## Abstract

We propose a scheme whereby a time domain fractional Fourier transform (FRFT) is used to post process the optical chaotic carrier generated by an electro-optic oscillator. The time delay signature of the delay dynamics is successfully masked by the FRFT when some conditions are satisfied. Meanwhile the dimension space of the physical parameters is increased. Pseudo random binary sequence (PRBS) with low bit rate (hundreds of Mbps) is introduced to control the parameters of the FRFT. The chaotic optical carrier, FRFT parameters and the PRBS are covered by each other so that the eavesdropper has to search the whole key space to crack the system. The scheme allows enhancing the security of communication systems based on delay dynamics without modifying the chaotic source. In this way, the design of chaos based communication systems can be implemented in a modular manner.

© 2014 Optical Society of America

## 1. Introduction

Chaos based optical communication systems have attracted considerable attention due to their superiority [1, 2]. It is worth noting that most recent results reported in the literatures focus on electro-optic chaotic systems with delayed dynamics [3–9]. In these systems, the message security relies essentially on the difficulty of identifying the hardware parameters of chaotic transmitters. However, many different techniques have been proposed to estimate the parameters of chaotic system. Of all the parameters in delayed dynamical systems, time delay is of great importance, since it is the key to generate high dimensional chaotic signals which is the foundation of chaos based communications. An effective type of attack is to recover the time delay by using some statistical methods, such as autocorrelation function (ACF), delayed mutual information (DMI), extrema statistics and filling factor [10–13]. On the other hand, since U. Parlitz suggested exploiting the adaptive synchronization of chaotic systems to solve parameter estimation problem, various synchronization-based schemes have been studied by researchers [14–16].

To resist these attacks, several methods attempt to conceal the delay signature by adjusting the delay time itself. In [17], the delay time is chosen to be close to the laser relaxation period. However, chaos complexity is weak in this regime. Moreover, in [18], it has been demonstrated that the delay signature can still be retrieved from the phase time series, even in the presence of noise. Time delay modulation [19] has also been considered as a theoretically feasible way to prevent the time delay extraction, but it is difficult to implement practically. These methods are also threatened by synchronization-based parameter identification methods [20]. Therefore efforts have been made to modify the system structure. A cross feedback configuration is introduced in semiconductor ring lasers [21]. With such a feedback, the time delay signature can be eliminated both in the intensity and the phase dynamics. In [22], the time delay is successfully hidden by combining all-optical and electro-optical schemes. Another currently effective method is proposed by R. M. Nguimdo, et al. [8, 9]. Both serial and parallel configurations of the electro-optic phase chaos system with two feedback loops have been considered in their schemes. An external PRBS was mixed within the chaotic carrier to perform time delay concealment.

In this paper, we propose an alternative way to enhance the secure strategy for electro-optic chaotic system by using a post-processing technique, without modifying the chaotic source. Similar concept has been proposed in [23]. A non-linear, non-invertible transmission function (implemented by a Mach-Zehnder (MZ) modulator) is used to improve the security of chaotic signal. Although time delay concealment is not one of the concerns in the literature, it gives us an inspiration. In our scheme, the transform adopted should be invertible, since a non-invertible transform may have some difficulties in synchronization. Another concern is that the transform should be an encryption transform. Otherwise the original chaotic carrier will easily be obtained by an illegal receiver. At last, the transform should be implementable in optical field.

Motivated by the above discussion, we found that the FRFT in time domain fits our requirements perfectly. FRFT has been widely used in signal processing and image encryption [24–29]. Thanks to the principle of space-time duality theory [27], FRFT becomes a new way to analyze and process optical time signals [28, 29]. In this work we introduce, time domain FRFT is used as an encryption transform to post process the electro-optical chaotic carrier. As we will show below, when certain conditions are satisfied, the time delay signature can be masked perfectly. Last but not least, PRBS is also introduced in our scheme. It is noteworthy that the PRBS is not necessary for time delay mask, which makes our method conceptually different from the method proposed in [8], but still, it is indispensable for resisting brute force attack.

## 2. System setup

Here we propose a configuration built on electro-optic chaotic system and time domain FRFT. The proposed setup is illustrated in Fig. 1. Both the transmitter and the receiver consist of an electro-optic chaotic system and a FRFT (IFRFT) module, connected in serial. In the electro-optic chaotic system [3], a continuous-wave (CW) laser diode (LD) at the telecom wavelength of 1550 nm is seeded to a MZ modulator whose radio-frequency (RF) and direct-current (DC) half-wave voltages are *V _{π}* and

*V*. After an optical coupler (OC) and an optical fiber delay line (DL) of delay time

_{πdc}*T*, the optical signal is detected by a photodiode (PD). Before fed back into the MZ modulator, the generated electrical signal is amplified by a RF driver. Intransmitter, the message is added inside the loop by the OC. And in receiver, the message is obtained via canceling the chaotic carrier. The dynamical modeling of chaotic system in the transmitter and the receiver can be described in Eqs. (1) and (2), which were proposed and studied in [4–7].

