Capacity Limit of Queueing Timing Channel in Shared FCFS Schedulers AmirEmad Ghassami

Xun Gong

Negar Kiyavash

ECE Department Coordinated Science Laboratory University of Illinois at Urbana-Champaign [email protected]

Google Inc. [email protected]

ECE and ISE Departments Coordinated Science Laboratory University of Illinois at Urbana-Champaign [email protected]

Abstract—The capacity of a queueing timing channel in which a user modulates messages to another user via his pattern of access to a shared resource scheduled in an FCFS manner is calculated. One example of such a channel is the cross-Virtual Network (VN) covert channel in data center networks. In data center networks, software-defined-networks generate logically isolated virtual networks, across which direct data exchange is impossible. However, since packet flows belonging to different VNs inevitably share underlying network infrastructure, it is possible to transfer data across VNs through timing channels resulting from the queueing effects of the shared resource. Index Terms—Queueing Timing Channel, Capacity Limit, FCFS Scheduler.

Shared Scheduler

Transmitter Side Receiver Side

I. I NTRODUCTION Timing channels are mediums in which information is conveyed through inter-timing of events (for instance, packets). Such channels have been studied in the context of communication with queueing disturbance [1], [2]. Traditionally, timing channels are synonymous with covert channels, wherein parties communicate despite not being allowed to do so. In the context of packet networks, there is a large body of literature on this type of channels [3], [4], [5], [6], [7]. More recently, timing channels have been exploited to implement side channel attacks. In [8], Gong et al. proposed an attack where a remote attacker learns about a legitimate user’s browser activity by sampling the queue sizes in the downstream buffer of the user’s DSL link. In these types of queueing timing side channels, information is inevitably conveyed through delays experienced by the users of a shared resources, and hence the leakage depends on the scheduling policy. The information leakage of a queueing side channel in an FCFS scheduler is analyzed in [9]. The analysis for more general work conserving policies has been done in [10]. In this paper, we study a cooperative version of a queueing timing channel in which a user accesses the shared resource in a manner that allows him to modulate messages to another user with whom he shares the resource. One example is the crossVirtual Network covert channel in data center networks. In data center networks, software-defined-networks are frequently used for load balancing [11], which generates logically isolated virtual networks, preventing direct data exchange. However, since packet flows belonging to different VNs inevitably share underlying network infrastructure, such as a router or

978-1-4673-7704-1/15/$31.00 ©2015 IEEE

Fig. 1: System model

a physical link, it is possible to transfer data across VNs through timing channels resulting from the queueing effects of the shared resource. We consider the architecture depicted in Fig. 1 where a scheduler serves packets from two users: U1 and U2 . In this system, user 1’s goal is to send a message to user 2. However, there is no direct channel between user 1 and 2, i.e., the transmitter side and the receiver side of the system (the upper side (in green) and the lower sides (in blue) in Fig. 1). Physically, UiR is the node which receives UiT ’s packet stream for i ∈ {1, 2}. Despite this, a timing channel is created between the two users via the delays experienced by user 2 through the coupling of its traffic with user 1’s traffic due to the shared scheduler. More precisely, since U1T and U2T ’s packets share the same queue, U1T can encode messages in the arrival times of its packets, which are passed onto U2R via queueing delays. Therefore, effectively, the nodes U1T and U1R are on the transmitter side and the nodes U2T and U2R are on the receiver side of the channel of our interest (Fig. 1). In the next section, we explain the system model in more detail. The capacity of the introduced channel is calculated in Section III. Finally, our concluding remarks are presented in Section IV.

