Electro-optic phase chaos systems with an internal variable and a digital key

Romain Modeste Nguimdo; Pere Colet

doi:10.1364/OE.20.025333

1. Introduction

The possibility of encoding and decoding multi-gigabit sensitive data using broadband chaos, has been demonstrated theoretically [1, 2], experimentally [3, 4] and in realistic installed networks [5, 6]. For this paradigm of communications, security relies mainly on the difficulty of identifying the emitter parameters necessary to build an adequate receiver which can synchronize with it [7, 8]. For this purpose, flexibility and parameter concealment are necessary to achieve a good degree of security. In particular, the systems usually considered for chaos-based communications leverage on delay to generate high dimensional chaotic carriers on which the message is encoded. The concealment of the delay time is of great interest because, in some systems, its identification is enough to reconstruct the underlying chaotic dynamics [9, 10]. In other systems, e.g. [2, 11–16], while breaking the delay time does not directly allows to hack the message, it poses a security thread since the key space dimension (a sort of equivalent to a digital key size) is reduced, exposing the systems to brute-force-attacks. Unfortunately, it has been found that the delay time can be readily identified in these systems by applying statistical techniques to the transmitted signal [10, 17–19].

There has been several proposals to overcome this drawback. For instance, in Fabry-Perot semiconductor lasers with optical feedback it has been suggested that the time-delay signatures can be eliminated if the delay is chosen close to the relaxation period of the laser operating with moderate feedback [20, 21]. In this situation statistical quantifiers computed from the intensity of the transmitted field fail to identify the delay time. However it has been recently shown that applying the same statistical techniques to the phase dynamics time-delay signatures can be successfully retrieved due to the correlation between the phase and its delayed version in the dynamics [22]. A more sophisticated technique to conceal the delay time has been introduced recently leveraging on bidirectional semiconductor ring lasers in which the cross-feedback between the counter-propagating modes allows an efficient concealment of time-delay signatures both in intensity and phase time series [23].

In optoelectronic systems [2, 12, 14], attempts to conceal the delay time by choosing it close to a characteristic time of the system, such as the fast time-scale of the filter, will not be successful since in this parameter region the system is not chaotic. In a first attempt to conceal the delay time in electro-optical systems, a cascaded system consisting of a combination of an all-optical system [24] and opto-electronic phase-chaos system [2] has been proposed [25]. The results for the statistical quantifiers computed from the intensity time series shows that it is possible to conceal the time delay associated to the electro-optical system. However, as more sophisticated devices such as an optical 90° hybrid coupler [26, 27] can allow to detect the amplitude and the phase simultaneously, the approach of Ref. [22] can be used in this case to retrieve all the time delays since the overall transmitted phase is just a linear superposition of the all-optical and electro-optical system phases. Furthermore, the cascaded system becomes less chaotic than the original phase chaos system [2] for large values of the overall loop gain (≳ 3), rendering therefore the delay identification more vulnerable.

To provide better security to chaos-based communications, we have suggested recently [28] an advanced scheme that integrates a digital key in a phase-chaos electro-optical delay system consisting of two delay chains. The digital key allows to conceal the delay time in the phase dynamics, it adds a significant degree of flexibility to the system and, moreover, it increases the key space dimensionality, avoiding another typical limitation of hardware cryptography, namely the fact that its parameter space dimension is usually relatively low compared to algorithmic cryptography. Furthermore, the non-linear mixing of chaos and the digital key allows also to conceal the key. In the scheme introduced in [28] each delay chain has two electro-optic phase modulators (PM) seeded by a continuous-wave semiconductor laser. In each chain the first PM is driven by an external signal (the digital key in one of the chains and the message to be encoded in the other) while the second PM is driven by the output of the other chain. Each chain also includes a fiber delay line, generating a delay T_i, and an imbalanced Mach-Zehnder interferometer with differential delay δT_i. The Mach-Zehnder interferometer transforms in a nonlinear way the phase variations into intensity variations, which are finally detected by a photodiode. The electrical output of the photodiode after amplification is the input for the phase modulation of the other chain. Therefore the two delay chains operate in a serial configuration, and can be viewed as part of an overall delay loop. In this configuration time-delay concealment occurs only when the digital key is present and operates at a bit rate above a threshold given by the differential delay time of the chain in which the key is introduced.

