Mathematical Problems of Computer Science 55, 35–43, 2021.

UDC 519.72

The Role of Information Theory in the Field of Big

Data Privacy

Mariam E. Haroutunian1 and Karen A. Mastoyan2

1Institute for Informatics and Automation Problems of NAS RA
2Gavar State University

e-mail: armar@sci.am, kmastoyan@yandex.com

Abstract

Protecting privacy in Big Data is a rapidly growing research area. The first ap-
proach towards privacy assurance was the anonymity method. However, recent research
indicated that simply anonymized data sets can be easily attacked. Later, differential
privacy was proposed, which proved to be the most promising approach. The trade-off
between privacy and the usefulness of published data, as well as other problems, such
as the availability of metrics to compare different ways of achieving anonymity, are in
the realm of Information Theory. Although a number of review articles are available in
literature, the information - theoretic methods capacities haven’t been paid due atten-
tion. In the current article an overview of state-of-the-art methods from Information
Theory to ensure privacy are provided.

Keywords: Big data, Anonymization, Differential privacy, Entropy, Mutual infor-
mation, Distortion

1. Introduction

In recent years, Big Data has become a hot research topic, because it helps businesses and
organizations to improve the decision making power and provides new opportunities with
data analysis.

Big Data life cycle can be divided into the following stages: data generation, storage and
processing. Multiple parties are involved in these stages, hence, the privacy violation risks
are increased.

A number of privacy preserving mechanisms have been developed [1], [2], however, the
study on Big Data privacy issues are at a very early stage [3]. Modern technologies and
tools, such as social networks, search engines, hacking packages, data mining and machine
learning tools, cause a lot of problems to individual privacy.

In general, it is very hard to find a clear definition or a global measurement on privacy.
The studies on privacy can be separated into two classes: content privacy and interaction
privacy. So far, the majority of research on privacy protection is conducted in the context

35



36 The Role of Information Theory in the Field of Big Data Privacy

of databases. The goal of a privacy preserving statistical database is to enable the user to
learn properties of the population while securing the personal information.

The practically dominant privacy protection strategy is the use of cryptography. One
way to protect data is to encrypt it in such a way that only the owner can decrypt it. The
task of machine learning is to find the dependency in the data. An idea proceeds: why not
to train the model on encrypted data? The problem with this approach is that when we
encrypt data, the dependencies in it are lost, because this is the aim of encryption - to change
the data so that the dependencies cannot be discovered. The degree of entropy in the data
after encryption prevents models from capturing these dependencies. Therefore, encrypting
data and training models on them do not work. The other disadvantage of cryptography for
privacy is the limited computing power of mobile devices for safe encryption and decryption
algorithms. That is why other methods for privacy preserving are required.

The main research categories of privacy are the data clustering and the theoretical frame-
works. Anonymization is a key component of data clustering. Anonymization is the pro-
cess of removing personal identifiers, both direct and indirect, that can lead to a person’s
identification. A person can be directly identified by name, address, zip code, telephone
number, photograph or image, or other unique personal characteristics. A person can be
indirectly identified if certain information is linked to other sources of information, including
workplace, job title, salary, zip code, or even the fact of having a specific diagnosis or condi-
tion. Data cannot be completely anonymous and useful. Generally speaking, the richer the
data, the more interesting and useful it is. This has led to the concepts of anonymization
and removal of personally identifiable information, by which it is hoped that sensitive parts
of the data can be suppressed to maintain the confidentiality of the records, while the rest
can be published and used for analysis.

The early approach in data clustering direction is the k-anonymity method (1998), then
its extension as l-diversity was suggested in 2007 and later the t-closeness method was
developed in 2010. In the second category of privacy frameworks the differential privacy
(DP) and its developments are included (Fig. 1). DP neutralizes linkage attacks, since it
is a property of the data access mechanism and is not related to the presence or absence of
auxiliary information available to the attacker.

Fig. 1. The categories of privacy study [3].

