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

 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

∑





of the secret

values at each party P

. 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

is the number

of rounds for which the defaulter ‘i’ is ousted in

Policy 1 and incr

is the increment added to y

Consider c

is the maximum number of chances

given to the defaulter to enter the setup, n

is the no.

of rounds that the party i has defaulted, rating

is the

current rating of the i

player in policy 3 and life

the number of lives left with the i

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

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513

moderator will just ensure that the parties are ousted

properly and are given their due punishment.

Input: p, N, n

, policy used, y

, incr

c, rating

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 = 1 to p)do

3: each party computes the share of

other party P

, where share(S

(x)(random polynomial)

4: for t = 1 to p do

5: send share(S

, P

) to party P

6: receive the shares share(S

, P

)

7: end for

8:compute Sum(x

)=q

)..+q

)

9: for t = 1 to p do

10: send Sum(x

) to party P

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

= 0)

14: defaulter(d) is removed from the

setup for ‘y

’ rounds; n

++;

15: Goto Step 3 for ‘p – D’ parties

16: Else

17: defaulter(d) is removed from the

setup for ‘g

’ rounds where g

= y

incr

; y

++;

18: if n

=c; the party P

permanently ousted

19:

Goto Step 3 for ‘p – D’ parties

20: If policy 2 is chosen by the

setup

21: If (n

= 0 )

22: defaulter is removed for 1 round

until a limit of ‘c’ rounds; n

++;

23: if n

=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

= c;

27: Party with lowest rating sends

share first

28: If (party sends share)

29: Increment the rating

by 1 Else

30: {Decrement rating

and Life

by 1

and no shares are sent to that party

in that round.

31: If Life

= 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 .

∑







secret values.

36: end for

37: Each party gets the value of

∑







38: If policy 1 or 2 is followed

39: After y

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

does not send its shares

first

44: Decrement rating

and Life

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

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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

i,t

), u

i,t

) are the utilities

or payoff where the i

(0 < i ≤ p) party sends and

receives corresponding t

sum of shares. w

i,s

and w

i,R

are the weights of these utilities respectively. u

(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:

(σ

, σ

–i

}) = w

* u

) + w

* u

)

(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

) 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

i,t

) and

i,t

) as the number of shares sent and received

where the u

i,t

) is negative as none of the rational

parties want to send their shares willingly and the

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

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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

1,2,3 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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