4. Chaotic OFDM encryption with PAPR reduction

and N is a large number of samples, the CCDF of PAPR of the signal is [14]

PðPAPR > zÞ ¼ 1 � PðPAPR ≤ zÞ ¼ 1 � FðzÞ

Digital chaos has recently attracted numerous applications in OFDM-PONs [4–9], especially for data encryption, which is mainly due to the chaotic characteristics including pseudorandomness, ergodicity, and high sensitivity to the initial values. Secure optical OFDM transmission is achieved by digital chaos, in which a huge key space is generated and predetermined by chaotic equations. Since the initial values and the other control parameters are utilized as the secure keys between OLT and ONUs, it provides a huge key space, which guarantees the physical-layer confidentiality. At ONUs, the same chaotic sequences are gener-

The fundamental properties of digital chaos can be described, for example, via a 4-dimensional

x\_ ¼ að�x þ yÞ þ yzu y\_ ¼ bðx þ yÞ � xzu z\_ ¼ cy � u þ dxyu u\_ ¼ �eu þ xyz

where a, b, c, d, and e are constant parameters. When a = 35, b = 10, c = 80, d = 0.5, and e = 10, these appropriate initial values bring the system into chaotic zones. Eq. (5) can be solved by Runge-Kutta method with a time step of h = 0.001. The solutions of Eq. (5) output the attractor diagrams of the 4D chaos, as plotted in Figure 2, where excellent chaotic behaviors in terms of

In digital chaos, the chaotic state is very sensitive with respect to the initial values, thus even a tiny change or modification of the original initial values will let it enter into another different chaotic state. In Figure 3(a), the variation curve is illustrated for the digital chaotic sequence {yi}, under a slight change (1 · 10�15) in the initial value of y0. The auto- and cross-correlation functions Rac(τ) and Rcc(τ) are plotted in Figure 3(b) and (c) respectively, which reveal the high sensitivity associated with the chaotic initial values, where τ is the time lag. The good quality of randomness observed in Figure 3 is essential to guarantee high-level security reliability for

**-20 -10 <sup>0</sup> <sup>10</sup> <sup>20</sup> -40**

**x**

**-20 -10 <sup>0</sup> <sup>10</sup> <sup>20</sup> -10**

**x**

**u**

**z**

3. Characteristics of digital chaos

320 Optical Fiber and Wireless Communications

(4D) hyper chaos [15]

data encryption.

ated using the same keys for data recovering after reception.

pseudo-randomness and phase dynamics are illustrated.

**-20 -10 <sup>0</sup> <sup>10</sup> <sup>20</sup> -10**

Figure 2. Chaotic attractor diagrams in phase planes of (x, y), (x, z), and (x, u).

**x**

**y**

8 >><

>>:

<sup>N</sup> <sup>¼</sup> <sup>1</sup> � ð<sup>1</sup> � <sup>e</sup>

�z Þ

<sup>N</sup> <sup>ð</sup>4<sup>Þ</sup>

ð5Þ

A variety of chaotic encryption schemes has been proposed for the physical-layer security in OFDM-PON [4–9], in which digital chaos has been incorporated into OFDM signal generation or modulation to enhance data security during transmission. However, in the existing chaotic secure schemes, the encryption is generally achieved without considering the improvement of transmission performance through the effective PAPR reduction in OFDM signals.

This section presents some systematic novel approaches to enhance the physical-layer security and jointly improve the OFDM transmission performances. The key idea behind these schemes is the combination of chaos-based encryption along with PAPR reduction; these approaches mainly belong to two categories of PAPR reduction: signal scrambling and matrix precoding, such as


#### 4.1. Chaotic signal scrambling

Signal scrambling is one of the traditional approaches in PAPR reduction. Chaotic sequences can be employed to scramble an input data block of OFDM symbols randomly, and then one of the scrambled OFDM signals with minimum PAPR is selected for final transmission. With the use of chaotic sequences in OFDM symbol scrambling, the physical-layer security is enhanced since the initial keys are required for correct recovery of the original OFDM data via correct counter-scrambling. At the same time, the transmission performance is improved simultaneously, owing to the effective reduction in PAPR via signal scrambling.

#### 4.1.1. Chaotic partial transmit sequence (PTS)

The block diagram of OFDM data encryption using chaotic PTS is plotted in Figure 4 [17]. After serial-to-parallel (S/P) conversion, a pseudo-random binary sequence (PRBS) is mapped onto QAM subcarriers, and then is sent for data encryption, which is applied via the following threefold encryption: chaotic training sequence (TS) insertion, chaotic random partition generation, and chaotic phase weighing factors generation.