In this work we introduce, and study theoretically and numerically, a new system with two delay loops that operate in parallel so that only the output of one loop is transmitted to the receiver while the other loop remains internal. At a difference with [28] this configuration allows to conceal the internal time delays which are critical for decoding without the need for a external digital key. As we will show below, this intrinsic concealment capability comes from the fact that when loops are coupled in parallel and each loop has a different differential delay time the dynamics of the internal loop is uncorrelated to the transmitted signal. This is a mechanism not present in the serial configuration in which the dynamical variables describing both loops are always correlated so that without digital key all the delay times can be readily identified in the transmitted signal.

On the downside, while in the serial scheme synchronization between a matched emitter and receiver pair is always achieved (unconditional synchronization) [28], in the parallel configuration considered here the fact that the internal variable is not transmitted implies that it must be regenerated at the receiver. In this situation synchronization is not always possible even in the ideal case of no mismatch between emitter and receiver. Nevertheless we show that if the gain of the internal loop is not too large an excellent degree of synchronization can be achieved.

Last but not the least, the new system also allows for the introduction of a digital key. While it is not required for time delay concealment, it is still useful to increase the parameter space. When included, the digital key plays a critical role in synchronization. In fact it turns out that the new system based on parallel loops is five times more sensitive to a key-mismatch than the serial system considered in [28] thus increasing the security.

2. System

The proposed setup is illustrated in Fig. 1. Both emitter and receiver consist of two nonlinear delayed differential processing loops, connected in parallel. The sub-indices i = 1, 2 refer to a given loop. An electro-optic phase modulator (PM₁) seeded by a continuous-wave (CW) telecom semiconductor laser (SL) is phase-modulated by a voltage proportional to x₁(t). The output of PM₁ is then split into two parts. One part is sent to the receiver while the second part is successively phase modulated by a voltage proportional to x₂(t) and by the digital key R(t), generated as a Pseudo-Random Bit Sequence (PRBS). After the double phase modulation the resulting optical signal is divided into two parts. Each part is fed to a fiber delay line which delays the signal by a time T_i and then fed to an Mach-Zehnder interferometer (MZI_i) with imbalance time δT_i, which converts phase variations into intensity variations. The intensity variations are detected by a photodiode (PD_i) and amplified by an RF driver with an effective gain G_i. The output of each amplifier, proportional to x_i, is applied to the respective RF electrode of PM_i to close the loop i. The message m(t) is encoded as an additional phase modulation using another PM placed in between the SL and PM₁ (as shown in Fig. 1) or alternatively just after PM₁ and prior to the split of the signal to be transmitted to the receiver. At this point we would like to note the following points: first, only the output of PM₁ is transmitted to the receiver, so loop 2 can be considered as internal. Second, a total of four phase modulations (two chaotic proportional to x₁(t) and x₂(t) + pseudorandom + message) are successively applied to the optical signal delivered at the SL output before its undergoes phase-to-intensity conversion. Third, this system requires less components than the previous one [28] since it uses a single light source for emitter-receiver system instead of three.

Fig. 1 Transmitter and receiver setup in the parallel configuration: SL: semiconductor laser, PM: phase modulator, MZI: imbalanced Mach-Zehnder interferometer, PD: photodiode, x₁(t) and x₂(t) are the dimensionless output voltages of the RF drivers for the external and the internal loops while R(t) and m(t) are the pseudo-random bit sequence and message, respectively. Sub-index 1 refers to the loop whose output is transmitted while 2 refers to the internal loop.

