A Game Theory based Repeated Rational Secret Sharing Scheme for
Privacy Preserving Distributed Data Mining
Nirali R. Nanavati and Devesh C. Jinwala
Sardar Vallabhbhai National Institute of Technology, Surat, India
Keywords: Privacy, Game Theory, Secure Multiparty Computation, Rational Secret Sharing, Distributed Data Mining.
Abstract: Collaborative data mining has become very useful today with the immense increase in the amount of data
collected and the increase in competition. This in turn increases the need to preserve the participants’
privacy. There have been a number of approaches proposed that use Secret Sharing for privacy preservation
for Secure Multiparty Computation (SMC) in different setups and applications. The different multiparty
scenarios may have parties that are semi-honest, rational or malicious. A number of approaches have been
proposed for semi honest parties in this setup. The problem however is that in reality we have to deal with
parties that act in their self-interest and are rational. These rational parties may try and attain maximum gain
without disrupting the protocol. Also these parties if cautioned would correct themselves to have maximum
individual gain in the future. Thus we propose a new practical game theoretic approach with three novel
punishment policies with the primary advantage that it avoids the use of expensive techniques like
homomorphic encryption. Our proposed approach is applicable to the secret sharing scheme among rational
parties in distributed data mining. We have analysed theoretically the proposed novel punishment policies
for this approach. We have also empirically evaluated and implemented our scheme using Java. We
compare the punishment policies proposed in terms of the number of rounds required to attain the Nash
equilibrium with eventually no bad rational nodes with different percentage of initial bad nodes.
1 INTRODUCTION
The utility and sharing of huge amounts of data has
become possible today with the development of
network, data collection and storage technologies.
The huge amount of extracted knowledge among
different parties does have the issue of loss of
privacy of the participants. Hence a number of
algorithms have been proposed to resolve the issue
of privacy in distributed data mining. These
algorithms are actually based on the concept of
coopetation(Pedersen et al., 2007) among the parties.
[m,m] Secret Sharing for the PPDARM semi
honest model in a mesh topology which is our focus
of study has been proposed recently by (Nanavati
and Jinwala, 2012); (Ge et al., 2010). However in a
realistic formulation of PPDARM these parties
would be rational. According to (Abraham et al.,
2006); (Maleka et al., 2008); (Miyaji and Rahman,
2011) these rational agents will have an inclination
to not send their shares as each of them would first
prefer getting the secret and secondly prefer that
fewer of the other agents that get it, the better.
Hence we propose an extension to the [m,m] Secret
Sharing scheme for Privacy preservation in
Distributed Data Mining (PPDDM) for such rational
agents.
(Kargupta et al., 2007) does explain multiparty
PPDDM as a game but it does not cater to Secret
sharing and only explains the Secure sum protocol in
a ring topology unlike the mesh topology used by us.
(Maleka et al., 2008) introduces the concept of
Repeated Rational [m,n] Secret Sharing but cannot
be used for SMC. It is a model with mediators unlike
our model which is without mediators. (Miyaji and
Rahman, 2011) is a recent work that explains the
application of game theory but only to the set
intersection protocol in PPDDM. (Abraham et al.,
2006) has a novel version of Rational Secret Sharing
that explains collusion and cheap talk but does not
consider secret sharing as a repeated game and does
not give details of the punishment strategy or the
application to PPDDM.
Hence we introduce a novel algorithm to the best
of our knowledge which models Secret Sharing
(Shamir and Adi, 1979) in PPDARM as a Repeated
512
R. Nanavati N. and C. Jinwala D..
A Game Theory based Repeated Rational Secret Sharing Scheme for Privacy Preserving Distributed Data Mining.
DOI: 10.5220/0004525205120517
In Proceedings of the 10th International Conference on Security and Cryptography (SECRYPT-2013), pages 512-517
ISBN: 978-989-8565-73-0
Copyright
c
2013 SCITEPRESS (Science and Technology Publications, Lda.)
Rational Secret Sharing scheme using cheap talk. It
is modeled for rational parties without mediators.
This model is applicable to all the recent work done
in PPDARM and distributed clustering using [m,m]
secret sharing(Nanavati and Jinwala, 2012); (Ge et
al., 2010)(Patel, Garasia and Jinwala, 2012) without
mediators to compute the secure sum in a mesh
topology unlike most other [m,n] – threshold secret
sharing algorithms that aim to share a secret and not
perform secure sum computation.
