodes, then the average oca group side ength of s g n) = Θ ) nm wi be enough to decode a the contents. Notice that siiar oca group side ength is found in [9] for the case of uncoded caching. A. Capacity of proactive routing approach In this section we assue that any node in each oca group is copetey aware of a the fies cached in its oca group. Nodes in oca groups are cooperating with each other to transfer a requested content. In uncoded caching scenario it has been proved [9] that if M < nm, a capacity of M/ ) Θ is achievabe. Our proposed decentraized coded caching approach achieves a capacity of Θ M/ ogn))) whie providing perfect secrecy as wi be proved subsequenty. Theore 2. In decentraized coded caching with a proactive routing schee the foowing network throughput capacity is achievabe ) M λn) = Θ. 5) ogn) Proof. Coroary 1 shows that on average E[N] = Θ/M) nodes are required to reconstruct a the contents in decentraized coded caching. Reark 1 shows that each oca group of side ength Θ ) nm contains this any nodes. As shown in Figure 1, a these nodes cooperate to transit the content to the requesting node. Since there are Θogn)) nodes in each square-et, the tota nuber of transissions for one

5 5 ) ) 2 sgn) request in a oca group is equa to Θ ogn) sn). Therefore, the tota nuber of fie transissions to satisfy a content requests in each oca group has the order of ) ) 2 sgn) Θ ogn) sn) M. On the other hand, in each oca ) 2 sgn) group we can have Θ sn) siutaneous transissions. This ipies that a network throughput of ) M λn) = Θ = Θ 6) ogn) ogn) is achievabe. ) 2 sgn) sn) sgn) sn) ) 2 M B. Capacity of reactive routing approach Reactive routing usuay requires ess overhead but incurs higher deays in content deivery. However, one of the advantages of our proposed coded caching is that we can seect any rando direction and decode the desired content with the sae optiu nuber of nodes. Such a scenario is depicted in the upper eft corner of Figure 1. For sipicity of our capacity anaysis, we assue that a contents are of equa size with each having Q bits. Assue that λn) is the axiu achievabe network throughput. This ipies that with a probabiity cose to one, the network can deiver nλn) contents per second. Therefore, with a probabiity cose to one, network nodes can transit nλn)e[n]q 1 bits per second. There are exacty c 2c 1sn)) square-ets at 2 any tie sot avaiabe for transission. Hence, the tota nuber of bits that the network is capabe of deivering is W equa to c 2c 1sn)) where W is the tota avaiabe bandwidth. 2 Therefore, ) W λn) = ne[n]qc 2 c 1 sn)) 2 = Θ 1, 7) E[N] og n where W and Q are the tota avaiabe bandwidth and tota nuber of bits for each content respectivey. This suggests that to find the axiu achievabe network throughput, it is enough to find the average nuber of transission hops needed to deiver the contents. Theore 3. In decentraized coded caching with reactive routing, the network capacity is ) M λn) = Θ. 8) ogn) Proof. Using Lea 1, on average E[N] = Θ M ) nodes or equivaenty hops) are required to decode a content in decentraized coded caching. This aong with equation 7) proves the theore. In decentraized uncoded caching strategy, nodes cache contents with unifor distribution. Lea 2 coputes the average nuber of hops to retrieve a content. Lea 2. If a content is requested independenty and unifory at rando fro a set of contents, then the average nuber of requests to have at east one copy of each content is equa to E[] = i=1 1 i = H = Θ og)). Proof. This is the we-known coupon coector probe [7]. Lea 3. If each node stores M different fies unifory at rando during cache paceent phase, then the average nuber of nodes required so that each fie is cached in at east one node is between d, M) = H d, M) and 1 + M 1 j=0 H d, M) where j. 9) Proof. This probe is an extension of the coupon coector probe in Lea 2 as the fies in each node s cache are different. To find the average nuber of nodes to have one copy of each fie in at east one node, we start fro a cassic coupon coector probe. Assue that fies are chosen unifory at rando and as soon as M different fies are found, they are paced in a node s cache. Then the sae process is started over for the next node and after finding M distinct fies, the fies are paced in its cache. Assue that this process is repeated unti one copy of each fie is cached in at east one node. This is a geoetric distribution probe, then on average d, M) fies are required to fi up one node s cache. Based on Lea 2, we wi have a copy of each fie in at east one node s cache after an average H fie requests. Hence, the average nuber of nodes required to have one copy of each fie in at east one node is between and 1 + H d, M). H d, M) Theore 4. If >> M, the capacity of the network using decentraized uncoded caching is equa to ) M λ = Θ. 10) og) ogn) Proof. Lea 3 shows that in case of uncoded caching the average nuber of nodes needed to satisfy a requests is upper and ower bounded as H d, M) E[N] 1 + H d, M). 