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This set-up can be experimentally implemented using similar components as the original experimental set-up for electro-optical phase chaos [2, 6] although the implementation of the double loop requires two additional PMs and an additional MZI in both emitter and receiver. Still, as stated before, this configuration requires less components as the serial configuration [28].

The dynamical model can be described as follows. The electronic bandwidth of the loop is assumed to result from two cascaded linear first-order low-pass and high-pass filters. Considering the filter output voltages V_i(t) and proceeding as in [2, 29], the emitter dynamics can be described by the dimensionless variables x_i(t) = πV_i(t)/(2V_π,i) where V_π,i is the half-wave voltage of the modulator PM_i

x_{i} + τ_{i} \frac{d x_{i}}{d t} + \frac{1}{θ_{i}} u_{i} = G_{i} {cos}^{2} [Δ {(x_{1} + x_{2})}_{T_{i}} + Δ {(R + m)}_{T_{i}} + ϕ_{i}],

where du_i/dt = x_i, Δ(F)_t₀ = F(t − t₀) − F(t − t₀ − δt₀) and ϕ_i is the static offset phase of MZI_i. For numerical simulations, we consider the key physical parameters arbitrary chosen, within the range of experimentally accessible values [2, 28], as follows: T₁ = 15 ns and T₂ = 17 ns, τ₁ = 20 ps, τ₂ = 12.2 ps, θ₁ = 1.6 μs, θ₂ = 1.6 μs, δT₁ = 510 ps, δT₂ = 400 ps, ϕ₁ = π/4, ϕ₂ = π/8, G₁ = 5 and G₂ = 3. These parameters have been used for the original setup in [2, 6]. In practice the overall loop gain is limited, although values as large as 6 can be achieved.

3. Delay Time Concealment

The delay time can be extracted using the standard delay time identification techniques, e.g., autocorrelation function C(s), delayed mutual information (DMI), extrema statistics and filling factor [17–19]. Out of those, C(s) and DMI are robust to noise perturbations and therefore are suitable to crack the time delay in practical situations. For a time series x(t), C(s) is defined as

C (s) = \frac{〈 [x (t) - 〈 x (t) 〉] [x_{s} (t) - 〈 x (t) 〉] 〉}{{[〈 x (t) - 〈 x (t) 〉 〉]}^{2}},

where x_s(t) = x(t − s) and 〈...〉 stands for the time average. The DMI measures the information on x(t) that can be obtained by observing x_s(t)

D M I (s) = \sum_{x (t), x_{s} (t)} p (x (t), x_{s} (t)) ln \frac{p (x (t), x_{s} (t))}{p (x (t)) p (x_{s} (t)))},

where p(x(t)) is the probability distribution function of x(t) while p(x(t),x_s(t)) is the joint probability distribution function.

We do not take into account the message in this section (m = 0). The relevant delay times for the model are T₁, T₁ + δT₁, T₂ and T₂ + δT₂. Figure 2 displays the autocorrelation (a) and the DMI (b) without (solid line) and with a PRBS of amplitude π/2 at 3 Gb/s (dashed line), computed from a long series for x₁(t). Without PRBS, two relevant peaks are found both in the autocorrelation and in the DMI at delay times T₁ and T₁ + δT₁ as expected. What is more relevant is that no peak is found around the internal loop delay time positions, T₂ and T₂ +δT₂. We have also checked that using the time distribution extrema and the filling factor methods these delay time signatures remain concealed. Therefore the system fully conceals the internal loop delay times even without digital key.

Fig. 2 Autocorrelation function C(s) (a) and delayed mutual information DMI(s) (b) of x₁(t) without PRBS (red line), and with a PRBS of amplitude π/2 at 3 Gb/s (black). A time series of length 10 μs with 10⁷ data points was used.

