Secure Two-party Agglomerative Hierarchical Clustering Construction
∗
Mona Hamidi
1
, Mina Sheikhalishahi
2
and Fabio Martinelli
2
1
Dipartimento Ingegneria dell Informazione e Scienze Matematiche, Universita di Siena, Siena, Italy
2
Istituto di Informatica e Telematica, Consiglio Nazionale delle Ricerche, Pisa, Italy
Keywords:
Privacy, Hierarchical Clustering, Data Sharing, Secure Two-party Computation, Distributed Clustering.
Abstract:
This paper presents a framework for secure two-party agglomerative hierarchical clustering construction over
partitioned data. It is assumed that data is distributed between two parties horizontally, such that for mutual
benefits both parties are willing to identify clusters on their data as a whole, but for privacy restrictions, they
avoid to share their datasets. To this end, in this study, we propose general algorithms based on secure scalar
product and secure hamming distance computation to securely compute the desired criteria in constructing
clusters’ scheme. The proposed approach covers all possible secure agglomerative hierarchical clustering
construction when data is distributed between two parties, including both numerical and categorical data.
1 INTRODUCTION
Facing the new challenges brought by a continuous
evolving Information Technologies (IT) market, large
companies and small-to-medium enterprises found in
Information Sharing a valid instrument to improve
their key performance indexes. Sharing data with
partners, authorities for data collection and even com-
petitors, may help in inferring additional intelligence
through collaborative information analysis (Martinelli
et al., 2016) (Sheikhalishahi and Martinelli, 2017b).
Such an intelligence could be exploited to improve
revenues, e.g. through best practice sharing, mar-
ket basket analysis (Oliveira and Za
¨
ıane, 2002), or
prevent loss coming from brand-new potential cyber-
threats (Faiella et al., 2017). Other applications in-
clude analysis of medical data, provided by several
hospitals and health centers for statistical analysis
on patient records, useful, for example, to shape the
causes and symptoms related to a new pathology (Ar-
toisenet et al., 2013).
Information sharing, however, independently
from the final goal, leads to issues and drawbacks
which must be addressed. These issues are mainly re-
lated to the information privacy. Shared information
might be sensitive, potentially harming the privacy of
physical people, such as employee records for busi-
ness applications, or patient records for medical ones
(Martinelli et al., 2016). Hence, the most desirable
∗
This work has been supported by the H2020 EU funded
project C3ISP [GA #700294].
strategy is the one which enables data sharing in a se-
cure environment, such that it preserves the individual
privacy requirement while at the same time the data
are still practically useful for analysis.
Clustering is a very well-known tool in unsuper-
vised data analysis, which has been the focus of sig-
nificant researches in different studies, spanning from
information retrieval, text mining, data exploration, to
medical diagnosis (Berkhin, 2006). Clustering refers
to the process of partitioning a set of data points into
groups, in a way that the elements in the same group
are more similar to each other rather than to the ones
in other groups.
The problem of data clustering becomes challeng-
ing when data is distributed between two (or more)
parties and for privacy concerns the data holders
refuse to publish their own dataset, but still they are
interested to shape more accurate clusters, identified
on richer sets of data. For a motivating example sup-
pose that a hospital and a health center hold different
datasets with the same set of attributes. Both cen-
ters are interested to shape clusters on the whole data,
which brings the benefits of identifying the trends and
patterns of diseases on a larger set of samples. How
would it be possible to learn about clusters’ shapes
without disclosing patients’ records ?
To this end, in this study, we address the problem
of securely constructing hierarchical clustering algo-
rithms between two parties. The participating par-
ties are willing to model an agglomerative hierarchi-
cal clustering on the whole dataset without revealing
432
Hamidi, M., Sheikhalishahi, M. and Martinelli, F.
Secure Two-party Agglomerative Hierarchical Clustering Constr uction.
DOI: 10.5220/0006659704320437
In Proceedings of the 4th International Conference on Information Systems Security and Privacy (ICISSP 2018), pages 432-437
ISBN: 978-989-758-282-0
Copyright © 2018 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved

their own dataset. Both scenarios of data being de-
scribed in numerical and categorical attributes is ad-
dressed in this study. For each scenario, we propose
secure two-party computation protocols which can be
exploited as a general tool to construct securely all
possible agglomerative hierarchical clustering algo-
rithms between two parties.
At the end, each data holder finds the structure of
hierarchical clusters on the whole data, without know-
ing the records of the other party. In this study, as in
general cases, it is assumed that clustering on joint
datasets produces better result rather than clustering
on individual dataset.