The proposed model is advantageous as it avoids
the use of techniques like homomorphic encryption
and zero knowledge proofs that incur a high cost.
Our punishment policies do not aim at removing the
players but aim at getting the maximum possible
participation in the game.
2 PROBLEM FORMULATION
Our problem setup involves a co-opetative setup of
vertically or horizontally partitioned databases
where PPDARM is required (depending on the
application (Nanavati and Jinwala, 2012); (Ge et al.,
2010); (Nanavati and Jinwala, 2013) with ‘p’
rational parties collaborating to find the secure
global sum in their data privately. This problem
extends the [m,m] threshold scheme proposed for
rational agents.
We model secret sharing as a game among
rational agents represented by Γ(m,m). We also
consider it as a repeated game (Maleka et al., 2008)
where the same set of players come to play the same
game repeatedly. Our model is a model without
mediators unlike(Maleka et al., 2008). The players
are connected together in a mesh topology.We
assume that the players are normally concerned
about their future utilities. Among the three main
attacks possible in [m,m] secret sharing(Nanavati
and Jinwala, 2013), we try to resolve the two attacks
by a rational adversary who is not disrupting the
protocol by sending wrong shares but is trying to
selfishly get his own gain by witholding his shares
or sending them only to the collusion. Our
punishment policies resolve these attacks and
motivate the player to abide by the protocol. The
attack where wrong shares are send by malicious
parties can be resolved using the Verifiable Secret
Sharing Scheme.
3 GAME THEORY AND NASH
EQUILIBRIUM
Game Theory has today developed into an umbrella
term encompassing various scenarios in the real
world where the individuals want maximum benefit
after taking into account the actions of the other
parties involved in the setup. ‘Cheap talk’ in a game
theoretic framework refers to the direct and costless
communication among players
In game theory, the Nash equilibrium is
a solution concept of a game that comprises of two
or more players, and none of them has anything to
gain by changing only his own strategy alone. If
each player has chosen a strategy and no player can
benefit by changing his or her strategy while the
other players keep theirs unchanged, then the current
set of strategy choices and the corresponding payoffs
constitute a Nash equilibrium.
4 PROPOSED ALGORITHM
The main aim of PPDDM is to be able to compute
the global function on all the parties and hence they
want to have maximum participation. For the [m,m]
Repeated Secret Sharing game proposed by us; we
assume that all rational parties broadcast their shares
and then their sum of shares simultaneously to know
the value of the global sum
S

ij
of the secret
values at each party P
i
. It is an extension of the
algorithm based on Shamir’s secret sharing
technique(Shamir and Adi, 1979); (Ge et al., 2010).
We improve on the punishment strategies of
(Kargupta et al., 2007); (Maleka et al., 2008) as
mentioned in (Nanavati and Jinwala, 2013) using
[m,m] Secret Sharing so that all parties will have the
maximum utility and will attain the Nash
equilibrium state.
Considering the p rational partitions (where 0 < i
p) and each transaction has a subset of ‘N’ items.
Consider D is the no. of defaulters; y
i
is the number
of rounds for which the defaulter ‘i’ is ousted in
Policy 1 and incr
i
is the increment added to y
i
.
Consider c
i
is the maximum number of chances
given to the defaulter to enter the setup, n
i
is the no.
of rounds that the party i has defaulted, rating
i
is the
current rating of the i
th
player in policy 3 and life
i
is
the number of lives left with the i
th
player. The
algorithm proposed by us is given below in Figure 1.
The proper administration of the punishment
policies listed in section 4.1 will be monitored by a
moderator who is not a part of the protocol. The
AGameTheorybasedRepeatedRationalSecretSharingSchemeforPrivacyPreservingDistributedDataMining
513
moderator will just ensure that the parties are ousted
properly and are given their due punishment.
Input: p, N, n
i
, policy used, y
i
, incr
i,
c, rating
i
Output: Secure Sum using Repeated Rational
Secret Sharing for [m,m] scheme
1:for each transaction j = 1 to N do
2: for each party P
i
, (i = 1 to p)do
3: each party computes the share of
other party P
i
, where share(S
ij
,P
i
)=
q
i
(x)(random polynomial)
4: for t = 1 to p do
5: send share(S
ij
, P
i
) to party P
t
6: receive the shares share(S
ij
, P
t
)
7: end for
8:compute Sum(x
i
)=q
1
(x
i
)+
q
2
(x
i
)..+q
n
(x
i
)
9: for t = 1 to p do
10: send Sum(x
i
) to party P
t
11: If a party does not send a sum of
shares; initiate cheap talk.