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ISIT 2015

II. S YSTEM M ODEL We consider an FCFS scheduler, commonly used in DSL routers, serving packets from two users. Time is discretized into slots, and the scheduler is able to process a single packet in each time slot. We assume both users are allowed to issue packets of unit size per slot or stay idle. If the server was not limited to serve one packet per slot, then the following simple scheme would achieve the highest possible rate, that is, a capacity of 1 bit per time slot: U2T simply sends a packet in every time slot. Sending a packet by U1T in any time slot would then cause a delay in the arrival of the simultaneous packet of the second user and hence could be detected. Therefore, in each time slot, U1T could simply idle to signal a bit ‘0’ or send a packet to signal a bit ‘1’, resulting in the capacity of 1 bit per time slot. Of course, this scheme is not feasible in practice as it would destabilize the queue and result in severe packet drops. Thus, we analyze the capacity for coding schemes satisfying queueing stability; i.e., the total packet arrival rate does not exceed the service rate (which is one). Furthermore, throughout the paper, we assume that the second user’s packet has priority in the case of simultaneous arrivals. As mentioned earlier at each time slot, both U1T and U2T are allowed to send either one or no packets; hence, the input and output packet sequences of each user could be viewed as a binary bitstream, where ‘1’ and ‘0’ indicates sending and not sending a packet in the corresponding time slot, respectively. Assume message W chosen uniformly from the set ˆ is second user’s {1, 2, ..., M } is transmitted by U1T , and W estimate of the sent message. Our performance metric is the average error probability, defined as follows: ˆ)= Pe , P r(W 6= W

M X 1 ˆ 6= m|W = m). P r(W M m=1

We define the code, rate of the code, and the channel capacity as follows: Definition 1. An (n, M, )-code consists of a codebook of size M with binary codewords of length n which satisfies Pe ≤ . Since we have assumed a fixed length for time slots and the scheduler’s processing time, the information transmission rate of the code may be defined following the standard definition of the information rate in [12, p. 195] as the amount of conveyed information normalized by the number of uses of time slots, i.e., lognM . Definition 2. (Channel Capacity) The Shannon capacity, C, for a channel is the maximum achievable rate at which one can communicate through the channel when the average probability of error goes to zero. In other word, C is the largest rate R such that:  log M >R−δ n ∀δ > 0, ∃(n, M, n )-code s.t. n → 0 as n → ∞ [13, p. 84].

In order to transmit a message, U1T uniformly at random picks a message W ∈ {1, · · · , M } and encodes it into a length n binary sequence, {δ1 , · · · , δn }. At time slot i, U1T sends a packet if δi = 1, and stays idle otherwise. With the goal of eventually decoding this message, U2T sends a bit stream of length n to the scheduler during the same length n time period. User 2 will use this bit stream and the bit stream received at U2R , to decode the sent message. Note that since we have considered a deterministic FCFS scheduler and users can agree on the packet stream sent by U2T ahead of time, the feedback U1R is already available at the encoder. The following notations will be used through out the paper: n n • ∆1 : Random binary sequence with realization δ1 sent by T U1 , where:  if a packet has been issued  1 by U1T in time slot i δi =  0 otherwise r1 : U1T ’s packet rate. m T • A : U2 ’s packet arrival times during n time slots. m T • D : U2 ’s packet departure times. m T • r2 : U2 ’s packet rate = limn→∞ n . As mentioned earlier, we analyze the capacity and coding scheme in the timing covert channel under the restriction that the queue is stable, i.e., the total packet arrival rate does not exceed 1: r1 + r2 ≤ 1 (1) •

Furthermore, define: th • Ti = Ai+1 − Ai : inter-arrival time between i and (i + th T 1) packet of U2 . We denote a realization of T m by τ m. P Ai+1 −1 • Xi = ∆j : random variable representing the j=Ai number of U1T ’s packets sent between times Ai and Ai+1 . ˆ i : estimate of Xi by decoder (User 2). • X ˆ : decoded message. • W Therefore, the following Markov chain holds: ˆm → W ˆ W → ∆n → X m → X

(2)

In the following section, the capacity of the introduced system will be calculated. III. C ODING THEOREM In this section, a general formula for the capacity of the system introduced in the previous section is proposed. The achievability and converse arguments are used for the proof. Theorem 1. The capacity of the timing channel in a shared FCFS scheduler depicted in Fig. 1 is given by: ˜ 1 , 1) + (1 − α)H(γ ˜ 2, 1 ) C = sup αH(γ 2 α,γ1 ,γ2 (3) s.t. 1 α(1 + γ1 ) + (1 − α)( + γ2 ) = 1 2

