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Context-Aware Privacy Preservation in Network

Caching: An Information Theoretic Approach

Seyedeh B. Hassanpour, Abolfazl Diyanat, Ahmad Khonsari, Seyed P. Shariatpanahi, and Aresh Dadlani

Abstract—Caching has been recognized as a viable solution to

surmount the limited capability of backhaul links in handling abun-

dant network trafﬁc. Although optimal approaches for minimizing

the average delivery load do exist, current caching strategies fail to

avert intelligent adversaries from obtaining invaluable contextual

information by inspecting the wireless communication links and

thus, violating users’ privacy. Grounded in information theory, in

this paper, we propose a mathematical model for preservingprivacy

in a network caching system involving a server and a cache-aided

end user. We then present an efﬁcient content caching method that

maximizes the degree of privacy preservation while maintaining

the average delivery load at a given level. Given the Pareto optimal

nature of the proposed 𝜖-constraint optimization approach, we also

obtain the maximum privacy degree achievable under any given

average delivery load. Numerical results and comparisons validate

the correctness of the our context-oriented privacy model.

Index Terms—Content caching, context-oriented privacy, error

probability bound, Pareto optimal, information theory.

I. INTRODUCTION

BACKHAUL congestion due to the ever-rising data trafﬁc

growth is arguably the most prominent limiting factor

in 5G network performance. Network caching, where popular

contents are cached closer to the edge devices, has recently

been deemed as a feasible remedy to improve network capac-

ity and alleviate the trafﬁc load during peak hours [1]–[3].

Despite the beneﬁts from placing requested contents in close

proximity of interested users, proactive caching facilitates

adversary nodes to access the data and gather information on

users’ preferences and location. Preserving privacy in caching

systems is therefore, a great challenge as user data revelation

is usually against users’ conﬁdentiality agreements [4].

In general, privacy in networks can be preserved based

on two distinct aspects. In data-oriented privacy, the data

integrity is maintained while being transmitted over the net-

work such that accessibility to on-the-ﬂy content is prohibited

to malicious nodes [5]. Existing works on data-oriented pri-

vacy mostly emphasize on either proposing new encryption

methods or adding extra layers of encryption in the network

[6]. Contrarily, context-oriented privacy refers to preserving

the privacy of users and their requested ﬁles which can be

divulged by eavesdropping certain features of the transmitted

packets such as the time and location of creation, usage

pattern, and other ﬁle-related statistics that remain unguarded

in conventional protection mechanisms [7], [8].

With regard to cache content placement (CCP), an optimal

probabilistic caching strategy is investigated in [9] to max-

imize the cache hit probability and cache-aided throughput

in wireless D2D caching networks. In [10], a hybrid caching

scheme jointly optimized with the transmission schemes is

Server User

Adversary

Files Cache

f1

. . .

fM

0N0N

Fig. 1. A server with 𝑀ﬁles, each of size 𝑁bits, communicating with an

end user with a cache size of 𝑁bits in presence of an adversary.

proposed to handle the trade-off between the signal coop-

eration gain and the caching diversity gain. While neither

[9] nor [10] consider the presence of external adversaries,

there exist a handful of studies devoted to CCP strategies

that focus on context-oriented privacy. The authors in [11]

presented a protocol enforcing fair subdivision of limited

cache storage and context privacy provision. More recently,

a wireless caching network wherein multiple cooperative ad-

versarial nodes tap contextual information of users is studied

in [12]. The authors aimed to maximize the probability of

delivering all requested content within a given radius. To

add randomness to the eavesdroppers’ estimates, which are

based on the eavesdropped transmitted packets, they applied

probabilistic caching to obtain the optimal probabilities. By

virtue of the non-convex essence of their CCP optimization

problem, the authors adopted genetic algorithm to ﬁnd the

best caching to mislead the eavesdroppers. The works in [11]

and [12], however, do not address the trade-off between trafﬁc

load minimization and degree of context privacy.

The main contribution of this letter is to introduce an

information theoretical formulation for a new CCP protocol

that characterizes the trade-off between trafﬁc load and context

privacy of the system. In particular, we consider a strong

adversarial node with coverage over the entire area so as

to reduce the extra trafﬁc overhead imposed due to coop-

eration between eavesdroppers. Assuming that the adversary

is equipped with the best estimator while overhearing the

channel, we also derive analytical bounds for estimation error

in terms of the Fano-Kovalevskij inequality [13]. Finally,

using the proposed 𝜖-constraint CCP optimization model, we

minimize the average delivery rate over the channel for any

desired level of privacy through simulation results.