12: If policy 1 is chosen by the
setup
13: If (n
i
= 0)
14: defaulter(d) is removed from the
setup for ‘y
i
’ rounds; n
i
++;
15: Goto Step 3 for ‘p – D’ parties
16: Else
17: defaulter(d) is removed from the
setup for ‘g
i
’ rounds where g
i
= y
i
+
incr
i
; y
i
=g
i;
n
i
++;
18: if n
i
=c; the party P
i
is
permanently ousted
19:
Goto Step 3 for ‘p – D’ parties
20: If policy 2 is chosen by the
setup
21: If (n
i
= 0 )
22: defaulter is removed for 1 round
until a limit of ‘c’ rounds; n
i
++;
23: if n
i
=c; the party i is
permanently ousted
24: Goto Step 3 for ‘p – D’ parties
25: If policy 3 is chosen by the setup
26: Life
i
= c;
27: Party with lowest rating sends
share first
28: If (party sends share)
29: Increment the rating
i
by 1 Else
30: {Decrement rating
i
and Life
i
by 1
and no shares are sent to that party
in that round.
31: If Life
i
= 0
32: Party ‘i’ is permanently ousted
33: Goto Step 3 for ‘p – D’ parties}
Figure 1: Proposed Repeated Secret Sharing Algorithm
applicable to our scenario of rational parties.
35: solve the set of equations to find
the sum of .

ij
secret values.
36: end for
37: Each party gets the value of

ij
38: If policy 1 or 2 is followed
39: After y
i
or 1 round respectively
if party initiates cheap talk to enter
40: Allow it and goto 3
41: If policy 3 is followed
42: After each round there is a
ratings agreement which if satisfied;
the party with lowest rating is made
to send the shares first
43:If p
i
does not send its shares
first
44: Decrement rating
i
and Life
i
by 1
and no shares are sent to that party
in that round.
45:end for end for
Figure 1: Proposed Repeated Secret Sharing Algorithm
applicable to our scenario of rational parties (cont.).
4.1 Punishment Policies
Policy 1: Incremental Punishment Strategy
for repeated Rounds upto c Times: If a party
defaults; remove him from the game for ‘y’ rounds.
He is again given a chance to enter the setup after
‘y’ rounds and if he defaults again; he is removed
for an incremental ‘g’ rounds. This defaulting
behavior is allowed for only upto ‘c’ attempts.
Policy 2: Punishment Strategy allowing ‘c’
Attempts: Remove the party for 1 round ; if he
again defaults remove him again and give him ‘c’
chances where ‘c’ is the no. of chances you want to
give a defaulter and is decided by the coalition.
Policy 3: Ratings based Punishment Strategy
simulating a Real Game: The rating of the party
that follows the protocol is incremented by one and
the party that does not follow the protocol gets its
rating decremented by 1. These ratings are
maintained at all parties since we do not have a
mediator in our protocol. There is a round of ‘rating
agreement’ which is more of a cheap talk that needs
to take place before each round of secret sharing
which ensures that the bad rational parties do not
change the ratings. Now based on the ratings; the
party with the lowest rating is supposed to send the
shares first so as to ensure corrective behaviour. If
the party with lowest ratings fails to send its shares
first; its lives and ratings are decremented by 1 and it
does not receive the shares for that round. Also like
a real game; there are ‘c’ lives or chances which
SECRYPT2013-InternationalConferenceonSecurityandCryptography
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means that only ‘c’ decrements in rating are allowed
after which the party is ousted from the game.
If a new party wants to enter the setup; that party
is granted the minimum rating of the setup and is
made to send its shares first as the setup does not
trust this new party entirely.
To summarize if there is no penalty for cheating,
rational participants tend to behave dishonestly.
Hence this new scheme does help control the bad
behaviour of dishonest parties without a heavy cost.
5 THEORETICAL ANALYSIS
In a distributed setup where Secret sharing is
undertaken for PPDARM; the parties have different
roles like carrying out computations at their end,
sending the messages and receiving the messages.