The contribution of this paper can be summarized
as the following. A framework is proposed which
serves as a tool for two parties to detect the clus-
ter structures on the whole dataset, in terms of ag-
glomerative hierarchical clustering, without revealing
their data, in two different scenarios of data being de-
scribed either numerically or categorically.
Two secure computation protocols, named secure
scalar product and secure hamming distance proto-
cols, are exploited to propose new algorithms such
that each party is able to find the closest points (or
clusters) for agglomeration.
The rest of the paper is structured as follows. Re-
lated work, on the two concepts of privacy preserving
data clustering and secure hierarchical clustering con-
struction, is presented in Section 2. Section 3 presents
some preliminary notations exploited in this study, in-
cluding agglomerative hierarchical clustering, secure
scalar product, and secure hamming distance compu-
tation. Section 4 presents the system model in three
subsections: horizontal distributed data framework,
secure Euclidean distance computation, and finally
agglomerative hierarchical clustering. Finally Section
5 briefly concludes proposing future research direc-
tions.
2 RELATED WORK
The problem of privacy preserving data clustering is
generally addressed for the specific case of k-means
clustering, either when data is distributed between
two parties (Bunn and Ostrovsky, 2007) or more than
two parties (Jha et al., 2005).
In (Su et al., 2007), Document clustering has been
introduced, and a cryptography based framework has
been proposed to do the privacy preserving document
clustering among the two parties. It is assumed that
each has her own private documents, and wants to col-
laboratively execute agglomerative document cluster-
ing without disclosing their private content. The main
idea is to find which documents have many words
in common, and place the documents with the most
words in common into the same groups and then build
the hierarchy bottom up by iteratively computing the
similarity between all pairs of clusters and then merg-
ing the most similar pair. In the proposed approach,
differently from our technique, the problem when data
are described through numerical attributes, is not ad-
dressed. In (De and Tripathy, 2013) a secure hierar-
chical clustering approach over vertically partitioned
data is provided which increases the accuracy of the
clusters over the existing approaches. However, in our
study, we address the problem of hierarchical cluster-
ing construction when data is distributed horizontally.
In (Sheikhalishahi and Martinelli, 2017a), the
problem of secure divisive hierarchical clustering
construction is addressed when data is distributed be-
tween two parties horizontally and vertically. How-
ever, the criteria for divisive hierarchical clustering,
discussed in this study, is different from agglomera-
tive hierarchical clustering algorithms.
In all the aforementioned studies, differently from
our approach, or data is distributed between more than
two parties, or the problem is addressed for divisive
hierarchical clustering, or the problem is addressed
for vertical partitioned data. To the best of our knowl-
edge the problem of secure two-party data clustering
is a topic which is required to be explored deeper.
3 PRELIMINARY NOTATIONS
In this section, we present some background knowl-
edge which are exploited in our proposed framework.
3.1 Hierarchical Clustering
Clustering algorithm partitions a set of objects into
smaller groups such that all objects within the same
group (cluster) are more similar or close to each other
rather than the objects in different clusters (Jagan-
nathan and Wright, 2005). There have been a large
number of algorithms developed to solve the various
formulations of data clustering. Hierarchical clus-
tering, as a specific form of data clustering, gener-
ates a hierarchical decomposition of the given set of
data objects, which can be either agglomerative or di-
visive based on how the hierarchical decomposition
is formed. The agglomerative approach successively
merges the objects or clusters which are close to one
another, until all of the clusters are merged into one
or until a termination condition holds. The divisive
approach starts with all of the objects in the same
cluster. In each successive iteration, a cluster is split
Secure Two-party Agglomerative Hierarchical Clustering Construction
433

up into smaller clusters, until finally each object is
in one cluster, or until a termination condition holds
(De and Tripathy, 2013). Hierarchical methods, dif-
ferently from many other algorithms, have the ability
to both discover clusters of arbitrary shape and deal
with different data types.
3.2 Secure Scalar Product
Scalar product is a useful technique in data mining
such that many data mining algorithms can be reduced
to computing the scalar product.
For secure two-party scalar product computation, as-
sume that two parties, named Alice and Bob, each
has a vector of cardinality n, e.g X = (x
1
, .. . , x
n
) and
Y = (y
1
, . . . , y
n
), respectively. Then, both are inter-
ested in securely obtaining the scalar product of the
two vectors, i.e.
∑
n
i=1
x
i
· y
i
, without revealing their
own vectors. There are many different solutions with
varying degrees of accuracy, communication cost and
security. The solution which we have applied is the
one proposed in (Vaidya and Clifton, 2002), in which
the key is to use linear combinations of random num-
bers to make vector elements, and then do some com-
putations to eliminate the effect of random numbers
from the result. Algorithm 1 details the process.
