Straggling for Covert Message Passing on Complete Graphs
Pei Peng, Nikolas Melissaris, Emina Soljanin, Bill Lee, Huafeng Fan, Anton Maliev
SStraggling for Covert Message Passing on Complete Graphs
Pei Peng , Nikolas Melissaris , Emina Soljanin , Bill Lee, Anton Maliev, and Huafeng Fan Abstract — We introduce a model for mobile, multi-agent in-formation transfer that increases the communication covertnessthrough a protocol which also increases the information transferdelay. Covertness is achieved in the presence of a warden who has the ability to patrol the communication channels.Furthermore we show how two forms of redundancy can beused as an effective tool to control the tradeoff between thecovertness and the delay.
I. I
NTRODUCTION
In numerous circumstances, the very act of communicatingneeds to be hidden. In war time, not only the content ofmessages, but also the volume of communication to or fromsuspected parties can alert the adversary. In everyday life,revealing the identity of communicating parties, not onlythe exchanged information, affects the increasingly importantanonymity and privacy. An information theoretic approach toachieving covertness, adopted in several recent papers (seee.g. [1]–[3] and references therein), roughly speaking, relieson camouflaging messages as noise. We here propose covertmessage passing in a wireless mobile network environmentthat is complementary to and can be used in conjunction withthe previously proposed methods.The last decade has seen a wide variety of novel networkedsystems, with a growing trend towards distributed and mul-tiuser applications. Future 5G systems are supposed to hosthundred times more devices than current 4G networks, andone can potentially harness the resources expected to bebrought in by smart (everyday or battlefield) devices in thefuture Internet of (Battlefield) Things (Io β T) environments.The precise mathematical description of our problem willbe presented in the following section; we next give two highlevel examples. Consider a scenario where an agent Alicehas a message she wants to covertly send to her partnerBob who resides in the same (possibly occupied) city. Werepresent city streets as a graph. Information gathering anddissemination on graphs is a very interesting problem thatarises naturally and has recently seen active research in manydifferent contexts. Examples include: border control usingUAVs [4], measuring traffic, reporting road conditions andhelping with emergency response using UAVs [5], monitor-ing the ocean [6], measuring air pollution [7], and morerecently for timely exchange of information updates [8].The area over which Alice and Bob plan to communicatehosts a multitude of IoT objects capable of storing, sendingand receiving data. Alice can use some of these smart objects The authors are with the Department of Electrical andComputer Engineering, Rutgers University, the State University ofNew Jersey, USA { pei.peng,nikolas.melissaris,emina.soljanin } @rutgers.edu to store her data, and Bob can subsequently retrieve thestored data. The agents have one or both of the followingconcerns: 1) they do not want to be detected communicatingwith any of the relay objects and 2) the system shouldbe trustworthy even if some of the objects e.g., have lostpower, are unwilling to relay messages, or are otherwisecompromised in an adverse environment. Because of that,Alice decides to split her data into small chunks which shecan inconspicuously pass, one at the time, to relays (helpers)that appear in her proximity as she randomly moves throughthe area. Bob, who also randomly moves through the area,can then retrieve the stored data chunks. We assume thatthere is a warden Willie patrolling the city. Because theIoT objects are distributed in a wide area, Willie can onlyperiodically check if any of these objects is transmitting orreceiving data.Having to distribute and collect many chunks, as well asthe unpredictability of mobility and availability of helperscan cause large delays in such mobile information transfer.To increase persistence of information in the erratic envi-ronment, the agent may decide to make the data chunksredundant by using erasure correcting codes, which requiresthat more data chunks be distributed and collected. Onewould expect that that would further increase the delay inmobile information transfer. However, that is not necessarilythe case, and we will see that coding and some other formsof redundancy can in fact be used to reduce the delay, aspreviously shown to be the case in data download [9].There is another analogy for our communications scenario:Alice and Bob are at a party and Alice wants to send amessage to Bob without actually talking to him. Willie, whois also at the party, wants to detect whether Alice and Bobare communicating. Willie could easily detect if Alice talksdirectly with Bob, so Alice decides to divide the messageinto several pieces and whispers each piece to a differentmutual friend she has with Bob. Bob can then walk aroundin the party and collect these message pieces from the friendsto retrieve the message. Since Willie is unable to keep trackof all the participants in the party, Alice may be able sendthe message successfully to Bob without being detected.If the size of the message that Alice wants to send toBob is fixed, then the delay and the covertness of themessage transfer will depend on the number of chunks thatthe message is split into and the amount of redundancy(code rate) that is introduced. We here derive the expressionsthat describe this dependence under certain communicationmodel. The paper is organized as follows: In Sec. II, wepresent the system model and point out multiple tradoffs thatexist between the time to disseminate data, time to collect a r X i v : . [ c s . CR ] S e p ata, and the probability of covertness. In Sec. IV, we deriveexpressions for the covertness probability and dissemina-tion/collection delays. In Sec. VI, we present some numericalresults and analyze the maximum covertness probability andminimum delay by simulation.II. S YSTEM M ODEL
A. Communications Protocol & the Mobility Model
There are four (types of) participants in our communi-cation model: a source (Alice), r relays, a collector (Bob),and a warden (Willie). Alice and Bob walk randomly ona complete graph of s vertices. The r relays are placedon r different vertices of this graph. Alice has a messageof length m which she divides into k ≤ m chunks oflength (cid:96) = m/k , and then encodes the data chunks into n ≤ r coded chunks (of length still (cid:96) ). She transfers the n coded chunks to the first n relays she encounters asshe randomly walks through the graph. The collector Bobretrieves the message by collecting k chunks from the first k relays with data chunks he encounters as he randomlywalks through the graph. Note that we assumed without lossof generality that the encoding is such that any k out of n chunks are sufficient for message recovery. We assumea complete graph for our communication graph because itsimplifies the analysis and drives our point home withoutgetting into messy calculations. Other communication graphscan be used as well in future research, especially large regulargraphs or graphs with a few high degree hubs.Notice how the tradeoff between covertness and delaybecomes apparent here. On one hand, if Alice sets n = 1 ,meaning that she delivers the entire message in one chunk,then the probability of detection is small. This happensbecause the chance that Willie “sees” her is inversely pro-portional to the number of nodes. On the other hand, delayis increased because Bob will have to visit many nodes untilhe meets the relay that holds Alice’s message.At the other side of the spectrum if Alice splits themessage in many chunks, then it will take Bob less steps toretrieve it (since the probability of meeting a relay that holdsa message is high) but the probability that they are caughtincreases significantly because of the longer time that theyhave to spend walking on the graph. B. Chunk Transmission
We assume that the transmission time of a chunk (betweenAlice and a relay or Bob and a relay) consists of twoadditive components. The first is proportional to the chunklength (cid:96) , and the other is an independent random variablethat accounts for various disturbances (noise) in the system.We will assume that this random variable is exponentialwith parameter λ , and thus the transmission time is shiftedexponential with the shift value (cid:96) and the tail parameter λ . C. Warden Models
