This section is intended to investigate in which conditions the delay time can be concealed. Before investigating delay time concealment, let us clarify what are the relevant times for successful decoding, that is, whether synchronization depends on the individual delays T1 and T2 , or only on the total delay T = T1 +T2.
Assuming R′ =R and identical parameters for emitter and receiver except for the delay times T1 and T2, by Fourier transforming Eqs. (5.2), (5.4), (5.6), and (5.8) and subsequently dividing Eqs. (5.2) by (5.6), one obtains
Y2(ω)
W2(ω) =e−iω(T2−T2′), (5.9) whereF(ω) stands for the Fourier transform off(t). The relationship betweenY2(ω) and W2(ω) given by Eq. (5.9) indicates that
w2(t) =y2(t+ (T2−T2′)). (5.10) Thus, by replacing Eq. (5.10) into the right hand side of Eq. (5.5) and subsequently Fourier transforming Eqs. (5.1), (5.3), (5.5), and (5.7), it turns out that
X1(ω)
Z1(ω) =e−iω(T1+T2−T1′−T2′), (5.11) As the total delay time is T = T1 +T2 for the emitter and T′ = T1′ +T2′ for the receiver, it turns out that
X1(ω) =Z1(ω) (5.12)
for T = T′ even when the individual delay times are different. Therefore, for iden-tical parameters between the emitter and the receiver and for R′ = R, x1(t) will synchronize with z1(t) provided T =T′. We have also numerically checked that the synchronization still takes place at the receiver as long as the overall delay timeT is the same for emitter and receiver, even if individual delays T1 and T2 are different.
Since real systems are always noisy, another interesting point would be to in-vestigate the robustness of the quantifiers to noise. Appropriate ones for unmasking
Figure 5.4: (a) AC, (b) DMI, (c) TDE and (d) FF functions as a function of an embedding delays computed from 107data points corresponding to a 10µs time series after adding a Gaussian white noise of amplitude 3.5% of the carrier amplitude. The time series were generated from a system without PRBS (R(t) = 0). All the results are normalized to 1.
the delay time signatures are those robust to noise. To figure out this issue, let us first consider a noisy series by adding to the carrier generated without PRBS a Gaussian white noise of amplitude 3.5% of the carrier amplitude. Figure 5.4 shows the delay time identification computed using the methods described in chapter 3.
As it can be seen, despite the presence of noise in the time series, clear peaks are found at T = T1 +T2, T +δT1, T +δT2 and T +δT1 +δT2 both in the autocor-relation (a) and in the delayed mutual information (b), evidencing therefore that even with noise, the delay times can still properly be identified. On the contrary, no clear peaks are distinguishable at the different delay times when computing the time distribution statistics (c) or the filling factor (d) from the same noisy series.
Thus taking into account the fact that experimental time series are always noisy, we henceforth focus only upon AC and DMI methods since TDE and FF methods are so sensitive to noise that even just a small noise added to the carrier prevent them to work properly. Once the quantifiers are chosen, we can henceforth consider again a free-noise time series in our analysis since it is the ideal case for an eavesdropper to attempt the delay time identification.
As in the previous section, we consider that no message is transmitted (m(t) =
89
Figure 5.5: Autocorrelation functionC(s) (a) and delayed mutual information DMI(s) (b) ofx1(t) without PRBS (red line), and with a PRBS of amplitude of π/2 at 3 Gb/s (black line). A time series of length 10µs with 107data points was used.
0) to show the role of the PRBS in the delay time identification. The graphs in Fig. 5.5 display the autocorrelation (a) and the DMI (b) computed from the trans-mitted phase proportional to x1(t), when no PRBS is used (red line) and with a PRBS at 3 b/s with an amplitude of π/2 (black line). In the first case both func-tions show peaks at T = T1+T2, T +δT1, T +δT2 and T +δT1 +δT2, so that all relevant delay times can be readily identified. The delay time signature vanishes completely when the PRBS is included.