_{D}*x*

_{1,2}(

*t*) =

*πV*

_{1,2}(

*t*)/2

*V*,

_{π}*V*

_{1,2}(

*t*) are the input voltages for the MZ electrode.

*m*(

*t*) is the message signal,

*β*is the feedback strength of the loop, $\Phi $ is the offset phase,

*τ*and

*θ*are the characteristic response times of the loop, and

*T*is the propagation time of light between the transmitter and receiver.

_{C}FRFT is a generalization of the Fourier transform, and can be seen as the projection of a given signal between time and frequency axis. Next, let us consider the implementation of the FRFT in time domain. The FRFT of a given signal *x*(*t*) is defined as

*p*is the fractional order and 0<|

*p*|<2. The inverse transformation can be described by $x(t)={F}^{-p}[\tilde{x}(u)]$ due to the semigroup property of FRFT.

The FRFT (IFRFT) module consists of a phase modulator and two dispersive elements, as shown in Fig. 2. The FRFT (IFRFT) has a negative (positive) second order dispersion coefficient for the dispersive media and positive (negative) drive voltage for the phase modulator. The function of parabolic phase modulator can be written as

where*K*

_{1}is a constant, and

*c*

_{1}is the modulation coefficient. The function of dispersive media in frequency domain can be described aswhere

*K*

_{2}is a constant, and

*S*is the accumulated second order dispersion.

*S*= -

*DFλ*

^{2}/(2

*πc*), where

*D*is the dispersion coefficient,

*F*is the dispersion distance,

*λ*is the wavelength and

*c*is the velocity of light.

According to the principles and theoretical analysis reported in [28], let the relationships between *c*_{1}, *S* and the fractional order *p* be

*f*

_{0}is a scaling factor. Then, the FRFT of order

*p*in time domain is achieved.

The quadratic phase modulation can be approached by a sinusoidal phase modulation. Consider a sinusoidal waveform cos(*ω _{m}t*), where

*ω*= 2π

_{m}*f*,

_{m}*f*is the frequency of the sinusoidal waveform.

_{m}*T*= 1/

_{w}*f*is the time window of the transform. The sinusoidal signal can be expressed by the Taylor expansion as

_{m}_{3}phase modulator has an optical input, an optical output and an electrical input. Its transfer function can be given bywhere

*V*is the half-wave voltage. To realize the quadratic phase modulation, according to Eqs. (4) and (10), the drive voltage should beCombining Eqs. (9) and (11), the drive voltage can be expressed by

_{π}To analyze the security performance of the chaotic carrier after being processed by the FRFT, we consider that no message is added into the loop. The FRFT is used as a post-processing module in our scheme. Therefore it is independent to the delay dynamics. It can be seen as a “black box” to the input signal. As a result, the time delay signature of the input signal is eliminated in the output signal. The message amplitude is usually attenuated with respect to the carrier in order to be masked effectively. So whether the message signal is added into the chaotic dynamics will not affect the transformation as well as the simulation results.

The main parameters of the electro-optic chaotic system and the FRFT used in our simulations are given in Table 1.

## 3. Time delay mask

As stated before, statistical analysis and synchronization-based parameter estimation are the most competitive methods for breaking the delayed-chaotic systems. First we focus on the statistical analysis. These techniques, such as the ACF and the DMI, can be used to identify the time delay, which is a key parameter for the security of electro-optic chaotic systems. For a time series *v*(*t*), the ACF *C*(*s*) is defined as

*D*(

*s*) is defined as

*P*(

*v*(

*t*)) is the probability distribution function of

*v*(

*t*), and

*P*(

*v*(

*t*),

*v*(

*t*-

*s*)) is the joint probability distribution function.

Figures 3(a) and 3(b) display that the chaotic time series have a clear time delay signature before being transformed: obvious peaks appear in both the ACF and the DMI at *s* = *T _{D}* as expected. Then we set the order of FRFT as

*p*= 1. The time window of the transform

*T*is varied from 0.1ns to 4ns. Interestingly, we found that distinguishable peaks appear in ACF and DMI at

_{w}*s*=

*T*on condition that

_{D}*T*=

_{D}*MT*, where

_{w}*M*is an integer. For example,

*T*= 0.1ns(

_{w}*M*= 310), 0.2ns(

*M*= 155), 0.5ns(

*M*= 62), 1ns(

*M*= 31), etc. Without loss of generality, Figs. 3(c)–3(f) show the ACF and DMI when