790

where 0 ≤ α ≤ 1 and 0 ≤ γ1 , γ2 ≤ 12 and the function ˜ : [0, 1] × { 1 : n ∈ N} → [0, 1] is defined as: H n 1 1 ˜ )= sup H(X), H(γ, k k X∈{0,1,...,k}

processing inequality in Markov chain in (2). Therefore: 1 1 ˆ m , τ m )] + n log M ≤ [H(X m |τ m ) − H(X m |X n n 1 ≤ H(X m |τ m ) + n n m 1X ≤ H(Xj |τ m ) + n n j=1

k ∈ N, 0 ≤ γ ≤ 1 (4)

E[X]=kγ 1 ˜ Lemma 1. The function H(γ, k ) could be computed using the following expression:

1 1 ∗ ˜ H(γ, ) = [log2 (k + 1) − ψU (kγ) log2 e], k k k

(5)

where Uk is the uniform distribution on {0, 1, ..., k} and the ∗ function ψU (·) is the rate function given by the Legendrek Fenchel transform of the log moment generating function, ψX (·): ∗ ψX (γ) = sup{λγ − ψX (λ)}.

m

≤

where, in the maximization above, the mean of the distribution PXj |τ m is E[Xj |τ m ] and the set {E[X1 |τ m ], E[X2 |τ m ], ..., E[Xm |τ m ]} should satisfy the stability constraint r1 ≤ 1 − r2 , that is: m

1 X 1 E[Xj |τ m ] ≤ 1 − r2 m j=1 τj

(6)

λ∈R

1X max H(Xj |τ m ) + n n j=1 PXj |τ m

Denoting τj γj = E[Xj |τ m ], by the definition in (4), we have:

The distribution which achieves the optimum value in (4) is the tilted distribution of Uk with parameter λ, such that kγ = 0 ψU (λ). k See Appendix for a proof. Substituting (5) in (3) and solving it, the capacity of the timing channel in the shared FCFS scheduler of Fig. 1 is equal to 0.8114 bits per time slot, achieved by α = 0.177, γ1 = 0.43 and γ2 = 0.407. In the following, we give an sketch of our proof for the capacity rate.

max

PXj |τ m E[Xj |τ m ]=τj γj

˜ j, 1 ) H(Xj |τ m ) = τj H(γ τj

(7)

The distribution for each Xj which achieves the maximum value in (7) is the tilted distribution of Uτj with parameter λ, such that: 0 τj γj = ψU (λ), τ j

0 (.) ψU τj

is the derivative of log moment generating where, function of the uniform distribution on {0, 1, ..., τj }. Therefore, we will have: m

1 1X ˜ 1 log M ≤ τj H(γj , ) + n n n j=1 τj

A. Converse For the converse, we assume that U1T has the side information τ m , i.e., the realization of U2T inter-arrival times. For any (n, M, )-code we have: 1 (a) 1 log M = H(W ) n n (b) 1 = H(W |τ m ) n 1 ˆ |τ m ) + 1 H(W |W ˆ , τ m) = I(W ; W n n (c) 1 ˆ |τ m ) + n ≤ I(W ; W n (d) 1 ˆ m |τ m ) + n ≤ I(X m ; X n

such that the set {γ1 , γ2 , ..., γm } satisfies the stability constraint of the queue. The inter-arrival times take values in the set {1, 2, ..., n}. Therefore, in the summation above we can fix the value of inter-arrival time on the value τ and count the number of times that τj has that value. Defining mτ as the numberP of times that n the inter-arrival time is equal to τ (Note that n = τ =1 τ ·mτ ), we can break the summation above as follows: n

m

n

k=1 m

τ 1 1XX ˜ τ,k , 1 )] + n log M ≤ [ τ H(µ n n τ =1 τ τ 1X X ˜ τ,k , 1 )] + n = [τ H(µ n τ =1 τ

k=1

where (a) holds because W is a uniform random variable over the set of messages {1, ..., M }, (b) follows from the fact that the chosen message is independent of the inter-arrival time of decoder’s packets, (c) follows form Fano’s inequality with n = n1 (H(Pe ) + Pe log2 (M − 1)). (d) follows from data

mτ n X 1 1X 1 ˜ = [τ · mτ H(µτ,k , )] + n n τ =1 mτ τ k=1

where µτ,k is equal to the k

th

γj with τj = τ .