II. SY ST EM MO DE L DESCRIPTION

Consider the content delivery system shown in Fig. 1which

comprises of a server with 𝑀ﬁles, denoted by the set F=

{𝑓𝑘|1≤𝑘≤𝑀}, each of length 𝑁bits, and a cache-enabled

user with storage capacity of 𝑁bits. In the content placement

phase (CPP), the server preloads a ﬁle or a mixture of ﬁles into

the user’s cache using a given strategy until it is completely

2

Feasible region

PA

Pe

Pe= 1 −PA

Pe= 0

0.5

01

0.5Adversary without knowledge of the delivery load

Adversary with knowldege of the delivery load in optimal strategy

Fig. 2. The adversary best estimation error probability for F={𝐴, 𝐵 }.

ﬁlled. As a result, we have the constraint Í𝑀

𝑘=1|𝑓𝑐

𝑘|=𝑁on

the preloaded ﬁles, where |𝑓𝑐

𝑘|denotes the size of the cached

portion of ﬁle 𝑓𝑘. In the content delivery phase (CDP) that

follows, the user requests for a ﬁle from F. The server then

transfers the requested ﬁle only if it has not been cached earlier

at the user’s device. We assume that the user requests for ﬁle

𝑓𝑘with probability 𝑝𝑘. Since the user requests for at least one

ﬁle, we thus have Í𝑀

𝑘=1𝑝𝑘=1.

Moreover, we consider a passive adversary that eavesdrops

the communication between the server and the user. Without

any prior knowledge on the requested ﬁle, the adversary

attempts to detect the ﬁle from set Fin the CDP. We assume

that the adversary is armed with the best estimator. Let ˆ

𝑘

denote the adversary’s estimation of the index of the requested

ﬁle and 𝑃𝑒be the adversary’s estimation error in the sense of

maximum a posterior probability (MAP), deﬁned as:

𝑃𝑒=Pr[ˆ

𝑘≠𝑘],0≤𝑃𝑒≤1.(1)

We reserve the term adversary’s best estimation for the

estimation with minimum 𝑃𝑒among all possible estimations.

III. PROP OS ED SECURE CCP APP ROAC H

In this section, we will describe our approach towards

characterizing the trade-off between privacy and delivery

efﬁciency. For the sake of illustration, consider a server

that stores two distinct ﬁles, namely 𝐴and 𝐵, with request

probabilities 𝑃𝐴and 𝑃𝐵(𝑃𝐴≥𝑃𝐵), respectively. If the ﬁle

popularity is known to the adversary, then he/she can select

the most popular ﬁle as his/her estimation without having any

information about the CDP. In this case, 𝑃𝑒=1−𝑃𝐴which

corresponds to the dashed line in Fig. 2.

Intuitively, an efﬁcient CCP strategy is to minimize the total

number of transferred bits by considering the popularity of

the ﬁles requested by the user. The ﬁle popularity distribution

(𝑝𝑘)however, is known to all entities in the system, including

the adversary node. Additionally, the adversary is also aware

that the server in traditional network caching preloads the

most popular ﬁle into the user’s cache. In such a setting,

it becomes trivial to guess the index of the ﬁle (solid line

in Fig. 2). In what follows, we devise a CCP strategy that

achieves maximum ambiguity (or degree of privacy) for a

given delivery load.

A. Adversary Error Probability Bounds

To obtain the analytical upper and lower bounds on the ad-

versary’s best estimation error, we adopt the Fano-Kovalevskij

inequality for binary variables in our two ﬁle scenario, whereas

the results of [14] are used for 𝑀 > 2in the following theorem.