11) Therefore, for arge vaues of, the average nuber of nodes for decoding scaes as ) H E[N] = Θ. 12) d, M) To find a bound for d, M), notice that the series in the right hand side of equation 9) has M ters and the axiu ter M+1 corresponds to the case when j = M 1 and the iniu ter 1 corresponds to the case when j = 0. A ower bound and an upper bound on d, M) can be found by using the iniu and axiu ters respectivey. Therefore, M d, M) M M )

6 6 When >> M, then the ower and upper bounds of equation 13) converge to the sae vaue of M. ) og) E[N] = Θ 14) M Cobining 14) and 7) proves the theore. Reark 2. Theores 3 and 4 show that decentraized coded caching strategy can iprove the network capacity by a factor of og) in reactive routing. Figure 3 copares the capacity of coded caching with uncoded caching for both proactive and reactive routing agoriths. To pot this figure, we assued that the nuber of contents, grows poynoiay with the nuber of nodes n. Coded caching provides perfect secrecy as wi be discussed ater whie it perfors better worse) than uncoded caching for reactive proactive) routing agorith. Capacity Theoretica coparison of capacity uncoded caching, proactive routing uncoded caching, reactive routing coded caching reactive/proactive) Fig. 3: Capacity for coded and uncoded caching using proactive and reactive routing agoriths. n V. SECURITY This section evauates the security of coded caching strategy. Note that uncoded caching aows an adversary with a noiseess wiretap channe to perfecty receive the transitted fies. We prove that such an eavesdropper wi not be abe to reduce its equivocation about the transitted fies in coded caching approach when there is a arge nuber of fies. Therefore, asyptotic perfect secrecy can be achieved. This probe was originay studied by Shannon [24]. Our secrecy proof is appicabe to both proactive and reactive routing schees. As described earier, each node cobines its encoded fies as x i = M j=1 bi j ri j = M j=1 bi j vi jf and adds it to the previousy received fie and forwards it to the next hop. Therefore, the aggregate received fie by the requesting node N 0 is S r = N r i=1 xi = N r M i=1 j=1 bi j ri j. We assue that the encoding vectors vj i of the neighboring nodes are transitted to the requesting node N 0 through a secure ow bandwidth channe. When enough nuber of such vectors are gathered, N 0 coputes the decoding coefficients b i j in order to generate the desired fie F r. Then, it sends back the b i j coefficients through the ow bandwidth secure channe to its neighboring nodes. The secure channe is used ony to transit the encoding and decoding inforation. Transitting the arge encoded fies through secure channe woud be undesirabe due to ow bandwidth constraint. However, the encoding vectors vj i and the decoding gains bi j have uch saer sizes for transission through secure channe. If content F r is used in the encoding of at east one of the coded cached fies in N 0 i.e. a 0,j r = 1 for soe 1 j M), then the requesting node N 0 can generate the fie M M M x 0 r = b 0 jr 0 j = b 0 jvj 0 F = F r + b 0 j vj 0 F 15) j=1 j=1 j=1 fro its own encoded cached fies. Note that v req = M 0 j=1 b j vj 0 is a coding vector in -diensiona space. Node N 0 ony needs to receive S r = v req F and add S r to x 0 r in GF2) to retrieve F r. The requesting node N 0 uses the secure channe to coect enough nuber of encoding vectors in order to span v req. Then the requesting node N 0 finds the appropriate decoding coefficients b i j to span v req and sends these decoding coefficients back to the neighboring nodes through the secure channe. The neighboring nodes coaborativey create the right decoding fie S r and transit it to N 0. The second possibiity which is ess ikey to happen for arge vaues of M is that none of the encoded fies in N 0 contains F r i.e. a 0,j r = 0 for a 1 j M). In that case, N 0 generates a unique cobination of its encoded fies as x 0 r = M j=1 b0 j v0 j