When comparing syntactic and DP approaches, one can recall the trade-off between
privacy and data utility. Finally, databases are passed on to provide certain benefits (for
example, research knowledge, such as the effectiveness of a medical procedure). On the one



M. Haroutunian and K. Mastoyan 37

hand, in order to maximize the usefulness of the data, all data can be published untouched,
thus completely breaching privacy. On the other hand, not publishing any of the data
will lead to maximum privacy, but this empty database will be useless. Therefore, the
organization should seriously consider both its interests in data exchange and the risks they
are willing to accept. An organization seeking to exchange sensitive data should consider
the logistics issues involved in the exchange process and carefully balance the confidentiality
and usefulness of the published data. An important step in one of these considerations is the
choice of syntactic and DP. Such problems can be solved by information - theoretic methods
and tools.

Although, there are some survey/review - type papers introducing the concept of privacy
protection in Big Data, none of them provide detailed discussion regarding privacy with
respect to Information Theory. In the current paper we introduce a summary of up-to-date
methods on privacy assurance from Information Theory standpoints.

The rest of the paper is structured as follows. The DP is discussed in section 2. The
developments of privacy based on information - theoretic methods are presented in section
3. The paper is summarized in section 4.

2. Differential Privacy

The DP framework was suggested in 2006 [4], that offers privacy protection in the sense
of Information Theory. In recent years this topic has attracted attention and has been
researched in literature. Main results, among many others, are surveyed in [5] - [8].

DP examines impossibility paradox of obtaining any information about a specific person
by studying useful information about a multitude of people. Suppose the trusted party
contains a set of sensitive personal data (eg email usage data, movie watching data, medical
records) and wants to provide global statistical information about it. Such a system is called
a statistical database. By providing such aggregated statistical information about the data,
it is possible to disclose some information about individuals.

DP ensures that data about individuals from such a database cannot be retrieved, no
matter what additional datasets or sources of information are available to the attacker.
Such a guarantee is achieved due to the fact that the owner of the database uses such a
mechanism (algorithm) for providing data, in which the presence or absence of information
about a person in the database will not significantly affect the result of the request to it.

DP is designed to maximize the accuracy of queries from statistical databases while min-
imizing the possibility of disclosing the anonymity of records. The problem of analyzing
sensitive data has a long history spanning many areas. As data about people become more
and more detailed and technology allows more and more of this data to be collected and ana-
lyzed, there is an increasing need for a reliable, mathematically rigorous definition of privacy,
as well as a class of algorithms that satisfy this definition. Various approaches to anonymiz-
ing data have failed when researchers have been able to identify personal information by
combining two or more separate statistical databases. DP is the basis for the formalization
of confidentiality in statistical databases and was introduced in order to protect against such
methods of disclosing anonymity (deanonymization).

DP allows users to protect and maintain privacy when their data is in a specific database,
just as they would be safe, if the data were not in some database. After the publication
of human data in the database, in accordance with the differentiated confidentiality, the
likelihood of violation of the confidentiality of people should not increase. That is, the



38 The Role of Information Theory in the Field of Big Data Privacy

degree of secrecy can be assessed by the likelihood of damage. This is one of the practical
definitions of privacy.

DP can provide extremely strong guarantees of user privacy, but it does not guarantee
unconditional relief from all damages. And it doesn’t provide privacy where it didn’t exist
before. In general, DP does not guarantee that what a person considers his secret will
remain secret. It simply ensures that participation in the survey is not disclosed by itself,
that participation does not reveal any of the characteristics included in the survey. It is
possible that the results of the survey may reflect statistical data about a person. Health
screening for early signs of illness can produce strong, even convincing results. The fact that
these findings are valid for humans does not imply a breach of confidentiality. The person
may not even participate in the survey (DP ensures that these results are equally likely,
regardless of whether the person participated in the survey or not).

It is desirable that DP be endowed with the following qualities: protection against arbi-
trary risks, automatic neutralization of linkage attacks, quantification of privacy loss.

DP is based on introducing randomness into data. To realize this, there are different
mechanisms, e.g. the laplace mechanism, the exponential mechanism, mechanisms via α
-nets, etc. Due to the fact that differential privacy is a probabilistic concept, any of its
methods necessarily has a random component. Some of them, like Laplace’s method, use
the addition of controlled noise to the function to be calculated. Laplace’s method adds
Laplace noise, i.e. the noise from the Laplace distribution.