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The mechanism behind concealment of the internal loop time delays is the fact that the transmitted signal is practically uncorrelated with the internal loop dynamics. In what follows we address this issue in detail. In the serial configuration considered in [28], the dynamical variable describing one of the chains is driven only by the dynamical variable of the other chain delayed in time. The mutual driving through feedback generates a strong dynamical dependence that leads to a large correlation between the two variables. Therefore it is possible to unveil all the delay times by computing the quantifiers from only one variable. On the contrary, the parallel configuration considered here, has two particular characteristics that allow for decorrelation between the internal and the external variables. The first one is that the dynamics of each loop, besides being driven by the feedback from the other loop, includes self-feedback. This, by itself, would not be enough to preclude correlations since, as the RHS of Eq. (1) shows, both x₁ and x₂ are driven by x_sum = x₁ + x₂. The second characteristic is that the loops are driven by a differential delay Δ(x_sum)_{T_i} = x_sum(t − T_i) − x_sum(t − T_i − δT_i). For loop i, the differential delay mixes x_sum at two different times separated by δT_i. If δT₁ differs from δT₂ by an amount larger than the autocorrelation time of x_sum, then the result of the mixing in MZI₁ is practically uncorrelated from the one obtained in MZI₂. As a consequence in Eq. (1), x₁ and x₂ are driven by effectively uncorrelated chaotic signals. Notice that if each loop instead of having a differential delay feedback involving two times T_i and T_i + δT_i it had single delay time T_i, then one of the variables would be correlated with the other shifted in time by T₁ − T₂. Should that be the case, then the internal delays would appear in the statistical indicators of the transmitted signal. Thus differential feedback in each loop is necessary and furthermore the differential delay time of both loops must be different. Still, by itself, this is not sufficient, since in the serial configuration considered in [28] both dynamical variables are always strongly correlated despite the presence of different differential delay times. The interplay between the self-feedback and cross-feedback together with the presence of two different differential delays is what leads to decorrelation between internal and transmitted variables allowing for delay concealment.

To further discuss this issue within a mathematical framework, we consider the Fourier transform of Eq. (1),

X_{i} (ω) (1 + j ω τ_{i} + \frac{1}{j ω θ_{i}}) = G_{i} e^{- j ω T_{i}} FT {{cos}^{2} [\bar{Δ} {(x_{1} + x_{2})}_{δ T_{i}} + \bar{Δ} {(R)}_{δ T_{i}} + ϕ_{i}]},

where j² = −1, Δ̄(F)_δt₀ = F (t) − F(t − δt₀) and FT{z} stands for the Fourier transform of z. For δT₁ = δT₂ and ϕ₁ = ϕ₂, it turns out that

\frac{X_{1} (ω)}{X_{2} (ω)} = \frac{G_{1} (1 + j ω τ_{2} + \frac{1}{j ω θ_{2}})}{G_{2} (1 + j ω τ_{1} + \frac{1}{j ω θ_{1}})} exp [- j ω (T_{1} - T_{2})] .

Equation (5) establishes a linear relationship between x₁ and x₂. Consequently information on the internal variable dynamics can be easily retrieved from the transmitted variable x₁(t) and therefore for δT₁ = δT₂ one should expect that none of the time delays is concealed. And as shown in Eq. (5) this is certainly the case even if T₁ is different from T₂. In fact, even considering different values for the offset phases, ϕ₁ ≠ ϕ₂, we have numerically found that the delay times can be identified if δT₁ = δT₂. The numerical results for the autocorrelation and the DMI for δT₁ = δT₂ = 400 ps, ϕ₁ = π/4 and ϕ₂ = π/8 computed from the transmitted variable x₁ are shown in Fig. 3. For this specific case, we found that the maximum of the cross-correlation between x₁ and x₂ takes place at T₂ − T₁ (as predicted) and is quite large, 0.7. Peaks at T₂ and T₂ + δT₂ are apparent. Clear peaks also appear at T₂ − T₁ (out of the figure range). In fact, while typically the delay time signature is reduced when increasing the overall loop gain (which increases the complexity of the chaos), for δT₁ = δT₂, the delay time can always be identified even for G₁ = G₂ = 15, way beyond experimental limits.