*T*= 0.9ns and 1ns respectively. When

_{w}*T*≠

_{D}*MT*, peaks only appear at

_{w}*s*=

*T*and its multiples. The mechanism behind this phenomenon is that when

_{w}*T*≠

_{D}*MT*, the statistical feature of the chaotic signal in

_{w}*t*and

*t*+

*T*is masked by the windowing transformed waveform, the original delay properties are distorted in time axis. Next, we consider the acceptable range of fractional order

_{D}*p.*Figures 4(a) and 4(b) show the size of peaks found in

*C*(

*s*) and

*D*(

*s*) at

*s*=

*T*and

_{D}*s*=

*T*as a function of the absolute value of

_{w}*p*, the peak size in ACF and DMI at

*s*=

*γ*is defined as

*η*= 1ns. When |

*p*| is small, the peak sizes at the delay position are much bigger than the peaks induced by the FRFT, meaning that the delay characteristic of the chaotic signal has not been masked completely. When |

*p*| increases, the peak size at

*s*=

*T*decreases and the peak size at

_{D}*s*=

*T*increases, until the statistical properties of the transform predominate. As shown in the insets of Figs. 4(a) and 4(b), when 0.8<|

_{w}*p*|, the delay feature is completely masked. It is important that the condition of

*T*≠

_{D}*MT*should be maintained when varying the value of

_{w}*p*. According to our simulations, if

*T*=

_{D}*MT*, the time delay signature cannot be masked regardless of the value of

_{w}*p*. Therefore the conditions of

*T*≠

_{D}*MT*and 0.8<|

_{w}*p*| should be satisfied simultaneously. Another natural concern is whether the spectrum and dimensionality of the chaotic system affect the hiding behavior. According to the theoretical and experimental analyses in literatures [5–7], the spectrum and dimensionality are related to the feedback strength

*β*under current setup. When the time delay

*T*is away from the characteristic time and the feedback strength

_{D}*β*is larger than 3.5, the system enter the chaotic zone, and the upper bound of

*β*is restricted by physical devices. So we varying the feedback strength

*β*from 2.5 to 8 in our simulations, it turns out that the hiding behavior of time delay is not affected and the results are similar to those shown in Figs. 3 and 4. On the other hand, the parameters of FRFT are mainly restricted by the physical devices. For the dispersion media, fiber bragg grating (FBG) is a good candidate. The range of dispersion adjustment could reach up to 2000ps/nm, and this range can be enlarged by cascading multiple FBGs. The precision of the dispersion adjustment could reach up to 2ps/nm. The value of

*T*is mainly restricted by the frequency of the drive voltage [29], and the range can easily be from hundreds of ps to several ns. In our simulations,

_{w}*S*∈[300, 4000]ps

^{2}and

*T*∈[0.1, 4]ns are considered, and the results show that the masking conditions are not affected obviously (the threshold of |

_{w}*p*| may be slightly different from 0.8).

## 4. Synchronization

Several theoretical and experimental analyses have demonstrated the influence of chaotic parameters to synchronization [30–33], and the channel dispersion can be compensated by using dispersion-compensating fiber (DCF) or dispersion-shifted fiber (DSF). This problem has also been investigated comprehensively in the literature [34]. So we only focus on the FRFT here. We use the root-mean square synchronization error to quantify the quality of chaos synchronization, which is defined as

where <•> stands for time average and*δ*(

*t*) =

*x*

_{2}(

*t*)-

*x*

_{1}(

*t*). Figures 5(a) and 5(b) display the sensitivity of synchronization with respect to fractional order

*p*and dispersion

*S*respectively. σ grows fast from 0 when the fractional order

*p*’ and dispersion

*S*’ in the IFRFT are different from

*p*and

*S*in the FRFT. More important, we found that when the time window of the IFRFT (

*T*’) in receiver has a slight difference with the time window of FRFT (

_{w}*T*) in transmitter, the instant synchronization error |

_{w}*δ*(

*t*)|/<|

*x*

_{1}(

*t*)|> increase very fast along with time as shown in Fig. 5(c), which indicates that the synchronization degrades rapidly. This is because when there exists a mismatch of time window between the transmitter and receiver, an inevitable accumulation error of synchronization is produced over time due to the unmatched clock. For a legal receiver, the clock should be synchronize with the transmitter, and this can be achieved by extracting a clock from the decoded message signal. Similar technique has been used in optical OFDM system based on time domain Fourier transform [29]. Thus the chaos synchronization error will not accumulate over time and remains in an acceptable range. However, for a illegal receiver, such clock synchronization could be very hard to achieve in absence of the message.

The analyses above indicate that the original chaotic carrier of the transmitter is unacquirable without knowing the parameters of FRFT in transmitter, which makes synchronization based parameter estimation difficult to perform. And if an eavesdropper can only get the transmitted signal (the output of the FRFT), the individual will not be able to crack the parameters of the transformation without the input.