Figure 5(a)–(c) shows the different partition examples in PTS scheme. Compared with the adjacent and interleaved partitions, the random partition provides the weakest cross-correlation among the sub-blocks, which leads to the largest PAPR reduction after selecting the optimum phase weighing factors. The PAPR can be minimized after searching for the optimized OFDM symbols for chaotic (random) signal scrambling.

Figure 4. Schematic diagram of OFDM signal encryption using chaotic PTS.

Figure 5. Partition design in PTS, for (a) adjacent; (b) interleaved; (c) random partitions.

To generate digitized chaotic sequences for OFDM signal encryption, the chaotic sequences obtained in Eq. (5) are processed using [18]

4.1. Chaotic signal scrambling

322 Optical Fiber and Wireless Communications

4.1.1. Chaotic partial transmit sequence (PTS)

ation, and chaotic phase weighing factors generation.

symbols for chaotic (random) signal scrambling.

**Offline DSP**

**S/P**

**QAM Mapper**

Figure 4. Schematic diagram of OFDM signal encryption using chaotic PTS.

(1) *Xs*

(1) *Xs*

**(a)**

**(b)**

**(c)**

(1) *Xs*

Figure 5. Partition design in PTS, for (a) adjacent; (b) interleaved; (c) random partitions.

**Chaotic training** 

**sequence**

**PRBS**

Signal scrambling is one of the traditional approaches in PAPR reduction. Chaotic sequences can be employed to scramble an input data block of OFDM symbols randomly, and then one of the scrambled OFDM signals with minimum PAPR is selected for final transmission. With the use of chaotic sequences in OFDM symbol scrambling, the physical-layer security is enhanced since the initial keys are required for correct recovery of the original OFDM data via correct counter-scrambling. At the same time, the transmission performance is improved simulta-

The block diagram of OFDM data encryption using chaotic PTS is plotted in Figure 4 [17]. After serial-to-parallel (S/P) conversion, a pseudo-random binary sequence (PRBS) is mapped onto QAM subcarriers, and then is sent for data encryption, which is applied via the following threefold encryption: chaotic training sequence (TS) insertion, chaotic random partition gener-

Figure 5(a)–(c) shows the different partition examples in PTS scheme. Compared with the adjacent and interleaved partitions, the random partition provides the weakest cross-correlation among the sub-blocks, which leads to the largest PAPR reduction after selecting the optimum phase weighing factors. The PAPR can be minimized after searching for the optimized OFDM

**IFFT**

**X1 R1**

**IFFT**

···

**IFFT**

**Phase Weighing Factor** 

**x y u**

(2) *X <sup>s</sup>*

*s*

(2) *X <sup>s</sup>*

*s*

*s*

(2) *X <sup>s</sup>*

**P/S**

···

*s*

*s*

*s*

···

···

**Cyclic Prefix**

**+**

···

**RN**

**R2**

**Keys Chaotic System**

**PTS Partition**

**XN**

**z**

**X2**

**Encryption**

neously, owing to the effective reduction in PAPR via signal scrambling.

$$D\_{x,i} = \text{mod}(\text{Extract}(\mathbf{x}\_i, m, n, p), M) \tag{6}$$

where Extract(xi, m, n, p) outputs an integer, which is obtained by the mth, nth, and pth digits in the decimal part of xi, mod(a, b) outputs the remainder of a divided by b, M is the maximum digital value in the digital sequence, which is 256 in all of our schemes [4]. Similarly, the other sequences {yi}, {zi}, and {ui} can be digitalized into {Dy,i}, {Dz,i}, and{Du,i}. The details of chaotic PTS are given as follows.

The first sequence {Dx,i} is applied to generate the chaotic TS for time synchronization of OFDM symbols. Since the illegal ONU does not have the information of chaotic TS, it has to try (N0+N)/N0 times on average to realize correct synchronization and demodulation, where N<sup>0</sup> is the length of cyclic prefix (CP). Moreover, due to the pseudo-random characteristic of digital chaos, it produces an autocorrelation similar to the δ function, which is advantageous for accurate OFDM symbol synchronization, as shown in Figure 6(a). A legitimate ONU can correctly demodulate the original data; however, with a slight change in the initial keys, the peak does not appear, so that the data cannot be fully recovered, as illustrated in Figure 6(b).