Fig. 3 Autocorrelation function C(s) (a) and delayed mutual information DMI(s) (b) of x₁(t) without PRBS (red line), and with a PRBS of amplitude of π/2 at 3 Gb/s (black). Parameters as in Fig. 2 but with δT₁ = δT₂ = 400 ps.

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Figure 4 (a) shows the autocorrelation function for the variable x_sum = x₁ + x₂ for the parameters considered in Fig. 2. For times above 40 ps the autocorrelation is smaller than 0.1. Figure 4 (b) shows the cross-correlation between x₁ and x₂ as function of the relative mismatch in the differential delay times, ξ = (δT₂ − δT₁)/δT₁. When the feedback phases are identical the cross correlation starts at 1 for ξ = 0 as one can expect from the relationship given by Eq.(5). If the feedback phases are different, x₁ and x₂ are still strongly correlated at ξ = 0. As the difference between δT₂ and δT₁ is increased the cross correlation decreases and it finally decays to zero when this difference becomes of the order of the autocorrelation time for the variable x_sum. In practice, negligible values for the cross correlation are found for |ξ| larger than 10%.

Fig. 4 a) Autocorrelation function C(s) for the variable x_sum(t) = x₁(t)+ x₂(t) considering the same parameters as in Fig. 2(b) Cross correlation between x₁(t) and x₂(t) as function of the mismatch in the differential delay time ξ = (δT₂ − δT₁)/δT₁ for (•) ϕ₁ = π/4 and ϕ₂ = π/8 and (▵) ϕ₁ = ϕ₂ = π/4, considering δT₁ = 400 ps.

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This result is to be compared with the one shown in Fig. 5 for the dependence of the concealment on the mismatch in the differential delay times. It turns out that as |ξ| increases the peak sizes both in C(s) and DMI(s) decrease, achieving full concealment for a mismatch greater than 20%. In particular, the delay time signature is completely lost in C(s) already at a 10% mismatch in correspondence with the decay time of the autocorrelation function while the DMI decays even faster to a residual value, which, although small, remains distinguishable all the way up to 20% mismatch. The reason for this larger range of detection capability is that mutual information measures the relationship between variables beyond a linear correlation. In any case, for a differential delay mismatch above 20% not even mutual information is capable of finding traces of the internal delay times in the transmitted signal.

Fig. 5 Absolute value of the peaks in C(s) (a), and DMI (b), at T₂ (•), T₂ + δT₂ (∇) as a function of mismatch ξ = (δT₂ − δT₁)/δT₁ considering δT₁ = 400 ps. Solid line and bars correspond to the background mean value and standard deviation. A series of length 267 times T was used.

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We finally discuss the effect of the addition of a PRBS in the concealment of the delay times of the external loop, T₁ and T₁ + δT₁. As shown in Fig. 2 the addition of PRBS successfully conceals them for the autocorrelation function [Fig. 2(a)] but not for the DMI [Fig. 2(b)] (although the size of the peaks is significantly reduced). The fact that the PRBS does not completely suppress these peaks can be understood as follows. Without PRBS the size of the peaks signaling T₁ and T₁ + δT₁ is stronger than in the case of the serial configuration for the same parameters [28]. This indicates that the relationship between x₁(t) and its delayed version for the parallel configuration is stronger than for the serial one. The effective amplitude of the chaos driving the nonlinear term in Eq. (1) can be twice as large as that of the serial configuration since the signal delivered by the SL is successively modulated by x₁ and x₂. Since the mixing of the PRBS and the chaos is less balanced the delay time is not concealed. Despite that, the PRBS remains efficiently masked by the chaos as the cross-correlation between x₁(t) and R(t) is of the order of 10⁻³.