F. In order to decode F r, node N 0 needs to receive S r = F r + x 0 r. Hence, it uses the secure channe to coect enough nuber of encoding vectors vj i to be abe to construct S r. After soving the inear equation in GF2), it sends the decoding gains back to the neighboring nodes such that they can coaborativey create the encoded fie S r = F r +x 0 r which aows N 0 to retrieve F r. We cai asyptotic perfect secrecy for this approach is achievabe as ong as for each requested fie F r, the requesting node generates a different encoded cobination x 0 r which acts siiar to key and hence, the transitted signa is in fact an encrypted version of the essage F r. The intended receiver is indeed capabe of decrypting the essage by adding its secret key x 0 r to it. Notice that for each requested fie F r, a different key x 0 r is generated using the encoded cached fies in the requesting nodes. No other eavesdropper wi be abe to decode the essage as they do not have the secret key x 0 r. The ain advantage of the egitiate receiver is the inforation stored in its cache which aows it to create a unique key x 0 r for each requested fie F r. This is siiar to Shannon cipher probe [24]. In [24], Shannon introduced the Shannon cipher syste in which an encoding function e : M K C is apping a essage M M and a key K K to a codeword C C. In our probe, for each requested content by a user, the egitiate receiver uses a unique key K to recover the essage M. Even when a different user requests the sae fie, it uses a different key because each user caches different encoded fies. The unique key for each user depends on the coded fies that the node is storing and the coded fies fro neighboring nodes used for decoding the requested fie. Shannon proved that if a coding schee for Shannon s cipher syste achieves

7 7 perfect secrecy, then HK) HM) where H.) denotes the entropy. He proved that at east one secret key bit shoud be used for each essage bit to achieve perfect secrecy. If the sizes of essages, keys and codewords are the sae, there are necessary and sufficient conditions [1] to obtain perfect secrecy presented in the foowing theore. Theore 5. If M = K = C, a coding schee achieves perfect secrecy if and ony if For each pair M, C) M C), there exists a unique key K K such that C = em, K). The key K is unifory distributed in K. Proof. The proof can be found in section 3.1 of [1]. We wi use Theore 5 to prove that our approach can achieve asyptotic perfect secrecy. In either of the cases, the requesting node N 0 receives a codeword 5 S r = F r + x 0 r fro the ast node adjacent to N 0. Node N 0 uses XOR operation to decode F r fro S r using it s secret key x 0 r. Therefore, we have a Shannon cipher syste in which M = F r, K = x 0 r, C = S r and e denotes the XOR operation. To use this theore, first we prove that for arge enough vaues of, the key x 0 r is unifory distributed. Lea 4. The asyptotic distribution of bits of coded fies in caches tend to unifor. Proof. We assue that a fies have Q bits and they ay have a distribution different fro unifor. We wi prove that each coded cache fie wi be unifory distributed for arge vaues of. Let us denote the k th bit of fie F by f k where 1 k Q and 1. Assue that Prf k = 1) = p k = 1 Prf k = 0). Further, we assue that the bits of fies f k) are independent. The k th bit of the coded fie in the j th cache ocation of node i can be represented as ri,j k = a i,j f k, 16) =1 where a i,j is a binary vaue with unifor distribution and independent of a other bits. Using reguar suation not over GF2)) and denoting h i,j,k a i,j f k, we define Hi,j,k =1 hi,j,k. Therefore, Pr[ri,j k = 0] = Pr[H i,j,k 2 0]. Therefore, the k th bit of the coded fie is equa to 0 if an even nuber of ters in H i,j,k is equa to 1. The probabiity distribution of H i,j,k can be coputed using probabiity generating functions. Since h i,j,k is a Bernoui rando variabe with probabiity 1 2 pk, its probabiity generating function is equa to G i,j,k z) = pk ) pk z. 17) Since a i,j and f k are independent rando variabes, h i,j,k wi becoe independent rando variabes. Therefore, the probabiity generating function of H i,j,k denoted by G i,j,k H z) is the product of a probabiity generating functions. G i,j,k H z) = pk ) + 1 ) 2 pk z. 18) =1 5 This is true for both scenarios because a the operations are in GF2). Denoting the probabiity distribution of H i,j,k as h.), the probabiity of H i,j,k being even is =1 2 2 Pr[H i,j,k 2 0] = h2u) = h2u)z 2u z=1 