Using the Transfer Entropy to Build Secure Communication Systems ...

Using the Transfer Entropy to Build Secure Communication Systems Fabiano Alan Serafim Ferrari1 , Ricardo Luiz Viana1 , and Sandro Ely de Souza Pinto2 1

Federal University of Parana, Physics Department, post office box: 19044, Curitiba, Brazil 2 State University of Ponta Grossa, Physics Department, Secretariat - Bloco L Room 115, Av. Carlos Cavalcanti, 4746, Campus Uvaranas, Ponta Grossa, PR, Zip code: 84.030-900 {ferrari,viana}@fisica.ufpr.br, [email protected]

Abstract. We are livining in times in which is very hard to keep in secret documents and data. For this reason communication systems more safe and fast have been required and in this sense we proposed in this work some ways to build secure communication systems using transfer entropy in coupled map lattices. Keywords: Transfer entropy, coupled map lattice, secure communication systems.

1

Introduction

There are five basic elements to stablish a communication: the information source, where the message will be selected; the emitter, responsible to modify the message in such way that it can be transmitted over a channel; the channel, that is responsible to transport the message until the receiver; the receiver, that is responsible to modify the signal in a message to be readed by the addressee and the addressee that receive the sent message[1]. However to build secure communication systems are necessary other elements: the encipherer, that change the message in a cryptogram; a key, that allows the cryptogram be deciphered, and the decypher that using the key will recovery the message[2]. The possibility to use chaotic systems to develop secure communication systems starts with Pecora and Caroll’s work in which they show the criterious to chaotic systems reach the synchronization[3]. After that Cuomo and Oppenheim proposed a circuit exploring the synchronization effect and also applied it in communication[4]. Nowadays there are a huge number of models using chaotic systems to develop communication systems[5,6,7,8]. An intersting method to transmit information is the use of transfer entropy. The transfer entropy measures the transport of information and statiscal coherence between systems[9]. Using this idea Hung and Hu develop a system to transmit binary messages with unidirectional coupled map lattice in which the V.M. Mladenov and P.C. Ivanov (Eds.): NDES 2014, CCIS 438, pp. 92–99, 2014. c Springer International Publishing Switzerland 2014 

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coupling direction (clockwise and counterclockwise) represents the message[10]. Based on this idea in this work we will show how to expand the message size using the transfer entropy and replica of networks[11]. In the section II we show how to use the transfer entropy to detect coupling directions, then in the section III we show how to use this technique to build secure communication systems and in the section IV we present our conclusions.

2

Detecting Coupling Directions with Transfer Entropy

Be I and J two variables described by a chaotic map with a constant domain. We can divide this domain in P partitions with same size. So we define in and jn the respective states in the time n such i, j = 0, 1, 2, . . . , (P − 1). Then the transfer entropy can be defined as TJ→I =



p(in+1 , in , jn ) log

p(in+1 |in , jn ) , p(in+1 |in )

(1)

in which p(, ),p(|) are joint probability and conditional joint probablity respectively and the sum is over all the possible states[9]. If we have two variables X and Y described by xn+1 = (1 − ε)f (xn ) + εf (yn ), yn+1 = (1 − δ)f (yn ) + δf (xn ),

(2) (3)

where f (x) and f (y) are the local dynamic and ε, δ are the coupling strength. Then we have three possible situations for coupling: (i) the unidirectional coupling in which the X variable affects the Y variable but the opposite doesn’t occur, i.e., ε = 0.0 and δ = 0 as described by the figure 1 (a); (ii) the unidirectinal coupling in which the Y variable affects X variable but the opposite doesn’t occur, i.e., ε = 0 and δ = 0.0 as described by the figure 1 (b); (iii) the bidirectional coupling in which both variables affect each other, ε = 0 and δ = 0 as described by the figure 1 (c). For a system described by the equations (2) and (3) we can use as local dynamic the tent map,  2t if 0 < t < 0.5 f (t) = , (4) 2(1 − t) if 0.5 ≤ t < 1.0 and the time evolution for the unidirectional cases and bidirectional case will be undistinguishable as we can see in the figures 2 (a),(b) and (c). But if we evaluate the transfer entropy in both senses, TX→Y and TY →X , then will be possible to distinguish the cases because for each case we will have a different measure for the transfer entropy: Figure 2 (a) TX→Y = 0.0017 and TY →X = 0.0001 what indicates the coupling strength in one direction is bigger than other.

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F.A. Serafim Ferrari, R.L. Viana, and S.E. de Souza Pinto

δ X

Y

(a)

ε X

Y

(b)

ε X

δ Y

(c) Fig. 1. Three possible coupling configurations: (a) unidirectional coupling X → Y ; (b) unidirectional coupling Y → X and (c) bidirectional coupling

Fig. 2. Time evolution for three different coupling configurations: (a) δ = 0.1 and ε = 0.0; (b)δ = 0.1 and ε = 0.1; (c) δ = 0.0 and ε = 0.0. The color red represents the x variable and the color blue the y variable.

Figure 2 (b) TX→Y = 0.0072 and TY →X = 0.0072 what indicates the coupling strenghth is the same in both directions.

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Figure 2 (c) TX→Y = 0.0000 and TY →X = 0.0000 what indicates there isn’t coupling between the varibles. In this way, according with the coupling structure we can define a coupling direction and then explore this to build a communication system.

3

Using the Coupling Direction to Build Secure Communication Systems

3.1

Hung and Hu Method

The Hung and Hu model consist in build a coupled map lattice with a local and unidirection coupling[10]. In this model the message is transmitted according with coupling direction. If the coupling is in clockwise direction then the transmitted message is 1, as show in the figure 3 (a) and if the coupling is in counterclokwise direction then the transmitted message is 0, as show in the figure 3 (b).