Theorem 1. For any estimator ˆ

𝑘such that 𝑘→𝑌→ˆ

𝑘is a

Markov chain with 𝑃𝑒=Pr[ˆ

𝑘≠𝑘], we have:

Ψ≤𝑃𝑒≤𝐻(𝑘|𝑌)

2(2)

with Ψ4

=(𝐻−1(𝐻(𝑘|𝑌)),if 𝑀=2,

𝐻(𝑘|𝑌)−1

log2(𝑀−1),if 𝑀 > 2,(3)

where 𝑀=|F | and 𝑌is the random variable (r.v.) of adver-

sary’s observation corresponding to the number of bits trans-

mitted from the server to the user over the network during the

CDP. The conditional entropy 𝐻(𝑘|𝑌)is the total ambiguity

in 𝑘given observation 𝑌and 𝐻−1is the inverse of the binary

entropy function 𝐻(𝜗)=−𝜗log2(𝜗)−(1−𝜗)log2(1−𝜗).

Proof. See Appendix A.

With the bounds derived for the adversary’s best estimation

error in Theorem 1, we now maximize the lower bound on

𝑃𝑒in (3). Thus, exploiting the proposed approach guarantees

that 𝑃𝑒in the sense of MAP under any estimator will always

be greater than a particular threshold 𝜉∈R++, i.e. 𝑃𝑒=

Pr[ˆ

𝑘≠𝑘] ≥ 𝜉. As a result, maximizing the error probability

lower bound in (3) is equivalent to maximizing the conditional

entropy 𝐻(𝑘|𝑌). Indeed, this maximum error probability is

what we technically deﬁne as the privacy degree.

B. Exemplary Case (𝑀=2)

In this subsection, we consider the scenario depicted in

Fig. 2. We deﬁne the indicator r.v. 𝑘which is zero, if the user

requests for ﬁle 𝐴with probability 𝑃𝐴and one, if otherwise

with probability 𝑃𝐵=1−𝑃𝐴. We also let r.v. 𝑍represent

the number of bits of ﬁle 𝐴stored in the cache. For discrete

values of 𝑗, where 0≤𝑗≤𝑁, the probability mass function

(pmf) of 𝑍is given as 𝑝𝑧

𝑗,Pr[𝑍=𝑗]. Upon completion of

the CPP, when the user requests for one of the ﬁles, the server

transmits 𝑌bits to satisfy the demand (i.e. 𝑌=𝑁−𝑍). The term

𝐻(𝑘|𝑌)essentially captures the adversary’s ambiguity about

the identity of the requested ﬁle by knowing the size of the

transmitted data and determines its error as stated in Lemma 1.

Lemma 1. The term 𝐻(𝑘|𝑌)is calculated as follows:

𝐻(𝑘|𝑌)=𝐻(𝑘) + 𝐻(𝑍) − 𝐻(𝑌),(4)

where 𝐻(𝑘),𝐻(𝑍), and 𝐻(𝑌)are the entropies of 𝑘,𝑍, and

𝑌, respectively.

Proof. See Appendix B.

By letting 𝒑𝑍=[𝑝𝑧

0, 𝑝𝑧

1, . . . , 𝑝 𝑧

𝑗, . . . , 𝑝 𝑧

𝑁]be the distribution

of 𝑍, we observe that designing the CCP strategy is equivalent

to determining the vector 𝒑𝑍. Since 𝐻(𝑘)is independent of

𝒑𝑍, we therefore, arrive at the following optimization model

3

RN

Z1Z2... ZM

Fig. 3. Depiction of fraction of ﬁle 𝑓𝑘∈ F in an 𝑁−bit cache using r.v. 𝑍𝑘.

from (4) to achieve maximum ambiguity without imposing any

constraints on the delivery load:

𝒑∗

𝑍=argmax

𝒑𝑍

𝐻(𝑘|𝑌)=𝐻(𝑍) − 𝐻(𝑌)(5a)

s.t.

𝑁

Õ

𝑗=0

𝑝𝑧

𝑗=1,0≤𝑝𝑧

𝑗≤1,0≤𝑗≤𝑁. (5b)

Next, we prove in Proposition 1that an optimal solution

of (5) is uniformly distributed at the cost of transmitting 𝑁/2

bits.

Proposition 1. The uniform distribution is one of the optimal

solutions of (5).

Proof. See Appendix C.

To maximize the adversary’s ambiguity subject to a speciﬁc

delivery load constraint, we now augment the following con-

straint to the model in (5). That is to say, we maximize the

degree of privacy while keeping the number of bits transmitted

in the CDP below some constant 𝐶value:

𝒑∗

𝑍=argmax

𝒑𝑍

𝐻(𝑘|𝑌)=𝐻(𝑍) − 𝐻(𝑌)(6a)

s.t.