˜ ·) is a concave function of its Lemma 2. The function H(·, first argument.

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Using Lemma 2, by Jensen’s inequality, we will have:

Therefore, from Lemma 4 and (10), we have: 1 ˜ 1 , 1) + (1 − α)H(γ ˜ 2 , 1 ) + n log M ≤ αH(γ n 2 ˜ 1 , 1) + (1 − α)H(γ ˜ 2 , 1 ) + n ≤ sup αH(γ 2 α,γ1 ,γ2

n

1 1X ˜ τ , 1 )] + n log M ≤ [τ · mτ · H(µ n n τ =1 τ mτ 1 X µτ,k . mτ k=1 n X Using the equation n = τ · mτ , we have:

where µτ =

And the constraint in (9) reduces to: 1 α(1 + γ1 ) + (1 − α)( + γ2 ) = 1, 2 where 0 ≤ α ≤ 1 and 0 ≤ γ1 , γ2 ≤ 21 .

τ =1 n X

τ · mτ 1 ˜ τ , 1 )] + n log M ≤ [ Pn H(µ n τ τ · m τ τ =1 τ =1 n X

˜ τ , 1 )] + n = [πτ H(µ τ τ =1

Letting n → ∞, n goes to zero and:

which completes the proof of the converse.

τ · mτ . where πτ = Pn τ =1 τ · mτ Also, the constraints of the problem could be written as follows: n X πτ µτ = r1 , τ =1 n X

πτ

τ =1

1 = r2 . τ

Therefore, by (1), n X

πτ (

τ =1

1 + µτ ) ≤ 1. τ

(9)

1 ˜ − µ, 1 ). ˜ ) = H(1 Lemma 3. H(µ, k k From Lemma 3, we can rewrite (8) as follows: n

X 1 ˜ µτ , 1 )] + n , log M ≤ [ˆ πτ H(ˆ n τ τ =1

(10)

where µ ˆτ is: if 0 ≤ µτ ≤ 21 , if 12 ≤ µτ ≤ 1. Pn Lemma 4. For all vectors π ˆ1n and µ ˆn1 such that τ =1 π ˆτ = 1 P n and 0 ≤ µ ˆτ ≤ 12 , with τ =1 π ˆτ µ ˆτ = r1 , there exists 0 ≤ α ≤ 1 and 0 ≤ γ1 , γ2 ≤ 21 such that: 

µ ˆτ =

µτ 1 − µτ

αγ1 + (1 − α)γ2 = r1 , 1 α + (1 − α) = r2 = 1 − r1 , 2 and

n X

˜ µτ , 1 )] ≤αH(γ ˜ 1 , 1) [ˆ πτ H(ˆ τ τ =1 ˜ 2 , 1 ). + (1 − α)H(γ 2

˜ 1 , 1) + (1 − α)H(γ ˜ 2, 1 ) C ≤ sup αH(γ 2 α,γ1 ,γ2

(8)