In chaotic (random) PTS, the chaotic sequences {Dy,i}, {Dz,i}, and {Du,i} can be employed to generate the partition information and the phase weighing factors. First, the digitized chaotic sequence {Dy,i} is applied to generate the binary sequence {A1} using {mod(Dy,i,2)}. Similarly, the binary sequence can be generated by {Dz,i}. After bitwise logical operation of these two binary sequences, as shown in Figure 7, the chaotic random partition information {P(l)} is formed. The sub-blocks are then generated using X(l)= X� P(l), <sup>l</sup> =1, 2,���, <sup>L</sup>. where <sup>L</sup> denotes the total number of sub-blocks.

Figure 6. Effect of OFDM symbol time synchronization with (a) the right key; (b) a wrong key.


Figure 7. Generation of partition information (four sub-blocks) in chaotic PTS.

Second, the digitalized chaotic sequence {Du,i} is applied to generate the phase weighing factors Rl in PTS,

$$R\_l = \exp\left(\mathbf{j} \cdot 2\pi \cdot D\_{\mathbf{u},i}/M\right) \quad (\mathbf{i} = 1, 2, \dots, K) \tag{7}$$

where K is the total number of phase weighing factors. After multiplying the phase weighing factors and transforming from frequency into time-domain, the corresponding time-domain vector becomes

$$\mathbf{x} = \sum\_{l=1}^{L} \mathbf{R}\_{l} \mathbf{x}^{(l)} \tag{8}$$

$$\mathbf{x}\_n^{(l)} = \frac{1}{\sqrt{N}} \sum\_{s=0}^{N-1} X\_s^{(l)} \exp\left(j \frac{2\pi s n}{N}\right) \tag{9}$$

where x(l) is the IFFT of X(l), s and n denote the indices of OFDM subcarrier and symbol, respectively.

In PTS, the combinations of sub-blocks and phase weighing factors are calculated to find the minimum PAPR, thus the final PAPR is

$$PAPR\_{\text{FTS}} = \min\left(PAPR\left(\sum\_{l=1}^{L} R\_i x^{(l)}\right)\right) \tag{10}$$

The final OFDM vector becomes

$$\mathbf{x}\_{\text{pts}} = \sum\_{l=1}^{L} \mathbf{R}\_{l, \text{opt}} \mathbf{x}^{(l)} \tag{11}$$

where Rl,opt is the optimized phase weighing factors. At ONUs, the decryption and demodulation procedures are the inverse operations that are implemented at OLT.

Since the initial values in digital chaos serve as the secure keys to recover the original OFDM signals, only legal ONUs can generate the same chaotic scrambling information with the correct secure keys that applied at OLT. By introducing digital chaos into signal scrambling, not only the security confidentiality of OFDM signal transmission is greatly enhanced, but also a performance-improved transmission can be expected with a lower PAPR.

As for the computational complexity, it is the same for the chaotic PTS and conventional random PTS. If compared with the case of OFDM transmission without PTS, the computational complexity of chaotic PTS scheme will be L<sup>k</sup> times higher, which is inevitable for PAPR optimization. In our case, assuming four sub-blocks and two phase weighing factors, the computational complexity is 16 times higher for PAPR optimization; however, it guarantees both higher security and lower PAPR. Since the number of possible PTS partition states is 4<sup>N</sup> (4256 ≈ 10154 in our case), it is hardly to be broken through brutal-force attacks. However, chaotic PTS still requires sideband information to transmit the partition and the phase factor information, which reduces the overall spectral efficiency in transmission.

#### 4.1.2. Chaotic selected mapping (SLM)

Second, the digitalized chaotic sequence {Du,i} is applied to generate the phase weighing

where K is the total number of phase weighing factors. After multiplying the phase weighing factors and transforming from frequency into time-domain, the corresponding time-domain

> <sup>x</sup> <sup>¼</sup> <sup>X</sup> L

> > N X�1 s¼0 Xðl<sup>Þ</sup> <sup>s</sup> exp j

xðl<sup>Þ</sup> <sup>n</sup> <sup>¼</sup> <sup>1</sup> ffiffiffiffi N p

PAPRPTS ¼ min

tion procedures are the inverse operations that are implemented at OLT.

a performance-improved transmission can be expected with a lower PAPR.

information, which reduces the overall spectral efficiency in transmission.