𝑁

Õ

𝑗=0

𝑝𝑧

𝑗=1,0≤𝑝𝑧

𝑗≤1,0≤𝑗≤𝑁, (6b)

𝑃𝐴(𝑁−𝑍) + 𝑃𝐵𝑍≤𝐶, (6c)

where 𝑃𝐴(𝑁−𝑍) + 𝑃𝐵𝑍is the load delivered in the CDP and

𝐶corresponds to the effective capacity of the communication

link. Note that the augmented model in (6) is efﬁcient in the

Pareto optimality sense. Lemma 2proves an optimal solution

for this model. For 𝐶 < 𝑁/2, we obtain the optimal solution

numerically in Section IV.

Lemma 2. If 𝐶≥𝑁/2, then the uniform distribution is one

of the optimal solutions for the augmented model in (6).

Proof. See Appendix D.

C. General Case

We now extend our approach to any arbitrary value of 𝑀.

Suppose that the user chooses ﬁle 𝑓𝑘with probability 𝑝𝑘. We

deﬁne the random process Z={𝑍𝑘}, where 𝑍𝑘denotes the

fraction of ﬁle 𝑓𝑘(in bits) cached at the user’s end. Due to

the limited cache capacity, we have Í𝑀

𝑘=1𝑍𝑘=𝑁as depicted

in Fig. 3. Consequently, the pmf of the delivery load 𝑌can be

expressed as follows:

𝑃𝑌=Pr[𝑌=𝑗]=

𝑀

Õ

𝑘=1

𝑝𝑘×Pr[𝑍𝑘=𝑁−𝑗].(7)

f1f2f3f4f5f6f7f8f9f10 f11 f12

P5

k=1 pk≃1

3P8

k=6 pk≃1

3P12

k=9 pk≃1

3

Fig. 4. Example of a ﬁle set Fwith 𝑀=12 split into three subsets such

that the sum of popularity in each subset equals 1

3.

It should be noted that the r.v.s 𝑍𝑘are not independent and

their distribution is given by the following matrix:

𝑷𝑍=

𝑝𝑧

10 𝑝𝑧

11 . . . 𝑝𝑧

1𝑁

𝑝𝑧

20 𝑝𝑧

21 . . . 𝑝𝑧

2𝑁

.

.

..

.

.....

.

.

𝑝𝑧

𝑀0𝑝𝑧

𝑀1. . . 𝑝𝑧

𝑀 𝑁

,(8)

where element 𝑝𝑧

𝑘 𝑗 ,Pr[𝑍𝑘=𝑗]. As a result, designing the

CCP strategy corresponds to computing matrix 𝑷𝑍. Hence, the

optimal privacy-load trade-off can be formally characterized as

the following optimization model:

𝑷∗

𝑍=argmax

𝑷𝑍

𝐻(𝑍1, . . . , 𝑍 𝑀) − 𝐻(𝑌)(9a)

s.t.

𝑀

Õ

𝑘=1

𝑝𝑧

𝑘 𝑗 =1,0≤𝑝𝑧

𝑘 𝑗 ≤1,0≤𝑗≤𝑁 , (9b)

𝑀

Õ

𝑘=1

𝑝𝑘(𝑁−𝑍𝑘) ≤ 𝐶 , (9c)

𝑀

Õ

𝑘=1

𝑍𝑘=𝑁. (9d)

Note that 𝐻(𝑘)is independent of the optimization variable

and can be removed from the cost function in (9a). Constraint

(9c) ensures that the communication cost imposed on the net-

work remains below a threshold value 𝐶. Thus, by controlling

𝐶in (9c), and then attaining the corresponding privacy degree

by solving the optimization problem, the trade-off between

trafﬁc load and privacy degree can be managed. Evidently, (9)

reduces to (5) when 𝑀=2.