For the [m,m] secret sharing scheme we do not
discuss the attack where the parties send wrong
information as the aim of rational parties is not to
sabotage the protocol. Instead they may default by
withholding their sum of shares and/or sending them
just to the coalition. u
S
(S
i,t
), u
R
(R
i,t
) are the utilities
or payoff where the i
th
(0 < i p) party sends and
receives corresponding t
th
sum of shares. w
i,s
and w
i,R
are the weights of these utilities respectively. u
i
(σ
i
,
σ
–i
}) denotes the utility of the party’s strategy which
is interdependent on the other parties. Hence the
formulation of multiparty PPDDM as a strategic
game (Kargupta et al., 2007) for our approach is
proposed as:
u
i
(σ
i
, σ
–i
}) = w
i
,
s
* u
s
(S
i
,t
) + w
i
,
R
* u
R
(R
i
,t
)
(1)
Our game is based on the goal of getting maximum
shares and thus maximum participation and not on
the communication cost. Also if there is no penalty;
the gain would be maximum when no sum of shares
is sent by that bad rational party.
Hence the punishment policies we propose will
increase the utility of all the parties and the Nash
equilibrium is attained when the parties cooperate
honestly making it the optimum strategy.
5.1 Collusion Analysis
This algorithm can also effectively prevent collusive
behavior if the number of collusive parties C < p.
If the no. of collusive parties = p then the
protocol has no meaning. Now we consider D as the
no. of defaulters and C as the no. of colluding nodes
who send the sum of shares only in the colluding
group. For the collusion analysis; if C < D; then the
defaulting parties will not get the result back and so
that scenario has no meaning and is not a rational
behaviour. However if C = D = 1, then the party can
get the sum of values of the other parties. On the
other hand if C = p-1 or the general case where C=D
then the colluding parties can definitely get the value
of the sum polynomial Sum(x
i
) and hence can
predict the value of the non-collusive party. They
can have a high utility as they are not sending any
shares outside the collusion.
This can be avoided if we follow our punishment
strategy so that fewer parties will collude so as to
attain the maximum gain in future rounds. The
punishment policies are applied to the entire
collusion by assuming that if more than 1 party is
defaulting; there is possibility of a collusion.
6 RESULTS AND IMPLICATIONS
This section gives a brief summary of our
implementation. The platform for the prototype is
Java 7 SE on a 2.5 GHz i5 processor with 64 bit
Operating system and 6 GB RAM.
As a case study we show a comparison between
the no. of rounds of secret sharing and the
percentage of bad nodes for all our policies. We
have evaluated our results with a total of 10 nodes.
We assume that about at least 50% of the bad nodes
after being penalized prefer to join the game in the
coming round of which about at least 50% would
behave honestly.
Figure 2: Nash Equilibrium attained with the proposed
repeated RSS scheme for Punishment Policy 1.
The comparison shown above is based on
Equation (1). Also we consider the u
s
(S
i,t
) and
u
R
(R
i,t
) as the number of shares sent and received
where the u
s
(S
i,t
) is negative as none of the rational
parties want to send their shares willingly and the
w
i,R
is twice that of sending w
i,S
. according to the
preference of the rational agents. It validates the fact
that the parties after being ousted try and attain
better utilities. Also it compares the percentage of
AGameTheorybasedRepeatedRationalSecretSharingSchemeforPrivacyPreservingDistributedDataMining
515
bad nodes with the rounds of secret sharing.
Percentage of bad nodes = No. of nodes not
sending shares/ Total participating nodes
(does not include the permanently ousted
nodes)
(2)
The comparison shown in Figure 3 and Figure 4
below is based on punishment policy 2 and policy 3.
Figure 3: Nash Equilibrium attained with the proposed
repeated RSS scheme for Policy 2.
Figure 4: Nash Equilibrium attained with the proposed
repeated RSS scheme for Policy 3.
Our experiments indicate that irrespective of the
initial percentage of bad nodes; with the proposed
policies all the non cooperating nodes converge and
we get a Nash equilibrium where all the nodes are
honest. If the bad nodes do not change their
behavior; they are permanently ousted. This graph
would vary if the nodes behaved in a different
manner than the assumptions would still converge to
zero bad nodes after certain rounds.
Figure 5: Utility of output for Policy 1, 2 and 3 for initial
40% bad rational nodes.