We now introduce our very simple warden model. Webase covertness on the assumption that the warden Willieis able to monitor part of the vertices for some given time. A very easy and informal way to visualize this is to imaginethat the warden is stationed somewhere “in the middle” ofthe graph , on top of a lighthouse. This way, he can onlycheck the parts where the lighthouse sheds its light andcan’t see what is happening behind him. We assume thatno transmission covertness scheme is implemented, and thusif the warden happens to check a vertex of the graph whilethe chunk transmission is taking place, he will detect it withprobability 1. We further assume that when the transmissionstarts, the warden’s (monitoring) time follows a uniformdistribution U (0 , W ) . The information transfer stays covertiff the transmission of each chunk stays undetected. Thereare multiple warden models that can be considered in futureresearch. In this work, we observe two simple ones in orderto point out the tradeoff between covertness and delay thatour protocol manages to balance. The main idea separatingthe models is whether Willie can learn something aboutwhere meaningful transmission takes place just by observinghow Alice and Bob walk along the graph. More details aremade explicit in sections V-A and V-B.III. P ERFORMANCE M ETRICS
Our performance metrics of interest are covertness proba-bility , the dissemination time of the n coded chunks by Alice,the collection time of k coded chunks by Bob, and the totaldata transfer time (dissemination plus collection). We willsee, in the following sections, that each performance metricis optimized by a different set of system parameter values. A. Dissemination/Collection Time
We can consider the dissemination/collection time as acoupon collector’s problem: there are n stores at city squares,each one selling a different coupon, and there is a directroad between any two squares. A coupon collector walksrandomly in the city and want to buy j different coupons.If j = 1 , it is obvious the collector needs to visit only square get a coupon; If j = 2 , he needs to visit on average nn − squares; If j = 3 , the average visited squares is nn − + nn − . Following this pattern, we know that when j = n , the collector needs to visit on average nH n squares,where H n = (cid:80) ni =1 /i is the n -th harmonic number. Duringeach visiting, if he spends on average T ave in the store, thetotal average time is nT ave H n . The dissemination time andcollection time can be calculated in a similar way. B. Covertness Probability
The covertness probability is defined as the probability thatAlice transmits a message to Bob without being detectedby Willie. For example, assuming the message has datachunks, then Alice needs to transmit times to relays andBob also needs times to collect the chunks. If during eachtime, Willie will detect the transmission with a probability P d , then the covertness probability is P c = (1 − P d ) . Noticethat when Willie detects the transmission, it doesn’t means This is a very informal statement. There is no need for an exact “middle”,we just need the warden to be at a place where he can observe differentparts of the network at different times. e will get the content of the message. The message may becamouflaged as noise to avoid detection. But this is anothercovert communication problem, and won’t be studied in thispaper. IV. T
HEORETICAL P ERFORMANCE A NALYSIS
System Parameters s - number of graph vertices r - number of relays m - length of the message in bits k - number of message (data) chunks n - number of encoded chunks (cid:96) - length of the chunk in bits A. Covertness Probability
The detection probability of each single chunk transmis-sion is given by the following theorem:
Theorem 1.
If the transmission time between the source (col-lector) and a relay follows a shifted exponential distribution λe − λ ( t − (cid:96) ) and the Willie’s monitoring arrival time at thevertex has a uniform PDF U (0 , W ) , then the probabilitythat Willie arrives during the transmission (i.e., detects thetransmission) is P d = (cid:40) λW + mkW − e − λ ( W − m/k ) λW for W ≥ (cid:96) for W < (cid:96)
Proof: If W < (cid:96) = m/k , then Willie will definitely arrivebefore the transmission is complete, and thus the detectionprobability is P d = 1 .If W > (cid:96) , we calculate the detection probability as follows: P d = P ( t tr ≥ t ar ) = (cid:90) ∞ (cid:90) ∞ t ar f tr ( t tr ) f ar ( t ar ) dt tr dt ar = (cid:90) ∞ mk (cid:90) ∞ t ar f tr ( t tr ) dt tr f ar ( t ar ) dt ar + (cid:90) mk (cid:90) ∞ mk f tr ( t tr ) dt tr f ar ( t ar ) dt ar = (cid:90) ∞ mk e − λ ( t − m/k ) f ar ( t ar ) dt ar + mkW = (cid:90) W mk W e − λ ( t − m/k ) dt ar + mkW = 1 λW + mkW − e − λ ( W − m/k ) λW The entire message transmission will be undetected if eachchunk transmission is undetected. Recall that Alice needs todisseminate the n chunks and Bob needs to collect k chunks.Therefore the total number of chunk transmissions is n + k ,and thus the overall covertness probability is P c = (1 − P d ) n + k (1)From Theorem 1 and equation (1), we can see that when n increases, the covertness probability decreases, which meansthat coding redundancy hurts the covertness. However, it isless clear how changing the number of chunks k affects thecovertness probability, as we discuss below. V. C OMMUNICATION D ELAY