3.3 Secure Hamming Distance
Computation
In the case that Alice’s and Bob’s vectors are de-
scribed through categorical attributes, the distance
between vectors is computed with the use of secure
hamming distance. The secure communication be-
tween two parties for obtaining hamming distance is
on the base of oblivious transfer. A 1-out-2 oblivi-
ous transfer, denoted by OT
2
1
, is a two party proto-
col where one party (the sender) inputs n-bit strings
X
1
, X
2
∈ {0, 1}
n
, and the other party (the receiver) in-
puts a bit y. At the end of the protocol, the receiver
obtains X
b
but learns nothing about X
1−b
, while the
sender learns nothing about b (Bringer et al., 2014).
In (Bringer et al., 2013), the secure computation
of the Hamming distance has been presented based
on oblivious transfer. It is assumed that two parties ,
say Alice and Bob, hold bit strings of the same length
n, X = (x
1
, . . . , x
n
) and Y = (y
1
, . . . , y
n
), respectively.
Both are interested in jointly computing the Hamming
Distance between X and Y , i.e. D
H
(X,Y ) =
∑
n
i=1
(x
i
⊕
y
i
) without revealing X and Y . Algorithm 2 details the
process.
Algorithm 1: Sec.Scalar(): Secure Scalar Product.
Data: Alice and Bob have vectors X = (x
1
, . . . ,
x
n
) and Y = (y
1
, . . . , y
n
), respectively.
Result: Alice and Bob obtain securely S = X ·Y
1 initialization;
2 Alice and Bob together decide on random n ×
n
2
matrix C
3 for Alice do
4 Alice generates a random vector R of
cardinality
n
2
, R = (r
1
, . . . , r
n
2
)
5 Alice generates the n × 1 matrix Z, where
Z = C × R
6 Alice generates X
1
= X + Z
7 Alice sends X
1
to Bob
8 end
9 for Bob do
10 Bob generates the scalar product
S
1
=
∑
n
i=1
x
1
i
· y
i
11 Bob also generates the n × 1 matrix, where
Y
1
= C
T
×Y
12 Bob sends both S
1
and Y
1
to Alice
13 end
14 for Alice do
15 Alice generates S
2
=
∑
n
i=1
Y
1
i
· R
i
16 Alice generates the scalar product
S = S
1
− S
2
17 Alice reports the scalar product S to Bob
18 end
4 SYSTEM MODEL
In this section we explain in detail 1) what we mean
by horizontal distributed data, 2) how Euclidean dis-
tance can be computed securely between two vectors
owned by two different parties, and 3) how the pro-
posed algorithms can be exploited to construct any
agglomerative hierarchical clustering securely.
4.1 Horizontal Distributed Data
Suppose that two data holders are interested in detect-
ing the structure of clusters (through agglomerative
hierarchical clustering) on their datasets as a whole.
However, for privacy concerns, they are not willing to
publish or share the main dataset. As mentioned be-
fore, it is assumed that clustering on both datasets as
a whole (as in general cases) produces better results
comparing to clustering on individual dataset. In this
study it is assumed that data is distributed horizontally
between two parties. This means that each data holder
has information about all the features but for differ-
ent collection of objects. More precisely, let A =
ICISSP 2018 - 4th International Conference on Information Systems Security and Privacy
434

Algorithm 2: Sec.Hamming(): Secure Hamming
Distance Computation.
Data: Alice and Bob have n-bit strings
X = (x
1
, . . . , x
n
) and Y = (y
1
, . . . , y
n
),
respectively.
Result: Alice and Bob obtain securely the
hamming distance between X and Y .
1 initialization;
2 Alice generates n random values
(r
1
, . . . , r
n
) ∈
R
Z
n+1
and computes R =
∑
n
i=1
r
i
3 for each i = 1, . . . , n, Alice and Bob engage in
a OT
2
1
do
4 Alice acts as the sender and Bob as the
reciever
5 Bob’s selection bit is y
i
6 Alice’s input is (r
i
+ x
i
, r
i
+ ¯x
i
) where x is a
bit value and ¯x denotes 1 − x
7 The output obtained by Bob is
consequently t
i
= r
i
+ (x
i
⊕ y
i
)
8 end
9 Bob computes T =
∑
n
i=1
t
i
and sends T to Alice
10 Alice computes and outputs T − R
Figure 1: Hierarchical Clustering over Horizontal Parti-
tioned Data.