B. Achievability Achievability of information rate 0.8114 can be proved using rate splitting and time sharing. The sequence of steps in our achievability scheme is as follows: • Set α = 0.177. • Define P1 = [.57, .43] and R1 = H(P1 ). Generate an i.i.d. binary codebook C1 containing 2αnR1 sequences of length αn where P r(1) = 0.43 and P r(0) = 0.57. Define P2 = [.43, .325, .245] and R2 = 21 H(P2 ). Generate an i.i.d. ternary codebook C2 containing 2(1−α)nR2 sequences of length 12 (1 − α)n over symbols a0 , a1 and a2 with P r(a0 ) = 0.43, P r(a1 ) = 0.325 and P r(a2 ) = 0.245. Substitute a0 with 00, a1 with 10 and a2 with 11, so, we will have 2(1−α)nR2 binary sequences of length (1 − α)n. Combine C1 and C2 to get C, such that C has 2n(αR1 +(1−α)R2 ) binary sequences of length n where we concatenate ith row of C1 with j th row of C2 to make the ((i − 1)(2(1−α)nR2 ) + j)th row of C (note that 2(1−α)nR2 is the number of rows in C2 ). Rows of C are our codewords. In above, n should be chosen such that αnR1 , αn, (1 − α)nR2 and 21 (1 − α)n are all integers. T • Encoding: U2 sends the stream of all ones (one packet in each time slot) in the first αn time slots and sends bit stream of concatenated 10’s for the rest of (1 − α)n time slots. To send message m, U1T sends the corresponding row of C, that is, it sends the corresponding part of m from C1 in the first αn time slots and the corresponding part of m from C2 in the rest of (1 − α)n time slots. Note that this scheme satisfied the stability condition in (1). 1 • Decoding: Assuming the queue in not empty , since there is no noise in the system, the decoder can always learn the exact sequence sent by U1T . 1 Using the formula q = D − A − 1 (where the extra 1 is the time slot i i i needed to process the packet) the receiver side could always be aware of the queue length and prevent it from becoming zero by sending sufficient extra packets. Note that sending extra packets, increases the resolution and does not decrease the information transmission rate.

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A PPENDIX Proof of lemma 1. Suppose X is defined over the set {0, 1, ..., k}, H(X) =

Time

k X

PX (i) log

i=0

= Fig. 2: The packets above the horizontal axis indicate the arrivals at the scheduler. U2T ’s packets are shown in blue and U1T ’s packets are shown in red. The two blue packets below the horizontal axis correspond to the received packet at node U2R . If the queue is not empty in the start (here, with 2 packets buffered), U2R can detect that a packet was sent by U1T because it experiences an extra slot of delay (the gray block at the bottom).

k X

1 PX (i)

PX (i) log (k + 1)

i=0

=

k X

1 (k + 1)PX (i)

PX (i) log (k + 1) −

i=0

k X

PX (i) log

i=0

PX (i) 1 k+1

= log (k + 1) − D(PX ||Uk ) where Uk is the uniform distribution on {0, 1, ..., k}. Therefore, in order to maximize H(X), we need to minimize D(PX ||Uk ). But, it is known that: ∗ min D(PX ||Uk ) = ψU (kγ) log2 e, k

(12)

E[X]=kγ

Consequently, we will have: log2 2n(αR1 +(1−α)R2 ) n = αR1 + (1 − α)R2 1 = αH(P1 ) + (1 − α) H(P2 ) 2

∗ where ψU (kγ) is given in (6). k

C≥

(11)

The distribution which achieves the optimum value in (12) is the tilted distribution of Uk with parameter λ, such that 0 kγ = ψU (λ). k R EFERENCES

substituting the values in the expression above, we see that the rate 0.8114 timebitsslot is achievable. The main idea in this scheme is that assuming there are sufficient packets in the scheduler’s queue, the decoder can learn the number of user 1’s packets sent between two of his (U2T ’s) consecutive packets. Fig. 2 depicts an example of a condition where U1T ’s sent packet is detectable. It is assumed that there are 2 packets in the queue when U2T ’s first packet arrives and U2T sends 2 packets which are spaced by 3 time slots. Since U2R observes an empty gap (the gray block at the bottom in Fig. 2) in the corresponding received sequence, it can conclude that U1T has sent a packet between U2T ’s packets, yet, the exact location of that is not recognizable. Note that since in our achievable scheme, U2T ’s packets are spaced by either one or two time slots, it is enough to have one packet buffered in the queue.

IV. C ONCLUSION We studied the information transmission rate in a queueing channel which occurs through delays experienced by users who are scheduled to share a resource in the FCFS manner. The capacity of the channel is derived using achievability and converse arguments. We show that the information rate of 0.8114 bits per channel use is the highest achievable rate for such a channel. Capacity of extensions of the problem to other scheduling policies remains open.

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