D. Sub-Optimal Heuristic for Large 𝑀and 𝑁

When the number of ﬁles (𝑀) or the size of each ﬁle (𝑁) is

large, the optimization problem in (9) becomes computation-

ally expensive to solve. For large ﬁle sizes, we can split the

ﬁles into smaller portions called chunks, instead of splitting

them at the bit level. As for large number of ﬁles, in what

follows, we present a heuristic that makes solving the problem

feasible. Although this method yields a sub-optimal solution,

it is a quid pro quo for reduction in computational complexity.

The three steps of this library splitting heuristic are as follows:

•Step 1: Split the library Finto 𝑞equi-probable subsets,

such that the aggregated popularity in each subset almost

equals 1/𝑞. Fig. 4illustrates an example for 𝑞=3.

•Step 2: Split the cache of each user into 𝑞memory slots

such that each slot contains approximately 𝑁/𝑞bits.

•Step 3: Deﬁne a sub-problem solution as the optimal

caching of a subset into its corresponding memory

slot. Subsequently, solve the equivalent optimization sub-

problem 𝑞times for each subset.

4

0 10 20 30 40 50

0

0.2

0.4

0.6

0.8

1

.02 ∗j

j

The probability Z

Simulation results

Theoretical Formula

(a)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

30

35

40

45

50

55

60

65

70

Privacy degree, H(k|Y)

Delivery load

Numerical results

(b)

0.5 0.6 0.7 0.8 0.9

0

0.1

0.2

0.3

0.4

0.5

PA

Adversary error probability Pe

Proposed approach

Adversary without knowing delivery load

(c)

1 2 3 5 6 10 15 30

0

0.2

0.4

0.6

0.8

1

q

ψ

(d)

Fig. 5. (a) CDF of the solution of (5). (b) Pareto-optimal curve for (5)-(6c). (c) Adversary estimation error for 𝑀=2. (d) The difference between cost function

for optimal and sub-optimal problem (𝜓) versus 𝑞.

IV. SIMULATION RESULTS

The proposed CCP model is implemented using Matlab and

a95% conﬁdence level is adopted to demonstrate the accuracy

of the Monte Carlo simulation results. The setup comprises of

a server with library F={𝐴, 𝐵}and a user cache of size 𝑁=7

bits. We generate 20000 samples (0for requesting ﬁle 𝐴and

1for requesting ﬁle 𝐵) with 𝑃𝐴=0.7and 𝑃𝐵=0.3.

To validate Proposition 1, we use the fmincon function

in Matlab to compute an optimal solution of (5) in Fig. 5(a).

The plot shows the cumulative density function (CDF) of the

resulting distribution which is exactly same as the uniform

distribution.

Fig. 5(b) plots the curve of the delivery load constraint

(𝐶) against the degree of privacy (𝐻(𝑘|𝑌)). As evident in the

ﬁgure, 𝐻(𝑘|𝑌)=0when the delivery load constraint is 𝑃𝐵𝑁,

whereas we achieve maximum privacy (𝐻(𝑘|𝑌)=1bit)when

beyond 𝑁/2bits are transferred in the CDP.

We compare the adversary error probability (𝑃𝑒) with

respect to 𝑃𝐴in Fig. 5(c). This ﬁgure depicts the signiﬁcant

increment of lower bound of 𝑃𝑒(as compared to Fig. 2) and

thus, reaching the upper bound which is 1−𝑃𝐴. It is worth

noting that when 𝑃𝐴=0.5and 𝑃𝐴=1, the error probability

in our approach converges to that of an adversary with no

knowledge of the delivery load. As this ﬁgure clearly shows,

the maximum difference occurs roughly at 𝑃𝐴=0.7which

implies that our proposed approach achieves a relatively lower

degree of privacy as compared to reference points 𝑃𝐴=0.5

and 𝑃𝐴=1.

Finally, Fig. 5(d) plots the difference between the optimal

solution in (9) and sub-optimal solution discussed in Sec-

tion III-D for a library Fwith 𝑀=30 ﬁles. We denote this

difference by 𝜓and scale it between optimal (𝜓=0) and non-

optimal (𝜓=1) solution. When 𝑞=1, the optimal solution is

obtained from (9). But for example, if 𝑞=2, we divide the

library into two subsets and solve the optimization problem

for these two subsets independently. This ﬁgure shows that

we can reduce the time complexity of our problem by a factor

of 10 at the expense of obtaining a sub-optimal solution with

20% difference from the optimal solution.