4. Synchronization

The signal sent to the receiver is taken at the PM₁ output, so x₂(t) has to be generated at the receiver, through an internal closed loop. This makes the receiver to operate in semi-closed loop, which is known to be very sensitive to synchronization. The quality of the synchronization depends on several factors, including the coupling strength, parameter mismatch, noise, degradation due to fiber propagation effects. The latter has been considered in [29, 30] where it is shown that compensating the losses by in-lining erbium-doped fiber amplifiers (EDFAs) every 50 km and using dispersion management techniques, one can minimize the fiber effects to the very acceptable level. Here we neglect the effect of noise fluctuations and parameter mismatch and we focus on the conditions for synchronization depending on the internal loop gain.

Considering the setup shown in Fig. 1, and proceeding in a similar way as in Ref. [28] for the serial set up, one finds that the receiver dynamics can be described by

y_{i} + τ_{i} \frac{d y_{i}}{d t} + \frac{1}{θ_{i}} v_{i} = G_{i} {cos}^{2} [Δ {(x_{1} + y_{2})}_{T_{i}} + Δ {(R^{'} + m)}_{T_{i}} + ϕ_{i}],

where dv_i/dt = y_i. Since the message is encoded in the phase it has to be demodulated. This is done using a standard differential phase shift keying demodulator consisting in an MZI with an imbalance delay time δT_m and a photodetector [28]. The detected power is

P \propto {cos}^{2} [\bar{Δ} {(x_{1} + m - y_{1})}_{δ T_{m}}]

The final demodulated message m′ is obtained from P. In the ideal case of perfect synchronization y₁ = x₁ and m′ reproduces the original message m.

Considering G₂ = 0 in Eq. (6), y₂(t) decays to zero after a characteristic time $2 τ_{2} / [1 - \sqrt{1 - 4 τ_{2} / θ_{2}}] \approx θ_{2}$ . The receiver therefore operates in open loop and consequently the synchronization is unconditional for a matched receiver as shown in [2]. This is also the case for the serial configuration [28] since its receiver always operates in open loop. Thus starting from G₂ = 0, and disregarding the message m(t) = 0, we gradually increase G₂ in order to investigate the range of G₂ for which synchronization is possible. This can be done estimating the largest conditional Lyapunov exponent (LCLE) [31] which states that the stability of the synchronization in a delayed system can be determined by looking at the growth of state vector δ ∈ 𝕃 (where 𝕃 is a suitable space function) constructed in the interval [t − T,t]. Since the system has four different delay times, T₁, T₁ + δT₁, T₂, T₂ + δT₂, we should consider the largest one T_D = T₂ + δT₂. Defining δ_i = y_i(t) − x_i(t) the LCLE defined in [31] can be modified for this system as

λ_{L} = lim_{t \to \infty} \frac{1}{t} ln {\frac{{[\int_{- T_{D}}^{0} δ_{1}^{2} (t + t^{'}) d t^{'}]}^{1 / 2}}{{[\int_{- T_{D}}^{0} δ_{1}^{2} (t^{'}) d t^{'}]}^{1 / 2}}} .

Stable synchronization occurs for λ_L < 0. By subtracting Eq. (1) from (6) and linearizing for δ_i, one obtains

δ_{i} + τ_{i} \frac{δ_{i}}{d t} + \frac{1}{θ_{i}} ε_{i} = - G_{i} Δ {(δ_{2})}_{T_{i}} sin [2 Δ {(x_{1} + x_{2})}_{T_{i}} + 2 Δ {(R)}_{T_{i}} + 2 ϕ_{i}],

where dε_i/dt = δ_i. Thus δ₁(t) to be used in Eq. (8) can be obtained by numerical integration of Eqs. (1) and (9). Note that λ_L depends implicitly on the feedback strengths G₁ and G₂. Synchronization between the external variables x₁(t) and y₁(t) is only possible if internal variables do synchronize first, i.e. δ₂(t) = 0. Once δ₂ = 0 the dynamics of δ₁ decays to zero as

δ_{1} \propto exp [(\sqrt{1 - 4 τ_{1} / θ_{1}} - 1) t / 2 τ_{1}]

. This allows to estimate the value of λ_L when synchronization takes place

λ_{L} = \frac{\sqrt{1 - 4 τ_{1} / θ_{1}} - 1}{2 τ_{1}} \approx - \frac{1}{θ_{1}} .