DP works by adding statistical noise to data (or its inputs or outputs). Depending on
the location of the noise, DP is classified into two types: local DP and global DP (Fig. 2).

The most commonly used threat model in differential privacy is the global DP model.
The main component is a trusted data curator. Each source sends him his confidential data,
and it collects them in one place (for example, on a server). A repository is trusted if we
assume that it processes our sensitive data on its own, does not transfer it to anyone, and
cannot be compromised by anyone. In other words, we believe that a server with sensitive
data cannot be hacked. Within the central model, we usually add noise to query responses.
The advantage of this model is the ability to add the lowest possible noise value, thus
maintaining the maximum accuracy allowed by the principles of DP. The disadvantage of
the central model is that it requires a trusted store, and many of them are not. In fact, the
lack of trust in the consumer of the data is usually the main reason for using DP principles.

The local DP model allows you to get rid of the trusted data store: each data source
(or data owner) adds noise to their data before transferring it to the store. This means
that the storage will never contain sensitive information, implying there is no need for its
power of attorney. The local model of DP avoids the main problem of the central model: if
the data warehouse is compromised, then hackers will only have access to noisy data that
already meets the requirements of DP.

The local model is less accurate than the central one. In the local model, each source
independently adds noise to satisfy its own differential privacy conditions, so that the total
noise from all participants is much greater than the noise in the central model. Ultimately,
this approach is only justified for queries with a very persistent trend (signal). Apple, for
example, uses a local model to estimate the popularity of emoji, but the result is only useful
for the most popular emoji (where the trend is most pronounced). Typically, this model is
not used for more complex queries, such as those used by the US Census Bureau or machine
learning. The central and local models have both advantages and disadvantages, and now
the main effort is to get the best of them.



M. Haroutunian and K. Mastoyan 39

Fig. 2 Global Differential Privacy and Local Differential Privacy

3. Review of Information Theory - based Results in Privacy

The first problem is to have a metric to compare various ways of achieving anonymity. The
initial focus on analyzing the anonymity of messaging through mixed - based anonymity
systems in which all network communication is available for the attacker is given in [9]. An
information - theoretic metric based on the idea of anonymity probability distributions is
introduced. In the same paper it is demonstrated that if maximum route length in the mix
system exists, it is known to the attacker and can be used to extract additional information.
It means that more advanced probabilistic metrics of anonymity are needed.

An analytical measure of anonymity of routs in eavesdropped networks is proposed in [10]
using the information-theoretic equivocation. Cryptographic techniques prevent analysis of
packet content, however, information can be gained by analyzing the correlation of trans-
mission schedules of multiple nodes, as the packet timing information is easy to obtain in
wireless networks. For anonymity it is necessary for the routes to be undetectable using the
correlation across the transmission schedules, which results in a tradeoff between anonymity
and network performance. For this purpose a quantifiable metric is defined in [10] using
the uncertainty in networking information. The key result shows the equivalence between
anonymity - performance tradeoff and information - theoretic rate distortion.

It is often important to allow researchers to analyze data without compromising the
privacy of individuals or leaking confidential information outside the organization. In [11] it
is shown that sparse regression for high dimensional data can be carried out directly on a
compressed form of the data, in a manner to guard privacy in information - theoretic sense.
The suggested compression reduces the number of data records exponentially preserving
the number of input variables. These compressed data can then be made available for



40 The Role of Information Theory in the Field of Big Data Privacy

statistical analyses with the same accuracy as the original data. In this case the original
data are not recoverable from the compressed data, and the algorithms run faster, requiring
fewer resources. The privacy (the problem of recovering the uncompressed data from the
compressed one) is evaluated in information - theoretic terms by bounding the average
mutual information, which is connected with the problem of computing the channel capacity
of certain systems.

A new privacy measure in terms of Information Theory, similar to t-closeness is defined
in [12], which can be achieved by the postrandomization method in the descrete case and by
noise addition in the general case. The privacy criterion here is an average measure over a
divergence, and the privacy - distortion problem is strongly related to the rate - distortion
problem in the field of Information Theory, namely, the problem of lossy compression of
source data subject to a distortion criterion.