l¼1

where x(l) is the IFFT of X(l), s and n denote the indices of OFDM subcarrier and symbol,

In PTS, the combinations of sub-blocks and phase weighing factors are calculated to find the

� PAPR

xpts <sup>¼</sup> <sup>X</sup> L

l¼1

where Rl,opt is the optimized phase weighing factors. At ONUs, the decryption and demodula-

Since the initial values in digital chaos serve as the secure keys to recover the original OFDM signals, only legal ONUs can generate the same chaotic scrambling information with the correct secure keys that applied at OLT. By introducing digital chaos into signal scrambling, not only the security confidentiality of OFDM signal transmission is greatly enhanced, but also

As for the computational complexity, it is the same for the chaotic PTS and conventional random PTS. If compared with the case of OFDM transmission without PTS, the computational complexity of chaotic PTS scheme will be L<sup>k</sup> times higher, which is inevitable for PAPR optimization. In our case, assuming four sub-blocks and two phase weighing factors, the computational complexity is 16 times higher for PAPR optimization; however, it guarantees both higher security and lower PAPR. Since the number of possible PTS partition states is 4<sup>N</sup> (4256 ≈ 10154 in our case), it is hardly to be broken through brutal-force attacks. However, chaotic PTS still requires sideband information to transmit the partition and the phase factor

Rl ¼ exp ðj � 2π � Du,i=MÞ ði ¼ 1, 2,…, KÞ ð7Þ

2πsn N � �

�X L

l¼1

Rixðl<sup>Þ</sup> ��

Rixðl<sup>Þ</sup> <sup>ð</sup>8<sup>Þ</sup>

Rl,optxðl<sup>Þ</sup> <sup>ð</sup>11<sup>Þ</sup>

ð9Þ

ð10Þ

factors Rl in PTS,

324 Optical Fiber and Wireless Communications

vector becomes

respectively.

minimum PAPR, thus the final PAPR is

The final OFDM vector becomes

As shown in Figure 8, chaotic SLM is another alternative data encryption scheme based on signal scrambling using digital chaotic sequences. In chaotic SLM, the digitized chaotic sequences can be obtained by processing of the chaotic sequences from 2D Henon chaos [19],

$$\begin{cases} \mathbf{x}\_{i+1} = 1 - \eta \mathbf{x}\_i^2 + y\_i \\ y\_{i+1} = \psi \mathbf{x}\_i \end{cases} \tag{12}$$

The input OFDM sequence is denoted as X = [X0, X1,…, XN�1] in frequency domain, where Xk represents the complex data of the kth subcarrier, and N represents the total number of subcarriers. First, to implement the digital chaos into SLM scheme, the total number of the chaotic phase sequences is set to be V, and all of the phase sequences Pv = [Pv,0,Pv,1, …, Pv, N�1] (1 ≤ v ≤ V) are obtained from the chaotic sequence {Dy,i}, where Pv, <sup>k</sup> ¼ expðj � 2π � Dy, <sup>k</sup>=MÞ,ð0 ≤ k ≤ N � 1Þ. By component-wise vector multiplication of the input OFDM sequence X with the phase sequence fPvg, the input OFDM signals are encrypted fully via the chaotic sequences obtained in Eq. (12). The encrypted OFDM sequence {Yvg is

$$Y\_v = X \otimes P\_v = [X\_0 P\_{v,0\prime} X\_1 P\_{v,1\prime} \cdots, X\_{N-1} P\_{v,N-1}] \tag{13}$$

where ⊗ denotes the component-wise vector multiplication. Second, the IFFT is performed for all of the encrypted OFDM sequences {Yv}

$$\mathbf{x}\_v = \text{IFFT}(\mathbf{Y}\_v) \tag{14}$$

Finally, the OFDM signal sequence {xv} with minimum PAPR is chosen for OFDM transmission. It should be mentioned that in terms of computational complexity, better PAPR reduction can be expected with a larger V; however, the complexity will also increase due to the increased times of IFFT.

Figure 8. Schematic diagram of OFDM signal encryption using chaotic SLM.

#### 4.1.3. Experiment results of chaotic PTS and SLM

The experimental setup of encrypted OFDM-PON transmission using chaotic PTS/SLM is shown in Figure 9. The input data stream was mapped onto 129 OFDM subcarriers at OLT, where 64 subcarriers were in the format of 16-QAM, and one subcarrier was unfilled with DC. The other 64 subcarriers had to be set as the complex conjugate of the aforementioned 64 subcarriers, where Hermitian symmetry had to be satisfied to realize intensity modulation and direct detection (IM/DD) after IFFT. After TS insertion for OFDM symbol time synchronization, chaotic PTS/SLM optimization and parallel-to-serial (P/S) conversion, a CP of 1/8 length of OFDM symbol was added into each OFDM symbol.