V. CONCLUSION AND FUTURE WO RK

In this paper, we have proposed a caching strategy that

maximizes the adversary’s best estimation error while mini-

mizing the average delivery load, which is momentous in terms

of energy consumption and limited bandwidth in wireless

links. In the presented approach, we have formulated an 𝜖-

constraint optimization model to alter the statistical behavior

of the server so as to misguide the adversary. Furthermore, we

have maximized the Fano lower bound for the best adversary

estimation using information theory to reduce the adversary’s

accessibility to useful contextual information. Simulation re-

sults also validate the effectiveness of our approach. This

work can be further extended to investigate the same trade-off

considering multiple users where the adversary must estimate

both, the transmitted ﬁle and the requesting user.

APPENDIX

A. Proof of Theorem 1

For proof on the upper bound, we refer the reader to [13].

For the lower bound, we deﬁne the error event given estimator

ˆ

𝑘as:

𝐸=1 if ˆ

𝑘≠𝑘,

0 if ˆ

𝑘=𝑘. (10)

𝐻(𝐸 , 𝑘 |ˆ

𝑘)can be expanded as follows:

𝐻(𝐸 , 𝑘 |ˆ

𝑘)=𝐻(𝑘|ˆ

𝑘) + 𝐻(𝐸|𝑘, ˆ

𝑘)=𝐻(𝐸|ˆ

𝑘) + 𝐻(𝑘|𝐸, ˆ

𝑘).

If the selected ﬁle index (𝑘) and the estimated ﬁle index ( ˆ

𝑘) are

known to the adversary, then he/she can determine the error

without ambiguity, i.e. 𝐻(𝐸|𝑘, ˆ

𝑘)=0. Thus, we will have:

𝐻(𝑘|ˆ

𝑘)=𝐻(𝐸|ˆ

𝑘) + 𝐻(𝑘|𝐸, ˆ

𝑘)

(𝑎)

≤𝐻(𝐸) + 𝐻(𝑘|𝐸, ˆ

𝑘)(𝑏)

≤𝐻(𝑃𝑒) + 𝐻(𝑘|𝐸, ˆ

𝑘)

𝐻(𝑘|𝑌)(𝑐)

≤𝐻(𝑘|ˆ

𝑘) ≤ 𝐻(𝑃𝑒) + 𝐻(𝑘|𝐸 , ˆ

𝑘).(11)

Conditioning reduces entropy, so we have (a). The identity

(b) stems from the fact that 𝐸is a binary r.v.. For identity (c),

according to the Markov chain property, we have 𝐻(𝑘|𝑌) ≤

𝐻(𝑘|ˆ

𝑘). Therefore, we arrive at the inequality (11). We now

simplify (11) for caching two and more ﬁles as below:

1) For 𝑀=2:In this case, if the adversary knows which ﬁle

is selected, he/she can determine the error without ambiguity,

i.e. 𝐻(𝑘|𝐸, ˆ

𝑘)=0. Using this fact and (11), we arrive at:

𝐻(𝑘|𝑌) ≤ 𝐻(𝑃𝑒)=⇒𝐻−1(𝐻(𝑘|𝑌)) ≤ 𝑃𝑒,(12)

where 𝐻(𝑃𝑒)=−𝑃𝑒log2(𝑃𝑒) − (1−𝑃𝑒)log2(1−𝑃𝑒)and 𝐻−1(·)

is the inverse of 𝐻.

5

2) For 𝑀 > 2:We can write (11) as:

𝐻(𝑘|𝑌) ≤ 𝐻(𝑃𝑒) + 𝐻(𝑘|𝐸 , ˆ

𝑘)

(𝑎)

≤1+𝐻(𝑘|𝐸, ˆ

𝑘)(𝑏)

≤1+𝑃𝑒log2(𝑀−1).(13)

In (13), identity (a) follows from the fact that 𝐻(𝑃𝑒) ≤ 1. The

identity (b) is due to [14, Theorem 2.10.1]. Thus, we conclude

that:

𝐻(𝑘|𝐸, ˆ

𝑘)=Pr{𝐸=0}𝐻(𝑘|𝐸=0,ˆ

𝑘)

| {z }

Equal to zero

+

Pr{𝐸=1}

| {z }

𝑃𝑒

𝐻(𝑘|𝐸=1,ˆ

𝑘) ≤ 𝑃𝑒log2(𝑀−1).