A new measure for privacy of votes is proposed in [13], that relies on the notion of
entropy. Entropy is a natural choice to measure privacy in an information - theoretic setting,
and authors demonstrate how different formulations of conditional entropy answer different
questions about vote privacy. A theorem has been established that enables accurate analysis
of privacy offered by complex cryptographic voting protocols. Connections between two
existing privacy notions for votes have been established.

The study of DP from a rate - distortion perspective has been initiated in [14]. Rate -
distortion is applicable when the goal of the data collector is to publish an approximation
of the data itself. The case when the data collector is not trusted is considered, which leads
to using the local DP as a privacy measure. A robust rate-distortion setting is considered,
in which the source distribution is unknown, but comes from some class. The goal is to
look for a locally differentially private channel, that achieves minimum privacy risks while
guaranteeing distortion of the given level.

In [15] the relation between three different notions of privacy: identifiability, differential
privacy and mutual - information privacy is investigated. Identifiability guarantees indistin-
guishability between probabilities, DP guarantees limited additional disclosures, and mutual
information is the information - theoretic notion. Under a unified privacy - distortion frame-
work, where the distortion is the Hamming distance between the input and output databases,
some connections between these three privacy notions have been established.

Guaranteeing a tight bound on privacy risk often incurs a significant penalty in terms of
the usefulness of the published result. This privacy-utility tradeoff is studied in [16] in the
context of publishing a differentially private approximation of the full data set and measure
utility via a distortion measure.

4. Conclusion

In this article a general outlook on the current methods for estimating privacy of databases
from Information Theory perspectives is provided. A series of publications devoted to var-
ious problems of privacy solved by information - theoretic tools and methods is analyzed.
Research has shown that information-theoretic methods are effective for a wide range of
tasks ranging from anonymity to differential privacy.



M. Haroutunian and K. Mastoyan 41

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4 2 The Role of Information Theory in the Field of Big Data Privacy

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îêàçàëàñü íàèáîëåå ìíîãîîáåùàþùåé. Êîìïðîìèññ ìåæäó êîíôèäåíöèàëü-
íîñòüþ è ïîëåçíîñòüþ îïóáëèêîâàííûõ äàííûõ, à òàêæå äðóãèå ïðîáëåìû,
òàêèå êàê íàëè÷èå ìåòðèê äëÿ ñðàâíåíèÿ ðàçëè÷íûõ ñïîñîáîâ äîñòèæåíèÿ
àíîíèìíîñòè, îòíîñÿòñÿ ê ñôåðå òåîðèè èíôîðìàöèè. Íåñìîòðÿ íà
íàëè÷èå â ëèòåðàòóðå ðÿäà îáçîðíûõ ñòàòåé, âîçìîæíîñòÿì ìåòîäîâ òåîðèè
èíôîðìàöèè íå óäåëÿëîñü äîëæíîãî âíèìàíèÿ. Â ýòîé ñòàòüå ìû äàåì
îáçîð íîâåéøèõ ìåòîäîâ òåîðèè èíôîðìàöèè äëÿ îáåñïå÷åíèÿ êîíôèäåíöèàëü-
íîñòè. Àíàëèçèðóåòñÿ ñåðèÿ ïóáëèêàöèé, ïîñâÿùåííûõ ðàçëè÷íûì ïðîáëåìàì
êîíôèäåíöèàëüíîñòè, ðåøàåìûì ñ ïîìîùüþ èíñòðóìåíòîâ è ìåòîäîâ òåîðèè
èíôîðìàöèè. Èññëåäîâàíèÿ ïîêàçàëè, ÷òî òåîðåòèêî-èíôîðìàöèîííûå ìåòîäû
ýôôåêòèâíû äëÿ øèðîêîãî êðóãà çàäà÷, îò àíîíèìíîñòè äî äèôôåðåíöèàëüíîé
êîíôèäåíöèàëüíîñòè.

Êëþ÷åâûå ñëîâà: Áîëüøèå äàííûå, àíîíèìèçàöèÿ, äèôôåðåíöèàëüíàÿ
êîíôèäåíöèàëüíîñòü, ýíòðîïèÿ, âçàèìíàÿ èíôîðìàöèÿ, èñêàæåíèå


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