u=0 u=0 [ = 1 ] hu)z u + hu) z) u 2 u=0 u=0 z=1 = 1 2 Gi,j,k H 1) Gi,j,k H 1) = pk ) + 1 ) 2 pk = pk ) 1 ) 2 pk = 1 ) ) p k 2 Therefore, 1 ) ) i Pr[rk i,j = 0] = i p k 2 = =1 1 2 i ) 1 p k 1 = i 1 inf{p k } ) 1 = 2. =1 This proves the ea. =1 This ea paves the way to prove the foowing theore. Theore 6. The proposed coded caching strategy provides asyptotic perfect secrecy for the ast hop if is arge and < 2 M. Proof. To foruate this as a Shannon cipher probe, we assue that M = F r, K = x 0 r, and C = S r. The condition < 2 M ensures that a unique key exists for each requested essage since at ost 2 M possibe rando keys can be buit fro M cached fies. The encoding function is XOR operation. For any pair, C) M, C), a unique key K K exists such that C = + K which guarantees that M = K = C. Notice that the key K = x 0 r beongs to the set of a possibe bit strings with Q bits. Lea 4 proves that each coded cached content is unifory distributed aong a Q- bit strings. Hence each key which is a unique suation of cached encoded fies is unifory distributed aong the set of a Q-bit strings. In other words, regardess of the distribution of the bits in fies, x 0 r can be any bit string with equa probabiity for arge vaues of. Therefore, conditions of Theore 5 are et and asyptotic perfect secrecy is achieved. Reark 3. In this paper, we ony studied the security of ast hop counications in our approach and proved that even in the ost vunerabe ast) ink, secure counications is possibe. A ore genera security study for a inks reains as future work. Aso, the study of the security of this approach against cooperative eavesdroppers reains as future work. VI. CACHE HIT PROBABILITY This section is dedicated to coputation of cache hit probabiity when a node N 0 can access u other nodes N 1,..., N u or equivaenty, = um cache ocations. We copute the event that N 0 can decode any desired fie in the set F with this inforation. First, et s define the cache hit probabiity.

8 8 Definition 3. The cache hit probabiity for a contents is defined as the probabiity that any content can be retrieved using the cached inforation in nodes N 1,..., N u. We wi first study uncoded cache hit probabiity. A. Uncoded Caching In uncoded caching, each node is randoy choosing M different contents fro the set of contents. This can be odeed as a coupon coector probe with group drawings in which a coupon coector is coecting a nuber of different coupons in each tie and wants to find the probabiity that after u group coections, a the contents are coected. The resut studied before [10], [27] and is suarized beow. Theore 7. Assue that a coupon coector coects different coupons. Each tie a bunde of M different coupons are drawn unifory at rando. If u bundes are coected, then the probabiity that a the coupons are coected is equa to P a coected = M j=0 ) j )) u 1) j M j 19) M) Proof. The proof can be found in [27] a specia case of Theore 1) and aso in page 164 of [10]. Resut of Theore 7 is the cache hit probabiity for coecting a contents when uncoded caching is used. B. Coded caching In coded caching, we copute the probabiity that there exists ineary independent vectors within = um rando encoding vectors. If > = um, this probabiity is ceary zero. Therefore, without oss of generaity, we copute this probabiity when = um. This probe has been studied in iterature [16] and the resuts are suarized beow. Theore 8. Let 1 and s be positive integers and r =. If A = [a ij ] is an atrix whose eeents are independent binary unifor rando variabes and ρ ) is the rank of atrix A in GF2), then if we have P[ρ ) = s] 2 1 ss+r) 12 ) i i=s+1 r+s j=1 where the ast product equas 1 for s + r = j ) 1, 20) Proof. This is Theore in page 126 of [16]. Coroary 2. Let and A = [a ij ] be an atrix whose eeents are independent binary unifor rando variabes and ρ ) be the rank of atrix A in GF2). If, then P[ρ ) = ] 1 12 ) i. 21) i= +1 Equation 21) is the cache hit probabiity for coded caching approach. Reark 4. The cache hit probabiity for coded caching very quicky approaches 1 if is sighty arger than. In fact, there is a very sharp transitioning of the probabiity fro 0 to 1 in coded caching around the point =. However, in uncoded caching, shoud be uch arger than in order for the cache hit probabiity tends to 1 see Figure 5). Reark 5. This resut deonstrates that coded caching schee utiizes the cache space efficienty and avoids overcaching unike uncoded caching approach. VII. CACHE UPDATE ALGORITHM In this