## 5. The role of PRBS

At last, we discuss the role of PRBS in our scheme. As stated before, when certain conditions are satisfied (*T _{D}*≠

*MT*, 0.8<|

_{w}*p*|), time delay signature can be mask by the FRFT. However, we could not ignore the fact that the information of

*T*is exposed, which will cause potential security vulnerabilities. To overcome this drawback, an effective way is to vary the time window and the fractional order by using external PRBS, which will introduce another degree of freedom to enhance the security of the system.

_{w}To realize this, we only need to vary *f _{m}* and

*c*

_{1}along with time. Since the second order dispersion

*S*is fixed and

*f*

_{0}can be chosen arbitrary, given an expected

*p*,

*f*

_{0}can be calculated through Eq. (7) and

*c*

_{1}can be obtained by Eq. (6). Such implementation only needs one additional processing: varying the drive voltage for phase modulator, which will not cause remarkable system complexity. The updating period of

*T*and

_{w}*p*should be smaller than

*T*, since the statistical analyses against the time delay needs at least a time series with a length larger than

_{D}*T*. In our simulation, four different

_{D}*T*s (0.25ns, 0.35ns, 045ns, and 0.55ns) and four

_{w}*p*s (0.8, 1.0, 1.2 and 1.4) are chosen and controlled by two series of external PRBS in each side, and the mathematical model can be described as

*PB*1

*and*

_{i}*PB*2

*denote the two PRBS, and*

_{i}*T*is the updating period. We demonstrate the result of ACF and DMI in Figs. 6(a) and 6(b). The updating period is set as 20ns, and the total bit rate of the two PRBS are 100Mbps accordingly. Note that with the time window and fractional order switched by the PRBS, the signature of

_{U}*T*is eliminated in ACF and DMI, as well as

_{D}*T*.

_{w}To further explain the role of PRBS in our scheme, let us consider the following attack scenario: if the parameters of the FRFT are chosen carefully (let *T _{D}*≠

*MT*) and remain unchanged over time, an eavesdropper may still crack the time-delay by “random guess” method. The attacker could perform the IFRFT to the transmitted signal (chaotic carrier after being transformed) with an arbitrary parameter set (

_{w}*p*’,

*S*’,

*T*’), and then perform the ACF or DMI to the obtained waveform. If there is a peak appears at

_{w}*s*=

*T*, the time-delay is clearly identified. When we varying the parameters by using the PRBS, the simulations show that there are no obvious features appear in ACF and DMI (see Fig. 7) under the aforementioned attack. Peaks only appear at

_{D}*s*=

*T*’ and its multiples. These will not give the attacker any useful information. On the other hand, the cross correlation coefficient between the transmitted signal and the PRBS is under 10

_{w}^{−2}, which means the PRBS cannot be extracted from the chaotic carrier. To attack the system, the eavesdropper has to perform the exhaustive search of

*p*,

*S*,

*T*and the PRBS.

_{w}The PRBS used here can be produced in two different places separately as long as the initial states are the same. In our scheme, the initial states of PRBS along with the physical parameters are considered as the secret keys which can be distributed in other secure ways, i.e. the key exchange protocol base on public key technique [35] or quantum key distribution [36]. And the time synchronization of the PRBS can be achieved by extracting a clock signal form the decoded message. The PRBS can be generated by using conventional algorithm cryptography or digital chaos technique [37], which results in a large key space. One of the major restrictions is that the speed may be relatively slow due to the computation complexity. However, in our scheme, the bit rate of the PRBS only needs to be hundreds of Mbps, which is certainly acceptable in practice.

## 6. Conclusions

In summary, we have studied the role of FRFT in electro-optic chaotic secure communication systems. The FRFT is used as a post-processing module in our scheme which is independent to the delay dynamics. As a result, the time delay signature of the input signal is eliminated in the output signal when some conditions are satisfied. PRBS with low bit rate (hundreds of Mbps) are introduced to resist brute force attack. The chaotic carrier, FRFT parameters and the PRBS are covered by each other so that the eavesdropper has to search the whole key space, including the digital key and the analog one. Conceptually speaking, our method should be effective for any delayed chaos-based encryption. But we only perform the simulations based on Intensity chaos electro-optic delay system in the current paper, other types of chaotic system, e.g. phase chaos electro-optic system and all-optical chaotic system will be studied in our next step of work. From the designer’s viewpoint, we suggest using an optical encryption transform to resist the attacks against the parameters of chaotic systems, so the setup of chaotic sources can focus on the complexity. Thus, the design of chaos based secure communication systems can be implemented in a modular manner.

## Acknowledgments

This work was supported by the Fundamental Research Funds for the Central Universities, HUST: CXY12M005 and 2013TS049, and the National Nature Science Foundation of China (NSFC) under Grant 61307091.

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