The above processing steps were performed offline using Matlab programs. The encrypted signal was then loaded into an arbitrary waveform generator (AWG), which had a sample rate of 10 Gs/s of electrical OFDM signal generation. A continuous-wave (CW) laser (λ = 1550.3 nm) was applied as the optical carrier. After that, the electrical signal in AWG was changed into optical by an optical Mach-Zehnder modulator (MZM). Thus, the net data rate applied was actually at 8.9 Gb/s (10 Gs/s · 4 · 64/256 · 8/9), with the electrical bandwidth of 2.5 GHz (10 Gs/ s · 65/256). At ONU, the received data were recorded with a 20 Gs/s real-time oscilloscope after direct detection via a 10-GHz photodiode (PD). The waveform, electric and optical spectra of OFDM signals are measured and plotted in Figure 10.

The CCDFs of encrypted OFDM signals are plotted in Figure 11, where 10,000 OFDM symbols with 16-QAM modulation format were employed. From Figure 11 it can be noted that, by employing chaotic signal scrambling, significant PAPR reduction is achieved, with respect to the conventional OFDM transmission without any signal scrambling. The total number of subblocks and phase factors applied in PTS was L = 4 and K = 2 respectively, and the phase factor in SLM was V = 4.

Figure 9. Experimental setup of encrypted OFDM transmission using chaotic PTS/SLM.

Figure 10. Encrypted OFDM signals. (a) Waveform; (b) electric spectrum; (c) optical spectrum.

Figure 11. CCDFs of PAPR in chaotic (a) PTS; (b) SLM.

4.1.3. Experiment results of chaotic PTS and SLM

326 Optical Fiber and Wireless Communications

of OFDM symbol was added into each OFDM symbol.

OFDM signals are measured and plotted in Figure 10.

**Offline DSP**

**encrypted OFDM signal**

**0 5 10 15**

**Time [μs]**

**-1**

**0** 

**Amplitude [V]**

**1** 

**MZM**

Figure 9. Experimental setup of encrypted OFDM transmission using chaotic PTS/SLM.

**10 GSa/s AWG**

in SLM was V = 4.

OLT

**PRBS Keys**

The experimental setup of encrypted OFDM-PON transmission using chaotic PTS/SLM is shown in Figure 9. The input data stream was mapped onto 129 OFDM subcarriers at OLT, where 64 subcarriers were in the format of 16-QAM, and one subcarrier was unfilled with DC. The other 64 subcarriers had to be set as the complex conjugate of the aforementioned 64 subcarriers, where Hermitian symmetry had to be satisfied to realize intensity modulation and direct detection (IM/DD) after IFFT. After TS insertion for OFDM symbol time synchronization, chaotic PTS/SLM optimization and parallel-to-serial (P/S) conversion, a CP of 1/8 length

The above processing steps were performed offline using Matlab programs. The encrypted signal was then loaded into an arbitrary waveform generator (AWG), which had a sample rate of 10 Gs/s of electrical OFDM signal generation. A continuous-wave (CW) laser (λ = 1550.3 nm) was applied as the optical carrier. After that, the electrical signal in AWG was changed into optical by an optical Mach-Zehnder modulator (MZM). Thus, the net data rate applied was actually at 8.9 Gb/s (10 Gs/s · 4 · 64/256 · 8/9), with the electrical bandwidth of 2.5 GHz (10 Gs/ s · 65/256). At ONU, the received data were recorded with a 20 Gs/s real-time oscilloscope after direct detection via a 10-GHz photodiode (PD). The waveform, electric and optical spectra of

The CCDFs of encrypted OFDM signals are plotted in Figure 11, where 10,000 OFDM symbols with 16-QAM modulation format were employed. From Figure 11 it can be noted that, by employing chaotic signal scrambling, significant PAPR reduction is achieved, with respect to the conventional OFDM transmission without any signal scrambling. The total number of subblocks and phase factors applied in PTS was L = 4 and K = 2 respectively, and the phase factor

> **20Km SMF 1:n CW Laser PD**

> > **(a) (b) (c)**

**Frequency [GHz]**

**<sup>0</sup> <sup>1</sup> <sup>2</sup> <sup>3</sup> -3**

**-2 -1 0**

**Amplitude [dB]**

Figure 10. Encrypted OFDM signals. (a) Waveform; (b) electric spectrum; (c) optical spectrum.