section, a caching update agorith is described. Let s assue a new content F new shoud repace another content F k based on soe repaceent poicy such as Least Recenty Used LRU) or Least Frequenty Used LFU) poicy. The network controer uses a bit scrabing technique to create a fie F new with unifor bit distribution fro F new. Such bit scrabing techniques are widey used in counication systes to give the transitted data usefu engineering properties [8]. Notice that the bit scrabing technique akes F new equivaent of teporary secret key with unifor distribution. The network controer then generates F k + F new and broadcasts this fie to the network nodes. When node N i receives F k + F new, it wi add F k + F new to a of its cached encoded fies r i j which contain F k, i.e., a r i j s for which ai,j k = 1. In other words, if in the j th ocation of node i we have r i j = F k + a i,j F, 22) =1 k then F k +F new wi be added to r i j. This repaces F k with F new in a encoded fies that contains F k. If soe cached encoded fies does not contain F k, then no action is required for those encoded fies. Nodes can then decode F new using the sae decoding gains as for F k without any additiona overhead. When F new is decoded, then a de-scrabing agorith can be used to recover F new fro F new. A pseudocode representation of our caching update protoco is shown in Agorith 1. Agorith 1 Cache Update Agorith 1: procedure CACHE UPDATE 2: Find the content F k that shoud be repaced. 3: Scrabe F new to get F new with unifor bits. 4: Encode the new content F new with F k as F new F k. 5: Broadcast F new F k to a the nodes. 6: for the j th cache ocation of node i do 7: if F k is used in encoding r i j i.e. ai,j k = 1) then 8: Update the j th cache ocation of node i with F new F k r i j. 9: end if 10: end for 11: end procedure

9 9 Average nuber of hops Average nuber of hops versus cache size Coded Caching North Direction) Coded Caching South Direction) Coded Caching West Direction) Coded Caching East Direction) Uncoded Caching Probabiity of decoding any desired content Probabiity of decoding versus the nuber of cached contents Coded caching, = 100, a M, Siuation) Coded caching, = 100, a M, Theory) Uncoded caching, = 100, Siuation) Uncoded caching, = 100, Theory) M=20 M=10 M=25 M= Cache size M) Fig. 4: Average required nuber of hops nodes) for coded and uncoded caching schees to retrieve a content. Notice that during the caching update phase none of the contents is transitted and any eavesdropper woud ony receive encoded version of the fies. Therefore, with this caching update technique, the contents can be updated securey. VIII. SIMULATION This section verifies the anaytica resuts derived earier via siuations. Figure 4 copares the average nuber of hops required to decode the contents in decentraized coded and uncoded caching schees. We consider a wireess ad hoc network with n = 1000 nodes and = 100 contents. The siuation resuts ceary deonstrate that decentraized coded caching outperfors uncoded case particuary when the cache size is sa which is the ost ikey operating regie. For instance, with decentraized coded content caching, a cache of size 25 ony requires ess than 5 hops whie decentraized uncoded content caching needs around 20 hops for successfu content retreiva. This akes coded content caching uch ore practica copared to uncoded content caching. Note that capacity is inversey proportiona to the average hop counts. As can be seen fro Figure 4, for sa cache sizes, coded caching significanty reduces the nuber of hops required to decode the contents. This property is iportant for nodes with sa storage capabiity since arge nuber of hops can ipose excessive deay and ow quaity of service. Figure 4 proves another iportant resut that content retrieva can be optiay done in any rando direction. In this pot, we have used rando directions of east, west, south and north for content retrieva using coded caching and shown that the average nuber of hops in any of these rando directions is the sae. The four pots corresponding to these four directions is so cose that it is hard to distinguish the. Figure 5 shows the siuation resuts for different vaues of M when = 100. The cache hit probabiity is potted as a function of the nuber of cached contents and for different cache sizes M. However, for each fixed vaue of, the nuber of nodes µ = M wi be different depending on M. For instance, at = 400, M and µ are 25 and 16 respectivey. The siuation resuts