Finally we have another metric which evaluates
the “utility of the output” with respect to the number
of rounds in Figure 5. This metric varies inversely
with respect to the percentage of bad nodes. The
utility of output is the maximum when we have the
maximum participation from the parties. We observe
that the utility of output attains the maximum value
earlier for Policy 3 than Policy 1 and 2.
Utility of output = No. of good nodes/Total
No. of participants(does not include the
permanently ousted nodes)
(3)
Further we are considering an example where we are
considering four participants in the game theoretic
setup for our policies so that we can validate the
attainment of the Nash equilibrium with our policies.
We use the same assumptions as in the 10 node
scenario. Hence based on this, our payoff matrix for
repeated secret sharing is in Table 1 below where the
Nash equilibrium state is denoted by (NE) and the
good and the bad rational nodes are denoted by (G)
and (B) respectively.
It is clear from the Table 1 that if a party defaults
in the first round then; it only gets a good payoff if it
undertakes corrective behaviour. We also observe
from the Figure. 2-5 and Table 1 that Policy 1 needs
more rounds to attain the Nash equilibrium but the
advantage is that it is a harsher incremental policy.
Hence it might inspire more nodes to correct
themselves initially itself rather than not getting
shares for a large no. of rounds. Policy 2 is less
harsh as it ousts the party temporarily only for 1
round and helps attain the Nash equilibrium faster.
Finally Policy 3 replicates an actual game scenario
and instead of ousting temporarily; it decreases the
ratings of that player. Hence the punishment is not
harsh and the Nash equilibrium is attained faster.
7 CONCLUSIONS
In this paper we propose a novel scheme for
repeated rational secret sharing using the game
theoretic approach and cheap talk for various
approaches in PPDARM(Nanavati and Jinwala,
2012);(Ge et al., 2010). Our scheme can also be
applied to other privacy preserving approaches that
use Secret Sharing to undertake SMC among
rational agents without mediators.
We have proposed three novel and effective
punishment strategies for our game theoretic
approach. We also give the theoretical analysis and
empirical evaluation of our proposed scheme. We
compare the different policies proposed in terms of
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Table 1: Payoff Matrix for 4 party scenario showing Nash Equilibrium state as well.
Policy Used A,B,C,D R
1
R
2
R
3
R
4
R
5
R
6
R
7
1,2,3 G,G,G,G
3,3,3,3
(NE)
3,3,3,3 3,3,3,3 3,3,3,3 3,3,3,3 3,3,3,3 3,3,3,3
1,2 G,G,G,B 1,1,1,6 2,2,2,0
3,3,3,3
(NE)
3,3,3,3 3,3,3,3 3,3,3,3 3,3,3,3
3 G,G,G,B 1,1,1,6
3,3,3,3
(NE)
3,3,3,3 3,3,3,3 3,3,3,3 3,3,3,3 3,3,3,3
1 G,G,B,B -1,-1,5,5 1,1,0,0 2,2,2,0 1,1,1,6 2,2,2,0 2,2,2,0
3,3,3,3
(NE)
2 G,G,B,B -1,-1,5,5 1,1,0,0 2,2,2,0 1,1,1,6 2,2,2,0
3,3,3,3
(NE)
3,3,3,3
3 G,G,B,B -1,-1,5,5 2,2,2,0
3,3,3,3
(NE)
3,3,3,3 3,3,3,3 3,3,3,3 3,3,3,3
the no. of rounds taken to attain the Nash
equilibrium. For all these policies we have
considered different percentage of initial bad
rational nodes that refrain from sending their shares.
Among these policies our results show that the Nash
equilibrium is attained in the least no. of rounds for
Policy 3 which simulates an actual game setting.
Finally we conclude that our protocol works in
the favour of all the rational parties to attain a state
where all parties are honest ultimately and has the
maximum utility considering that they take their
future utilities in account.
However there are a few open research
challenges which include extending our scheme
using game theory to verifiable secret sharing as
well as to prevent the malicious adversaries. Another
extension would be to analyse the application of our
punishment policies to other privacy preserving
schemes in distributed data mining.
ACKNOWLEDGEMENTS
I would like to thank my student Neeraj Sen for the
discussions on this problem and also for assisting me
in the implementation. I would also like to thank my
husband Rohan for the endless talks and motivation.
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