**20 GSa/s Oscilloscope**

**<sup>1550</sup> <sup>1551</sup> -80**

**Wavelength [nm]**

Illegal ONU

ONU1 ONU2

**-60 -40 -20 0**

**Power [dBm]**

···

**Offline DSP**

The bit error ratio (BER) measurements of secure OFDM transmission using chaotic PTS/SLM are plotted in Figure 12. The OFDM signals can be correctly decrypted after transmission over 20 km standard single-mode fiber (SSMF) for legitimate ONUs with the correct keys, while any illegal ONU cannot decrypt OFDM signals with a wrong key, which is even a tiny discrepancy of 1 · 10<sup>15</sup> away from the correct key. The corresponding constellations are plotted as the insets in Figure 12. If compared with the conventional OFDM transmission, the BER performance was improved ~1 dB (BER@10-3) for encrypted OFDM signals, if compared with backto-back (B2B) signals, which can mainly be attributed to the effective reduction of PAPR through the signal scrambling schemes, either chaotic PTS or SLM.

#### 4.2. Chaotic precoding of OFDM signals

Matrix precoding is a set of typical approaches for PAPR reduction, which provides unique properties such as lower computational complexity and higher spectral efficiency (without requirement of sideband), if compared with the signal scrambling schemes. Actually, PAPR

Figure 12. BERs of encrypted OFDM transmission using chaotic (a) PTS; (b) SLM.

reduction is achieved through the proper selection and distribution of power between OFDM blocks. In this section, the digital chaos will be employed into the precoding schemes, aimed to construct or reconfigure the standard precoding matrices into alternative chaotic ones, so that the security can be enhanced during transmission via the introduction of digital chaos into the procedure of OFDM signal precoding, and jointly the PAPR can be reduced.

#### 4.2.1. Chaotic discrete Fourier transform spread OFDM (DFT-S-OFDM)

As depicted in Figure 13, the DFT-S-OFDM data encryption is implemented multi-fold, such as chaotic TS insertion, chaotic DFT matrix generation, and chaotic subcarrier allocation, all of which are predetermined by digital chaotic sequences [20]. The details of encryption are given as follows.

In the first fold of encryption, one chaotic sequence is used to generate TS, the same way as described in the previous chaotic signal scrambling schemes. The second fold of encryption is to generate the chaotic reconfigurable DFT precoding matrix. Assuming F is the standard M · M DFT matrix, the matrix element is given by

$$\mathbf{F}\_{\alpha,\beta} = \frac{1}{\sqrt{M}} e^{-j2\pi(\alpha - 1)(\beta - 1)/M}, \ 0 \le \alpha, \beta \le M - 1 \tag{15}$$

where α, β are the indexes of row and column, respectively. From Eq. (15), a reconfigurable matrix can be generalized

$$\mathbf{F}'\_{\mathbf{a},\mathfrak{P}} = \frac{1}{\sqrt{M}} e^{-j2\pi(a-m)(\mathfrak{P}-n)/M},\ 0 \le \alpha, \beta \le M-1 \tag{16}$$

where m, n are the real constants, which can be predetermined using chaotic sequences obtained in Eq. (5) as

$$m\_{y,i} = \dots + 10n\_1 + n\_2 + 0.1n\_3 + 0.01n\_4 + \dotsb \tag{17}$$

where nj donates the njth digit in the decimal part of y. Similarly, {nz,i} can be generated from {zi} in the same way as {my,i}.

Figure 13. Schematic diagram of chaotic encrypted DFT-S-OFDM signal transmission.