Figs. 5.6 (a) and (b) show the size of peaks found in C(s) and in the DMI at the relevant delay times as a function of the PRBS bit rate considering an amplitude of π/2. The peaks are clearly distinguishable for zero bit rate (no PRBS). When increasing the bit rate, the peak size decreases and approaches the background value of these functions (green line). The background mean and standard deviation are calculated in Fig. 5.6 using the highest 2000 spurious local maxima (e.g. excluding the peaks corresponding to real delay times). For low bit rates R(t) and R(t−δT1) take the same value most of the time, so ∆(R) vanishes most of the time and therefore its effect is small (see the concept of temporal non locality as introduced in [76] and as discussed in Sec. 2.3). Therefore the peaks both in the DMI and in C(s) can still be distinguished from the background standard deviation, shown with bars in the figure. When the bit rate reaches a value corresponding to the inverse of δT1 (∼1.97 Gb/s), ∆(R)T1 is typically non zero, and the PRBS plays a key role in the dynamics, concealing the delay time peaks.
The size of the peaks as function of the PRBS modulation amplitude is shown in Figs. 5.6 (c) and (d). An important remark is that the PRBS modulation
am-Figure 5.6: Absolute value of the peaks inC(s) (a,c), and DMI (b,d), atT (•),T+δT2(),T+δT1
(+) andT+δT1+δT2(H) vs. the PRBS bit rate (upper row) and PRBS amplitude (bottom row).
In (a) and (b) the PRBS amplitude isπ/2 while in (c) and (d) the PRBS bit rate is 3Gb/s. Solid line and bars correspond to the background mean value and standard deviation. A series of length 267 timesT was used.
plitude is a π-periodic function associated to the periodicity of cos2 in Eq. (5.1).
Thus, a PRBS of amplitudeπ has no effect since ∆(R)T1 only takes values 0 orπand both are equivalent in the cos2 term. Efficient concealment occurs for amplitudes betweenπ/3 and 2π/3 approximately. We have found that this range increases when increasing G1 and/orG2.
Remarkably enough, while the PRBS conceals the delay time in the chaotic carrier x1(t), the cross-correlation between x1(t) and R(t) is of the order of 10−3, meaning that the digital key itself is also concealed in the chaotic carrier. This is explained by the fact that the interplay between balanced amplitudes of the chaos and a PRBS is optimizing the mutual nonlinear mixing, resulting in an efficient mutual masking of each signal by the other.
On the other hand, it should be noticed that for δT1 = δT2, the interplay
91
(a) (b)
-0.01 0 0.01 0.02
30 31 32 33 34
C(s)
s [ns]
0.0001 0.0002 0.0003 0.0004 0.0005
30 31 32 33 34
DMI(s)
s [ns]
Figure 5.7: Autocorrelation functionC(s) (a) and delayed mutual information DMI(s) (b) ofx1(t) considering ∆T1 = ∆T2 = 400 ps. A time series of length 10 µs with 107 data points was used.
All the results are normalized to 1.
Figure 5.8: Absolute value of the peaks in C(s) (a), and DMI (b), at T (•), T+ 2δT1+τ1+τ2
(),T+ 2δT1(▽) as a function of mismatchη= (δT2−δT1)/δT1 consideringδT1= 400 ps. Solid line and bars correspond to the background mean value and standard deviation. A series of length 267 timesT was used.
between δT1 and δT2 leads to a resonance and consequently, pronounced peaks are observed at T, T +δT1, T + 2δT1. Figures 5.7 (a) and (b) illustrate this issue. For the autocorrelation the middle peak is indeed located at T +δT1. For the DMI the middle peak is composed of two peaks one atT+δT1 and the other atT+δT1+τ1+τ2. This splitting may explain the small size of the intermediate peak in the DMI.
While typically the delay time signature is reduced when increasing the overall loop gain, we have found that whenδT1 =δT2the delay time can always be identified even forG1 =G2 = 15, way beyond experimental limits. Therefore for efficient delay time concealment, one needs to takeδT1 6=δT2.
The dependence of the concealment on the mismatch between the two shorter
delay times η = (δT2 −δT1)/δT1 is investigated in Fig. 5.8. It turns out that as such mismatch increases, the peak sizes both in C(s) and DMI(s) decrease so that it becomes very difficult to identify the delay time for a mismatch greater than about 20%. In particular, we note that the delay time signature is lost in the autocorrelation already at a 10% mismatch while the DMI allows the identification of the delay time up to 20% mismatch.