Fig. 3. Description of the message according with the direction: (a) clockwise direction and message=1 and (b) counterclockwise direction and message=0

In a general form the coupled map lattice can be described as (i)

xn+1 = (1 − ε)f (xn(i) ) + εf (xn(i±1) )

(5)

where ± vary according with the message, + for clockwise direction, μ = 1 and − for counterclockwise direction, μ = 0. The function f (x) is the local dynamic, the tent map is a good option. We have observed that continuous maps work well. In this model the message is recovery using the transfer entropy to discovery the coupling direction. The security of this system consist in protect the time series of the network elements with a cryptogram then an eavesdropper can’t determine the transfer entropy and then can’t discovery the hidden message.

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F.A. Serafim Ferrari, R.L. Viana, and S.E. de Souza Pinto

Network Replica Method

While the Hung and Hu method allow us to send only binary messages we present here another method in which we create a network and then a replica of this network what enable us to send the same number of bits as the network size[11]. This method is divided in two stages: the pre-synchronization stage in which we create the replica network and the post-synchronization stage in which we send the message in a safe way. Pre-synchronization Stage. First of all we assume that the emitter and the receiver are networks. In the pre-synchronization stage the objective is make the emitter and receiver network become the same. To do that we define between the two networks a master-slave coupling where the emitter network (E) is the master and the receiver network (R) is the slave, so the networks can be described as (i)

en+1 = F (e(i) n ) (i) rn+1 (i)

= (1 −

(6)

γ)F (rn(i) )

+

γF (e(i) n )

(7)

(i)

in which en+1 and rn+1 are the time evolution rule for the elements of the E and R network respectively, in the figure 4 we show the network representation. (i) (i) (i−1) (i+1) ) + f (zn )] is the kernel The function F (zn+1 ) = (1 − ε)f (zn ) + 2ε [f (zn coupling, ε is the intranetwork coupling and f (z) is the local dynamic. The parameter γ is the internetwork coupling strength responsible to make the receiver network become equal to emitter network. However it is impossible to variables starting with different initial conditions become the same, for this reason when the equality between the ith element of the E network and the ith element of the R network is smaller than a reasonable precision, for example 10−14 , we truncate the time series with this precision allowing the networks become the same. When the network R become a replica of the E network then we turn off the coupling and due to the truncation they won’t be different anymore. If for the equations (6) and (7) we choose the tent map and γ = 0.5 and ε = 0.1 then the replica network can be build. When the networks become the same the second stage starts. Post Synchronization Stage. In this stage each element of the E network, e(i) , will transmit one bit of information. To do that we define a new variable S described by the equation sn+1 = (1 − β)sn +

N β  (i) (i) e m , η i=1 n

(8)

in which β ≡ (N − 1)/N , η is the sum of connected sites with S and m(i) is the binary message sent by the ith element of the E network. When the ith element of the E network intend to send the message 1 then this element is connected

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R

E

Fig. 4. Network representation. In the color green we have the receiver network (R) and in blue we have the emitter network (E). The lines in the colors green and blue represent the intranetwork coupling (ε). The lines in black represent the internetwork coupling, the coupling strength in the solid lines are equal to γ(1 − ε) while in the dashed lines are equal to γ 2ε .

with the S variable and m(i) = 1 and when the ith element of the E network send the message 0 it is not connected with S and m(i) = 0. The message is recovery when the transfer entropy between the ith elements of the R network and the S variable is measured. How the E and R network are the same then the transfer entropy will say if the ith element of the emitter sent or not the message. To differenciate the elements that are or not connected is necessary insert a filter to measure the transfer entropy such that when the measure is above the filter then the result is 1 and the site is connected and when the measure is below then the result is 0 and the site disconnected. In this case the filter is the standard deviation, σ(T ), related to whole transfer entropy measures. So the recovery message will be m(i) = Θ(Tr(i) →sn+1 − σ(T )), n

in which Θ is the Heaviside step function such  0 if x < 0, Θ(x) = 1 if x ≥ 0.

(9)

(10)

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F.A. Serafim Ferrari, R.L. Viana, and S.E. de Souza Pinto

In the figure 5 we have an example of transmitted message, in this case a message of 100 bits was sent. The security in our system is when the coupling is turn off because even if a eavesdropper acess the signal, in this case the S variable, he won’t be able to recovery the message because it depends on the relation between the E network and S variable. Another way to protect the message is to change the signal into a cryptogram then will become more difficult for an eavesdropper discovery the message.

Fig. 5. Example of a message of 100 bits using the replica network method. Above we have the sent message, in the middle the transfer entropy measured between all ) and below we have the the elements of the R network and the S variable (Tr(i) →s n n+1 received message. In the x-axis i is the index related to each network element. Here the local dynamic is the tent map, γ = 0.5 and ε = 0.1.

4

Conclusions

The transfer entropy allow us to determine coupling directions in coupled map lattices and this propertie can be used to build secure communication systems. However even with modifications in the Hung and Hu method the system is not fast enough but is safe. In our research we found that continuous map are better to develop this kind of communication systems. We also believe that mechanisms in which the security of the system is depedent of the relation between the emitter, receiver and the signal are important to keep a message in secret.

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Acknowledgments. This work was made possible through partial financial support from the following Brazilian research agencies: CNPq, CAPES, and Funda¸ca˜o Arauc´ aria. We acknowledge Romeu M. Szmoski for the relevant discussions.

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