Mathematically, the same basic features in the original standard F are kept for the reconfigurable DFT matrix F<sup>0</sup> . For example, the inverse matrix of F<sup>0</sup> is equal to its conjugate matrix. Moreover, all of the elements in F<sup>0</sup> are in geometric sequence with the same common ratio of e�j2π=<sup>M</sup> for the same row or column indexes.

reduction is achieved through the proper selection and distribution of power between OFDM blocks. In this section, the digital chaos will be employed into the precoding schemes, aimed to construct or reconfigure the standard precoding matrices into alternative chaotic ones, so that the security can be enhanced during transmission via the introduction of digital chaos into the

As depicted in Figure 13, the DFT-S-OFDM data encryption is implemented multi-fold, such as chaotic TS insertion, chaotic DFT matrix generation, and chaotic subcarrier allocation, all of which are predetermined by digital chaotic sequences [20]. The details of encryption are given

In the first fold of encryption, one chaotic sequence is used to generate TS, the same way as described in the previous chaotic signal scrambling schemes. The second fold of encryption is to generate the chaotic reconfigurable DFT precoding matrix. Assuming F is the standard

where α, β are the indexes of row and column, respectively. From Eq. (15), a reconfigurable

where m, n are the real constants, which can be predetermined using chaotic sequences

where nj donates the njth digit in the decimal part of y. Similarly, {nz,i} can be generated from

�j2πðα�1Þðβ�1Þ<sup>=</sup>M, <sup>0</sup> <sup>≤</sup> <sup>α</sup>, <sup>β</sup> <sup>≤</sup> <sup>M</sup> � <sup>1</sup> <sup>ð</sup>15<sup>Þ</sup>

�j2πðα�mÞðβ�nÞ<sup>=</sup>M, <sup>0</sup> <sup>≤</sup> <sup>α</sup>, <sup>β</sup> <sup>≤</sup> <sup>M</sup> � <sup>1</sup> <sup>ð</sup>16<sup>Þ</sup>

**P/S**

···

**Cyclic Prefix**

my,i ¼ ⋯ þ 10n1þn<sup>2</sup> þ 0:1n<sup>3</sup> þ 0:01n<sup>4</sup> þ ⋯ ð17Þ

**Size-N IFFT**

···

**ZP**

**M**

**N**

**ZP**

procedure of OFDM signal precoding, and jointly the PAPR can be reduced.

4.2.1. Chaotic discrete Fourier transform spread OFDM (DFT-S-OFDM)

M · M DFT matrix, the matrix element is given by

F 0 <sup>α</sup>,<sup>β</sup> <sup>¼</sup> <sup>1</sup>

<sup>F</sup>α, <sup>β</sup> <sup>¼</sup> <sup>1</sup>

ffiffiffiffiffi M p e

ffiffiffiffiffi M p e

**Keys Digital Chaos**

Figure 13. Schematic diagram of chaotic encrypted DFT-S-OFDM signal transmission.

**Synchronization**

**TS insert** **DFT matrix**

**Encryption**

**s1 s2**

**sM**··· **precoder**

**tM**

**t2**

**t1**

···

**x y u**

**z**

**Subcarrier**

**allocation**

as follows.

matrix can be generalized

328 Optical Fiber and Wireless Communications

obtained in Eq. (5) as

{zi} in the same way as {my,i}.

**PRBS**

**S/P**

**Offline DSP**

**QAM Mapper** To verify the capability of PAPR reduction of OFDM signals by the new DFT matrix F<sup>0</sup> , the upper bound of the peak factor should be considered, which is the square root of PAPR and depends on the aperiodic autocorrelation functions of the input symbols.

$$\gamma \le 1 + \frac{2}{N} \sum\_{n=1}^{N} |\rho\_n| \tag{18}$$

$$\rho\_n = \sum\_{l=1}^{N-l} a\_l a\_{l+n} \quad n = 1, \ 2, \ldots N-1 \tag{19}$$

where ρ<sup>n</sup> are the aperiodic autocorrelation coefficients, an are the input symbols (QAM, QPSK, etc.), and N is the total number of subcarriers. Eq. (18) shows that, if the autocorrelation coefficients in the corresponding input symbols are small, the peak factor of OFDM signals will be small.

Assuming the <sup>k</sup>th OFDM symbol is <sup>s</sup>ðk<sup>Þ</sup> <sup>≜</sup> <sup>½</sup><sup>s</sup> ðkÞ <sup>1</sup> ,s ðkÞ <sup>2</sup> ,⋯,s ðkÞ <sup>M</sup> � <sup>T</sup>, where T denotes the transpose of matrix, the DFT precoded OFDM signal becomes

$$\mathbf{r}^{(\mathbf{k})} \triangleq [t\_1^{(\mathbf{k})}, t\_2^{(\mathbf{k})}, \dots, t\_M^{(\mathbf{k})}]^T = \mathbf{F}'\mathbf{s}^{(\mathbf{k})} \tag{20}$$

As plotted in Figure 14(a), the same non-periodic autocorrelation functions are obtained for the OFDM symbol t (k) , either precoded with DFT matrix F or F<sup>0</sup> . From Figure 14(a), it should be noted that different shapes of sidelobe appear in the autocorrelation functions, while the sidelobe shapes of OFDM signals with DFT precoding are lower than those without DFT precoding, when the same modulation format is applied. This verifies that, effective PAPR reduction of OFDM signals is achieved by DFT precoding with the reconfigurable matrix F<sup>0</sup> , the same effect as the standard DFT matrix F.