{A
1
, A
2
, . . . , A
n
} be the set of n attributes all used to
express each record of data. Therefore, each record is
an n dimensional vector X
i
= (v
i
1
, v
i
2
, . . . , v
i
n
), where
v
i
j
∈ A
j
.
Figure 1 depicts a higher level representation of
hierarchical clustering construction on horizontal dis-
tributed framework. Alice and Bob, holding respec-
tively datasets D
a
and D
b
, are the two parties inter-
ested in constructing hierarchical clustering on D
a
∪
D
b
, without knowing the data information of the other
party. As it can be observed, the two tables are de-
scribed with the same set of attributes, but on dif-
ferent objects. To discover the structure of the de-
sired agglomerative clustering algorithm, Alice and
Bob communicate through secure computation pro-
tocols. Depending on if data in both sides is de-
scribed through numerical or categorical attributes,
secure Euclidean or Hamming distance computation
algorithms are exploited inside communication algo-
rithms, respectively.
4.2 Secure Euclidean Distance
Computation
Suppose Alice and Bob own vectors X = (x
1
, . . . , x
n
)
and Y = (y
1
, . . . , y
n
), respectively. We assume that
n > 2, and x
i
, y
i
∈ R for all i, i.e. both vectors con-
tain numerical elements. Both participated parties
are interested in obtaining securely the result of Eu-
clidean distance between X and Y , i.e. D
E
(X,Y ) =
(
∑
n
i=1
(x
i
− y
i
)
2
)
1
2
. Algorithm 3 details the process of
secure Euclidean distance computation with the use
of secure scalar product (presented in Algorithm 4).
Algorithm 3: Sec.Euclidean(): Secure Euclidean
Distance Computation.
Data: Alice and Bob have numerical vectors
X = (x
1
, . . . , x
n
) and Y = (y
1
, . . . , y
n
),
n > 2, respectively.
Result: Alice and Bob compute securely
Euclidean Distance of X and Y
D(X,Y ) = (
∑
n
i=1
(x
i
− y
i
)
2
)
1
2
=
(
∑
n
i=1
(x
i
)
2
+ (y
i
)
2
− 2(x
i
· y
i
))
1
2
1 initialization;
2 Alice reports X
2
=
∑
n
i=1
x
i
2
3 Bob reports Y
2
=
∑
n
i=1
y
i
2
4 X ·Y ← Sec.Scalar(X,Y)
5 return D
E
(X,Y ) = (X
2
+Y
2
− 2X ·Y )
1
2
Theorem 1. Euclidean distance computation as pro-
posed in Algorithm 3 does not reveal the information
of each party’s data, but the distance.
Proof. Due to the fact that it is assumed n > 2, hence
X
2
and Y
2
will not reveal something about the specific
amount of each component, but the squared result of
their vectors. Moreover, X · Y is computed through
secure scalar product protocol proven to be secure in
(Clifton et al., 2002) (Vaidya and Clifton, 2002).
4.3 Secure Agglomerative Hierarchical
Clustering
In the beginning of the agglomerative hierarchical
clustering process, each element is located in a clus-
ter by its own. Afterwards, the set of N objects
to be clustered are grouped into successively fewer
than N sets, arriving eventually at a single set con-
taining all N objects (Day and Edelsbrunner, 1984).
According to different distance measures between
Secure Two-party Agglomerative Hierarchical Clustering Construction
435

Table 1: Specification of Hierarchical Clustering Methods.
Hierarchical Clustering Lance-Williams Distance Metric
Single Link
α
i
= 0.5
β = 0
γ = −0.5
d(i ∪ j, k) =
1
2
d(i, k)
+
1
2
d( j, k)−
1
2
|d(i,k) − d( j, k)|
Complete Link
α
i
= 0.5
β = 0
γ = 0.5
d(i ∪ j, k) =
1
2
d(i, k)
+
1
2
d( j, k)+
1
2
|d(i,k) − d( j, k)|
Group average(UPGMA)
α
i
=
|i|
|i|+| j|
β = 0
γ = 0
d(i ∪ j, k) =
|i|
|i|+| j|
d(i, k) +
|i|
|i|+| j|
d( j, k)
Weighted group average
(WPGMA)
α
i
= 0.5
β = 0
γ = 0
d(i ∪ j, k) =
1
2
d(i, k) +
1
2
d( j, k)
Median method
α
i
= 0.5
β = −0.25
γ = 0
d(i ∪ j, k) =
1
2
d(i, k)
+
1
2
d( j, k)−
1
4
d(i, j)
Centroid method
α
i
=
|i|
|i|+| j|
β = −
|i|| j|
(|i|+| j|)
2
γ = 0
d(i ∪ j, k) =
|i|
|i|+| j|
d(i, k)
+
|i|
|i|+| j|
d( j, k)−
|i|| j|
(|i|+| j|)
2
d(i, j)
Ward method
α
i
=
|i|+|k|
|i|+| j|+|k|
β = −
|k|
|i|+| j|+|k|
γ = 0
d(i ∪ j, k) =
|i|+|k|
|i|+| j|+|k|
d(i, k)+
|i|+|k|
|i|+| j|+|k|
d( j, k)−
|k|
|i|+| j|+|k|
d(i, j)
clusters, all agglomerative hierarchical methods have
been divided into seven methods with the use of a
formula named Lance-Williams (Murtagh and Contr-
eras, 2012). These seven categories of agglomerative
hierarchical clustering algorithms are named single
link, complete link, average link, weighted average
link, Ward’s method, centroid method, and the median
(Gan et al., 2007).