are vaidating the theoretica resuts in Theore 7 and Coroary 2. As can be seen fro this figure, the cache hit probabiity for coded caching is uch higher Nuber of cached contents Fig. 5: Cache hit probabiity for any desired content when = 100. than that of uncoded caching. Aso, it is cear fro this pot that the cache hit probabiity of coded caching approaches 1 rapidy when starts to be greater than. For uncoded caching, speciay when is arge, the receiver shoud access a uch arger nuber of cached contents in order for the cache hit probabiity to approach 1. Therefore, in networks with a arge nuber of contents our coded caching approach wi quicky achieve a cache hit probabiity cose to one with a uch saer nuber of cache ocations. This is a significant benefit of our technique in reducing overcaching. IX. CONCLUSIONS This paper introduces a nove decentraized coded caching strategy in wireess ad hoc networks. The capacity of this approach is copared with that of uncoded caching for proactive and reactive routing protocos. Whie with proactive routing protoco, uncoded caching outperfors coded caching, coded caching perfors better with reactive routing protoco. Interestingy, it was shown that by choosing any rando direction, cose to optiu nuber of hops can be obtained to retrieve any content in coded caching. It has been shown that coded caching approach provides asyptotic perfect secrecy during fie transission. We have aso studied the cache hit probe and shown that the cache hit probabiity for any desired content wi be significanty higher in the proposed technique copared to uncoded caching. This technique reduces the probe of overcaching in the networks. An efficient and secure cache update agorith is aso proposed and the resuts are vaidated with siuations. The paper considered a static network, however our resuts can be extended to obie or vehicuar networks when the transission tie is uch saer than the network dynaics. Further, the size of the fie chunks can be adjusted based on the network dynaics to adapt to the rate of change in the network evoution. This paper considered rando inear fountain codes to achieve asyptotic perfect secrecy and better cache hit probabiity and capacity resuts. Other fountain coding choices for cache paceent ay reduce the decoding copexity whie reducing the capacity and cache hit probabiity. Seection of

10 10 appropriate fountain code depends on any factors such as decoding copexity, deay and capacity requireents. REFERENCES [1] Matthieu Boch and Joao Barros. Physica-ayer security: fro inforation theory to security engineering. Cabridge University Press, [2] Ning Cai and Rayond W Yeung. Secure network coding. In Proceedings of IEEE Internationa Syposiu on Inforation Theory, ISIT, page 323. IEEE, [3] Ning Cao, Shucheng Yu, Zhenyu Yang, Wenjing Lou, and Y. Thoas Hou. LT codes-based secure and reiabe coud storage service. In Proceedings of the IEEE INFOCOM 2012, Orando, FL, USA, March 25-30, pages , [4] Aexandros G. Diakis, Brighten Godfrey, Yunnan Wu, Martin J. Wainwright, and Kannan Rachandran. Network coding for distributed storage systes. IEEE Trans. Inforation Theory, 569): , [5] Aexandros G. Diakis, Vinod M. Prabhakaran, and Kannan Rachandran. Distributed fountain codes for networked storage. 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Foundations and Trends in Networking, 12), Mohsen Karizadeh Kiskani received his B.S. degree in Mechanica Engineering and M.S. degree in Eectrica Engineering fro Sharif University of Technoogy in 2008 and 2010, respectivey. He is currenty a Ph.D. candidate in Eectrica Engineering departent at University of Caifornia, Santa Cruz. His research interests incude wireess counications, inforation theory, and the appication of fountain codes, LT codes and Raptor codes in wireess networks and storage systes. He is aso interested in copexity study of Constraint Satisfaction Probes CSP) in Coputer Science. In June 2016 he obtained a M.S. degree in Coputer Science fro University of Caifornia Santa Cruz. Haid Sadjadpour S 94 M 95 SM 00) received Ph.D. degree in Eectrica Engineering fro University of Southern Caifornia at Los Angees, CA. In 1995, he joined AT&T Research Laboratory in Forha Park, NJ as a Technica Staff Meber and ater as a Principa Meber of Technica Staff. In 2001, he joined University of Caifornia at Santa Cruz, CA, where he is currenty a Professor. His research interests are in the genera areas of wireess counications and networks.