Figure 6(a) displays the LCLE as a function of G₂ for G₁ = 5 which corresponds to a relatively high gain in the external loop. Stable synchronization is found for

G_{2} < G_{2}^{th} \approx 3.2

. Furthermore, it can be seen that for all the values of G₂ for which x₁(t) and y₁(t) synchronize, i.e.

G_{2} < G_{2}^{th}

, the LCLE takes always the same value

θ_{1}^{- 1}

as predicted. This also corresponds to the characteristic time that the system would take to resynchronize after an eventual desynchronization. Beyond

G_{2}^{th}

, any small perturbation δ₁(t) or δ₂(t) grows in time and therefore λ_L becomes positive indicating desynchronization between the emitter and receiver. We have found that even setting R = 0, the range of values for G₂ for which synchronization takes place remains the same. Similar values for the synchronization threshold

G_{2}^{th}

are obtained for other values of the external loop gain G₁ provided G₁ > 3. Therefore in what follows we will consider only

G_{2} < G_{2}^{th}

.

Fig. 6 (a) Largest conditional Lyapunov exponent (LCLE) versus G₂ considering G₁ = 5, (b) Synchronization error σ in logarithmic scale.

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The quality of the synchronization between x₁(t) and y₁(t) can also be evaluated through the root-mean square synchronization error $σ = \sqrt{〈 δ_{1} {(t)}^{2} 〉 / 〈 x_{1} {(t)}^{2} 〉}$ . Figure 6(b) shows σ in logarithmic scale as function of G₂. As expected from the LCLE analysis there is perfect synchronization (σ < 10⁻¹³ corresponding to the numerical accuracy) up to $G_{2} = G_{2}^{th} \approx 3.2$ . Beyond this threshold value for G₂, the synchronization rapidly degrades as indicated by an error of order 1.

5. Effect of the PRBS on Synchronization

The PRBS will play a key role in parameter space dimension if the system is sensitive to PRBS mismatch. This sensitivity can be better appreciate by considering identical parameters between the emitter and the receiver. Thus, for R′ ≠ R the dynamics of δ_i(t) are given by

\begin{array}{l} δ_{i} + τ_{i} \frac{d δ_{i}}{d t} + \frac{1}{θ_{i}} ε_{i} = & - G_{i} sin [Δ {(δ_{2})}_{T_{i}} + Δ {(R^{'} - R)}_{T_{i}}] \\ \times sin [2 Δ {(x_{1} + y_{1})}_{T_{i}} + Δ {(δ_{2})}_{T_{i}} + Δ {(R + R^{'} + 2 m)}_{T_{i}} + 2 ϕ_{i}] . \end{array}

These equations indicate that for R′ ≠ R synchronization is degraded both on the internal and the transmitted variables since neither δ₂ nor δ₁ decay to zero.