Rearranging the terms results in 𝐻(𝑘|𝑌) −1

log2(𝑀−1)≤𝑃𝑒. This com-

pletes the proof.

B. Proof of Lemma 1

𝐻(𝑘|𝑌)can be written as follows [14, §2.2]:

𝐻(𝑘|𝑌)=𝐻(𝑘, 𝑌 ) − 𝐻(𝑌).(14)

To obtain the entropy 𝐻(𝑘|𝑌), we calculate the joint entropy,

𝐻(𝑘 , 𝑌), and 𝐻(𝑌)separately. If ﬁle 𝐴is requested and 𝑁−𝑗

bits of ﬁle 𝐴are stored in the cache, then the server should

transmit the remaining 𝑗bits in the CDP. To compute 𝐻(𝑘 , 𝑌 ),

we need the joint distribution Pr[𝑘=𝑖 , 𝑌 =𝑗]for 𝑖∈ {0,1}and

𝑗∈ {0,1,2, . . . , 𝑁 }, as given below:

Pr[𝑘=𝑖 , 𝑌 =𝑗]=Pr[𝑌=𝑗|𝑘=𝑖] × Pr [𝑘=𝑖],

Pr[𝑌=𝑗|𝑘=0]=Pr[𝑍=𝑁−𝑗],

Pr[𝑌=𝑗|𝑘=1]=Pr[𝑍=𝑗].(15)

After some mathematical manipulations, we get:

𝐻(𝑘 , 𝑌)=−𝑃𝐴log2𝑃𝐴−𝑃𝐵log2𝑃𝐵(16)

=−𝑃𝐴

𝑁

Õ

𝑗=0

Pr[𝑍=𝑁−𝑗]log2Pr [𝑍=𝑁−𝑗]

−𝑃𝐵

𝑁

Õ

𝑗=0

Pr[𝑍=𝑗]log2Pr [𝑍=𝑗]=𝐻(𝑘) + 𝐻(𝑍).

Substituting (16) in (14) eventually yields (4), which com-

pletes the proof.

C. Proof of Proposition 1

We ﬁrst suggest the uniform distribution as a possible

solution and then prove that this solution maximizes the cost

function in (5a) and satisﬁes (5b). For observation 𝑌, we have:

Pr[𝑌=𝑗]=𝑃𝐴Pr[𝑍=𝑁−𝑗] + 𝑃𝐵Pr[𝑍=𝑗].(17)

Since conditioning always reduces entropy [14], we have:

𝐻(𝑘|𝑌) ≤ 𝐻(𝑘)using Lemma 1

=============⇒𝐻(𝑍) ≤ 𝐻(𝑌).(18)

Now, we prove that for the uniform distribution, 𝐻(𝑍) −

𝐻(𝑌)becomes zero and 𝐻(𝑘|𝑌)attains its maximum value.

Furthermore, 𝑍achieves its maximum entropy (𝐻(𝑍)=

log2(𝑁+1)). Using the deﬁnition of entropy and (17), we

obtain the following:

𝐻(𝑌)=−

𝑁

Õ

𝑗=0𝑃𝐴+𝑃𝐵

𝑁+1log2𝑃𝐴+𝑃𝐵

𝑁+1=log2(𝑁+1).(19)

According to (19), we have:

𝐻(𝑍) − 𝐻(𝑌)=log2(𝑁+1) − log2(𝑁+1)=0.(20)

The uniform distribution is thus, one of the optimal solutions

of (5) and not the unique solution. This completes the proof.

D. Proof of Lemma 2

Suppose the r.v. 𝑍follows a uniform distribution. Hence,

E[𝑍]=

𝑁

Õ

𝑘=0

𝑘 𝑝𝑘=

𝑁

Õ

𝑘=0

𝑘

𝑁+1=𝑁

2.(21)

According to (21), we have:

E[𝑍]=𝑁

2≥𝑃𝐴𝑁−𝐶

𝑃𝐴−𝑃𝐵

=⇒𝐶≥𝑁

2.(22)

We already know that the uniform distribution is a feasible

(and not unique) solution for (5), i.e., the model without any

delivery load constraint. This completes the proof.

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