The message transfer from Alice to Bob consists threestages: 1. the meeting time steps: the source and collectorneed to meet enough relays; 2. The discovering time: thesource and collector need to find out if there is a relayon the same vertex; 3. The transmission time: the dissemi-nation/collection time between the source/collector and therelay. As we know, the transmission time relates to the sizeof the data chunks. It’s reasonable to assume that the largerdata chunk needs to spend a longer transmission time.To calculate the communication delay in covert com-munication, we also need to consider the Willie. Basedon different Willies’s detection patterns, we provide twocommunication delay models.For the first model, if Willie can’t observe the movementsof Alice and Bob, then he can not simply detect the transmis-sion by comparing the time Alice/Bob spends on each vertex.Therefore, we assume that during the source’s random walkwe can have two cases: Either the source does not meet arelay and the discovering time is or the source meets arelay on a vertex and we have a transmission time t tr .As described in Sec. IV-A, we model the transmission time t tr as a shifted exponential random variable with the tailparameter λ and the shift parameter m/k .For the second model, if Willie can observe the movementsof Alice and Bob, he may learn which nodes they arestopping for longer times at, and conclude that they holdvaluable information. To prevent this, one idea is to spendan equal length of time at each visited node, and eventransmitting an empty signal with no valuable information onnon-relay vertices. This way, even if Willie can track Alice’sand Bob’s movements, he does not gain any additionalinformation about their communication. Now, every nodevisited by Alice or Bob takes m/k units of time. A. Model 11) Dissemination Time:
In the dissemination stage, thesource Alice needs to disseminate n chunks to r relays.When Alice randomly walks on a complete graph with s vertices, the probability that she meets a relay is r/s . AfterAlice deposits the first data chunk in one of the r relays,the second chunk can only be stored in one of the remaining r − relays. The probability of meeting an occupied relaydecreases as the number of occupied relays grows. Therefore,in order to get the total dissemination time T dis , we need tofind dissemination time T i where i = { , , . . . , n } for eachencoded chunk of the message. Lemma 1.
The dissemination time T i to transmit the i th data chunk to any one of r − i + 1 relays is T i = t tr + 1 + a for a probability (1 − p r − i +1 ) a p r − i +1 (2) Where a ∈ { , , , ... } is the number of steps the sourcespent to meet a relay, p r − i +1 = r − i +1 s .And then the expectation of T i is E [ T i ] = 1 λ + mk + 1 p r − i +1 (3) roof: Since for time T i , the value of p r − i +1 is a constant.Let’s assume p = p r − i +1 . E [ T i ] = ∞ (cid:88) a =0 E [ t tr + 1 + a ] (1 − p ) a p = p ( E [ t tr ] + 1) ∞ (cid:88) a =0 (1 − p ) a + p (1 − p ) ∞ (cid:88) a =1 a (1 − p ) a − = E [ t tr ] + 1 p = 1 λ + mk + 1 p r − i +1 Lemma 2.
The total dissemination time T dis is that thesource transmits all n chunks to any n out of r relays. Thenthe expectation of time T dis is E [ T dis ] = nλ + nmk + s ( H r − H r − n ) (4) Proof: E [ T dis ] = n (cid:88) i =1 T i = n E [ t tr ] + n (cid:88) i =1 p r − i +1 = n E [ t tr ] + n (cid:88) i =1 sr − i + 1= nλ + nmk + s ( H r − H r − n ) It’s easy to see that n reaches optimal at n = k , whichmeans the less redundancy, the lower dissemination delay.
2) Collection Time:
During the collection step, the col-lector needs to collect any k out of n data chunks torecover the message. Since the transmission time betweenthe collector and relay follows the same distribution as thetime in dissemination step. Then as in Lemma 2, we can findthe total expectation of collection time T col in Lemma 3. Lemma 3.
The total collection time T col is the time spentby the collector to retrieve any k chunks from n relays. Thenthe expectation of time T col is E [ T col ] = kλ + m + s ( H n − H n − k ) From Lemma 3, E [ T col ] is decreasing when n is increas-ing.
3) Joint time:
After getting the dissemination time andcollection time, the joint time is just the sum of these twotimes.
Theorem 2.