Figure 7(a) shows the mean square synchronization error σ as a function of PRBS mismatch for different values of the internal loop gain G₂ while Fig. 7(b) shows the bit error rate (BER) of the recovered message. For G₂ = 0, there is no internal variable and therefore synchronization degradation relies on the effect of the PRBS on the transmitted variable. The synchronization error grows faster with the mismatch and just a mismatch fraction η = 1% in the PRBS leads to a synchronization error of 25% which corresponds to a quite poor synchronization. The BER grows linearly with the PRBS mismatch. The results obtained for G₂ = 0 coincide with those obtained in the serial configuration when both loops have a relatively large gain, G₁ = G₂ = 5 [28]. The reason for having coincident results is that in both cases the PRBS acts only on one of the variables. In fact in the serial loop configuration the PRBS acts always only on one of the variables. On the contrary, the parallel setup considered here allows for a single PRBS modulator to act simultaneously on both loops, leading to a much stronger effect as soon as the internal dynamics is switched on. As shown in Fig. 7 in the parallel configuration when increasing G₂, the degradation becomes stronger both in synchronization error and BER. As an illustration, for G₂ = 3 the degradation for η = 0.4% (i.e ≈ 131 mismatched bits in the receiver PRBS for a key 2¹⁵ = 32768 bit long) is equivalent to that obtained for η = 2% (i.e ≈ 655 mismatched PRBS bits) when G₂ = 0. In other words, the PRBS mismatch sensitivity for G₂ = 3 is 5 times larger than that obtained in the serial configuration with G₂ = 5 [28]. Using PRBS of different lengths leads to similar results, namely, the relevant parameter is the fraction of mismatched bits between emitter and receiver PRBS. Note that for bit rates lower than 1/δT_i, the effect of the PRBS is largely reduced. This is because at those low bit rates R(t) and R(t − δT_i) have the same value most of the time. The same happens for R′(t) and R′(t − δT_i). Therefore Δ(R′ − R)_{T_i} = R′(t − T_i) − R′(t − T_i − δT_i) − R(t − T_i) + R(t − T_i − δT_i) vanishes even if R and R′ are different.

Fig. 7 Influence of the PRBS-mismatch ratio η on (a) Synchronization evaluated through the root-mean square synchronization error σ without the message, and (b) on the BER for a 10Gb/s message. We have considered a PRBS R(t) of length 2¹⁵ = 32768 bits generated at 3 Gb/s, G₁ = 5 and G₂ = 0 (□), G₂ = 2 (▵), G₂ = 3 (•).

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6. Conclusions

We have studied an electro-optic phase chaos system with digital key based on two parallel electro-optic phase-chaos loops. This allows for the generation of two phase-chaos variables, one of which is transmitted to the receiver while the other remains internal. A suitable receiver is organized in a semi-closed loop configuration since it contains both an open loop for the transmitted variable and a closed one to regenerate the internal variable. Synchronization takes place even for moderate values of the internal loop gain up to G₂ ≈ 3.2. We have shown that the nonlinear dynamics of the system allows for a decorrelation between the internal variables and the transmitted signal so that the system intrinsically conceals the internal delay times. This was not possible in the serial configuration introduced before [28] which had to rely on an external digital key to conceal the delay times. The key ingredients for the intrinsic concealment are the parallel coupling of the loops and the operation using different differential delay feedback times for each loop.

Besides, the introduction of a digital key decreases the signature corresponding to the two delay times of the external loop although it does not completely suppress them. Interestingly, the parallel configuration allow for a single digital key to act on both dynamical variables leading to a much stronger effect on the synchronization degradation when the key is not matched as compared with the serial configuration [28]. Therefore the parallel configuration besides providing intrinsic time delay concealment also allows for the introduction of a digital key in a very effective way to increase the parameter space dimension. Both aspects contribute in a very significant way to enhance the confidentiality in chaos-based communications.

Acknowledgments

The authors thank L. Larger for valuable discussions. Financial support from MINECO, Spain, and Feder under Projects FIS2007-60327 (FISICOS) and TEC2009-14101 (DeCoDicA), by the EC Project PHOCUS (FP7-ICT-2009-C-240763) and from Comunitat Autònoma de les Illes Balears is acknowledged. R.M.N. acknowledges fellowship BES-2007-14627 under the FPI program of MINECO. Also, he acknowledges the Research Foundation Flanders (FWO) for project support and fellowships, and the Belgian Interuniversity Attraction Pole Network photonics@be.

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Electro-optic phase chaos systems with an internal variable and a digital key

Abstract

1. Introduction

2. System

3. Delay Time Concealment

4. Synchronization

5. Effect of the PRBS on Synchronization

6. Conclusions

Acknowledgments

References and links

Cited By

Figures (7)

Equations (11)

Optics Express