More precisely, in agglomerative hierarchical
clustering algorithms, the Lance-Williams formula is
used to calculate the dissimilarity between a cluster
and the other cluster formed by merging two other
clusters (Gan et al., 2007). Formally, if objects i and
j are agglomerated into cluster i ∪ j, then the new dis-
similarity between the cluster and all other objects of
cluster k is required to be specified as the following:
d(i ∪ j, k) =α
i
d(i, k) + α
j
d( j, k)
+ β d(i, j) + γ |d(i, k) − d( j, k)| (1)
where α
i
, α
j
, β and γ defines the agglomerative pa-
rameters (Murtagh and Contreras, 2012). The value
for each of these coefficients in different algorithms
has been listed in Table 1. Thence, if Alice and Bob
are able to find these different distance metrics se-
curely, they are both able to construct all possible ag-
glomerative hierarchical clustering algorithms with-
out revealing their own data.
As it can be observed from Table 1, it is enough
that the two participating parties obtain securely the
distance of each pair of elements. This means that it
is required for them to construct securely the dissim-
ilarity matrix on whole elements without disclosing
the data.
Algorithm 4 presents how Alice and Bob are able
to construct securely the dissimilarity matrix on all
records with the use of secure Hamming and Eu-
clidean distance computations, presented respectively
in Algorithms 2 and 3.
Algorithm 4: Sec.Matrix(): Secure Distance Matrix
Computation.
Data: Alice and Bob have information of
records X
1
, . . . , X
k
and X
k+1
, . . . , X
N
,
respectively.
Result: Secure distance matrix computation
1 initialization;
2 for 1 ≤ t,t
0
≤ k do
3 if X
t
, X
t
0
are numerical then
4 Alice reports (M(t,t
0
) ← D
E
(X
t
, X
t
0
))
5 else
6 Alice reports (M(t,t
0
) ← D
H
(X
t
, X
t
0
))
7 end
8 end
9 for k + 1 ≤ s, s
0
≤ N do
10 if X
s
, X
s
0
are numerical then
11 Bob reports (M(s, s
0
) ← D
E
(X
s
, X
s
0
))
12 else
13 Bob reports (M(s, s
0
) ← D
H
(X
s
, X
s
0
))
14 end
15 end
16 for 1 ≤ t ≤ k do
17 for k + 1 ≤ s ≤ N do
18 if X
t
, X
s
are numerical then
19 M(t,s) ← Sec.Euclidean(X
t
, X
s
)
20 else
21 M(t,s) ← Sec.Hamming(X
t
, X
s
)
22 end
23 end
24 end
25 for 1 ≤ t, s ≤ N do
26 return M(t, s)
27 end
Theorem 2. Algorithm 4 reveals no information to
other party except the distances of all elements.
Proof. The proof is straightforward from secure com-
putation of Euclidean and Hamming distance proven
to be secure in Theorem 1 and in (Bringer et al.,
2013), respectively.
5 CONCLUSION
In this work we proposed a framework which can be
exploited for two parties to construct any agglomer-
ative hierarchical clustering algorithm on their data
as a whole, without revealing the original datasets.
To this end, secure two-party computation algorithms
are proposed to obtain the required criteria for de-
tecting the clusters on the whole data. Two scenar-
ICISSP 2018 - 4th International Conference on Information Systems Security and Privacy
436

ios of data being described numerically and categor-
ically have been addressed. In future direction we
plan to analyze the efficiency of proposed approach
on benchmark dataset clustering to evaluate commu-
nication cost in reality. Moreover we plan to address
the problem when data is distributed among more than
two parties either horizontally or vertically.
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