Now we can easily calculate the expectation ofthe joint time: E [ T tot ] = n + kλ + (cid:16) nk + 1 (cid:17) m + s ( H r + H n − H r − n − H n − k ) From the conclusions in dissemination and collectionsteps, we can see that there must be an optimal n whichminimize the E [ T tot ] . To calculate the optimal n is compli-cated, so we will show result by simulation in next section. B. Model 21) Dissemination time:
Lemma 4.
The total dissemination time T dis is the timeneeded to transmit n code blocks to any n out of r relays.Then the expected value of T dis is: E [ T dis ] = (cid:18) λ + mk + 1 (cid:19) s ( H r − H r − n ) (5) Proof:
We follow the same reasoning as for the originalmodel. The expected number of nodes visited can be com-puted by modeling the relay traversal as coupon-collecting:vising n out of a given r nodes out of s total nodes. Theexpected number of visits needed is s ( H r − H r − n ) . At everynode visited, 1 unit of time is needed to check if the nodeis a relay, and the transmission takes an expected time of /λ + m/k .
2) Collection time:
Lemma 5.
The total collection time T col is the time neededto collect k code blocks from any k out of n relays. Thenthe expected value of T col is: E [ T col ] = (cid:18) λ + mk + 1 (cid:19) s ( H n − H n − k ) (6)This result is obtained in the same manner as the dissem-ination time, except that Bob needs to visit any k out of n with data on a graph with s nodes.
3) Joint Time:
Theorem 3.
The expected value of the total time needed fordissemination and collection is E [ T tol ] = (cid:18) λ + mk + 1 (cid:19) s ( H r + H n − H r − n − H n − k ) For this model, we next derive an expression for theoptimal value of n that minimizes the total transmission time,and omit the simulations.For given k , λ , m , s and r , we denote A n = H n − H r − n − H n − k . Then we have A n − A n +1 = − n +1 − r − n + n +1 − k .If A n − A n +1 < , we have − n + 1 − r − n + 1 n + 1 − k < ⇔ n + 2 n + 1 − rk − k > ⇔ n > √ rk + k − Similarly, if A n − A n +1 > , then n < √ rk + k − . Thuswe can optimize the total delay by producing (cid:6) √ rk + k − (cid:7) or (cid:4) √ rk + k − (cid:5) code blocks.VI. N UMERICAL AND S IMULATION A NALYSIS
A. Numerical Covertness Probability Analysis
Since it is easy to see that the covertness probabilityincreases with the redundancy n , we only discuss howthe covertness probability changes with k . We consider anexample where the distribution of transmission time is shiftedexponential with the tail parameter and shift parameter ig. 1. The covertness probability vs. k for 5 different values of n . Thecovertness probability increases rapidly with k when k is small. Then thebenefits of increasing k get smaller and even negative. /k . The Willie’s arrival time follows uniform distribution U (0 , .Figure 1 plots the covertness probability vs. k for 5different values of n for this example. We see from the figurethat, when n is small (e.g. n = 3 ), the covertness probabilityrapidly increases with k , and it is better to select k as largeas possible. As n is increasing (cf. n = 10 ), the covertnessprobability essentially stops increasing with k after somepoint (around k = 6 ). At that point, there is no need toincrease k . When n is large (e.g. n = 15 ), the covertnessprobability decreases with k after some point ( k = 12 inthe figure). Thus there is an optimal k . Also we can seewhen n = 1 , which means no redundancy is introduced,the covertness probability reaches the maximum. Howeverwhen n = 2 or n = 3 Since the redundancy may providemay reduce the communication delay between the source andcollector, it is worth to select a lower covertness probabilityscenario.We also notice that the covertness probability in the figureis very low. It is because of the simulation parameters’ valueswe selected. If we change the values, e.g. the Willies’s arrivaltime follows uniform distribution U (0 , , the covertnessprobability will increase significantly. Since our covert com-munication model is a new model, we don’t know the exactparameters’ values in practice. Besides, in this simulation,we want to study how covertness probability changes withthe number of message chunks. Therefore the changing ofcovertness probability is more important than its values. B. Minimum Delay Analysis by Simulation
We simulated our message passing protocol on a completegraph with 50 vertices. Again, the transmission time followsshifted exponential distribution with scale parameter andshift parameter /k . We are interested in seeing how theredundancy parameter n affects the average message transfertime for 1) different values of r and 2) different values of Fig. 2. The average time changes with redundancy n under different r .The number of message chunks k = 3 . Both simulation and theoreticalresults are provided. k . We obtained the results by simulation and also computedthe corresponding theoretical results by using Theorem 2.
1) Delay vs. n for Different Values of r : Figure 2 showsboth the simulated and the theoretical results, which closelymatch each other. From the figure, we can draw severalconclusions. First, when we consider each curve in the figure,we can see that introducing proper redundancy can signifi-cantly reduce the average time, but too much redundancy canhurt the performance. The optimal redundancy is reachedat n = 5 . Second, when we compare all the simulationresults or theoretical results, increasing the number of relays r can reduce the average time. This conclusion is not hardto understand. As we know the optimal redundancy is thetrade-off between the dissemination time and collection time.When we have more relays, the dissemination time willget reduced and the collection time will not be affected.Then we may increase dissemination time to reduce thecollection time. This can be easily derived from the equationin Theorem 2.
2) Delay vs. n for Different Values of k : Figure 3 showsboth the simulated and the theoretical results, which closelymatch each other. From Fig. 3, we can draw similar con-clusions as from Fig. 2. Note that k = 1 , n = 2 result inthe minimum average time. Therefore, splitting the messageintroduces delay.Figure 4 shows the results of the second communicationdelay model. Comparing with figure 3, more redundancyshould be introduced to get a lower average time, and when k = 3 and n = 5 the average time reaches the minimum.Now we can compare the figures from both covertnessprobability and minimum delay. If we compare figure 1 and3, we can find that when n = 1 and k = 1 , which means themessage is not divided and no redundancy is introduced, thecovertness communication can get the maximum covertnessprobability and a small enough average time. However itdoesn’t mean that the redundancy and chunk transmission ig. 3. The average time changes with redundancy n under different k .The number of relays r = 10 . Both simulation and theoretical results areprovided.Fig. 4. The average time changes with redundancy n under different k .The number of relays is r = 10 . Both simulation and theoretical results areprovided. are useless. In fact, in the covertness probability simulation,we assume the warden’s arrival time follows U (0 , andthe message length is m = 10 , which shows W = 50 isalways larger than l . However if the warden arrives morefrequently, for example the arrival time follows U (0 , , then W = 8 is smaller than l when k = 1 . In this case, when n = 1 and k = 1 , the covertness probability is . To get theoverall optimal values of covertness probability and delay,we must introduce some redundancy and divide the messageinto more data chunks.If we simultaneously consider both Fig. 1 and Fig. 4, wesee that for n = 1 and k = 1 the probability of covertness ishigh, but so is the expected delay. On the other hand, when n = 5 and k = 5 , the average time is low, but there is aprice to pay in the covertness probability. VII. C ONCLUSIONS AND F UTURE W ORK
We introduced a model for covert message exchange inmobile multi-agent environments. There are four (types of)participants in our communication model: a source (Alice), r relays, a collector (Bob), and a warden (Willie). Themodel stipulates the participants’ mobility patterns and thecommunications protocols, and defines performance metricsto be the probability of maintaining covertness and thetotal message transfer time. We obtained expressions forthese performance metrics by theoretical derivations and/orsimulation, and showed the tradeoff between them as afunction of the system parameters.Many more models for such systems seem reasonable, buthave not been studied yet. Future directions could includethe exploration of many different ways that the wardencan operate but also many different mobility patterns. Forexample, Willie might need to spend a certain amount of timebefore being able to detect a transmission at a node, or hecould also be adaptive, meaning that he can over time learnwhich nodes do not have relays and systematically avoidchecking them. Regarding, the mobility patterns, randomwalks on regular graphs or some other area traversal modelsmay be more practical.A CKNOWLEDGMENTS
This research is based upon work supported by the Na-tional Science Foundation under Grant No. SaTC-1816404.R
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