Figure 14. PAPR reduction using chaotic DFT, (a) normalized aperiodic autocorrelation function; (b) CCDFs of PAPR.

To directly evaluate the PAPR reduction, the CCDF curves is plotted in Figure 14(b), for the cases of OFDM signals with and without the reconfigurable DFT precoding, where 100,000 OFDM symbols are applied in the formats of QPSK, 16-QAM, and 64-QAM, while Nyquist sampling rate is adopted. It can be observed that, if compared with the cases without DFT, the OFDM signals precoding with the reconfigurable matrix DFT can reduce the PAPR, regardless of the modulation formats, while the differences at the probability of 10�<sup>4</sup> are about 3, 2.3, and 1.9 dB respectively for QPSK, 16-QAM, and 64-QAM. The CCDF curves in Figure 14(a) prove that, the new reconfigurable DFT does significantly reduce the PAPR of OFDM signals.

Finally, the fourth chaotic sequence {ui} is used to generate the scrambling criterion for subcarrier allocation before ZP. After an array {wi} (i=1,…,M) is obtained from {ui}, where M is the total number of subcarriers (as shown in Figure 13), the elements in {wi} are swapped into {wi 0 } in the new order with respect to their values, for instance, from the lowest value to the highest. Then the new index order of subcarriers is rearranged as that of {wi 0 }, details as shown in Figure 15. For example, assuming an original chaotic array of {0:8, 0:3, 0:5, 1:2, 1:1}, it is swapped from the small value to the big one, and then the new array f0:3, 0:5, 0:8, 1:1, 1:2g can be obtained. In the original OFDM symbol, the first subcarrier is changed into the third position of the new OFDM symbol; similarly, the second is assigned to the first position, and so on. Following this scrambling criterion, a key space of M! is created for OFDM data encryption.

The setup of IM/DD OFDM transmission experiment was the same as applied in the previous signal scrambling schemes; however, with the FFT points of 1024 and the number of subcarriers of 384, consequently the data rate was 13.3 Gb/s (10Gs/s · 4 · 384/1024 · 8/9). The BER measurements were shown in Figure 16, for the encrypted 16-QAM OFDM signals. The encrypted signals with DFT-S-OFDM precoding were fully recovered by the legal ONUs. Due to the effective PAPR reduction via DFT precoding, the BER performance was improved ~2 dB (BER@10�<sup>3</sup> ) for the encrypted DFT-S-OFDM signals, if compared with the cases of original OFDM transmission without DFT precoding. The power penalty was only 0.9 dB (at BER@10�<sup>3</sup> ) between the case of B2B and transmission after 20 km SSMF. Moreover, for any illegal ONU with a wrong key, even a very small deviation of 1 · 10�15, it was not possible to decrypt the original OFDM signals. The corresponding constellations for correct and incorrect chaotic keys are plotted as insets in Figure 16, which verified the security feasibility of chaotic DFT-S-OFDM signal transmission.

To quantitatively evaluate the performance dependence on the mismatches of row or column parameters Δm, Δn in the reconfigurable DFT matrix F<sup>0</sup> , the BERs are shown with respect to the parameters Δm, Δn in Figure 17, where the modulation format is 16-QAM, and the received optical power is �9 dBm. In Figure 17(a), when Δm > 0:01, the BER is significantly deteriorated

Figure 15. Subcarrier allocation in OFDM symbol using digital chaotic sequences.

Figure 16. BERs of chaotic encrypted DFT-S-OFDM transmission.

To directly evaluate the PAPR reduction, the CCDF curves is plotted in Figure 14(b), for the cases of OFDM signals with and without the reconfigurable DFT precoding, where 100,000 OFDM symbols are applied in the formats of QPSK, 16-QAM, and 64-QAM, while Nyquist sampling rate is adopted. It can be observed that, if compared with the cases without DFT, the OFDM signals precoding with the reconfigurable matrix DFT can reduce the PAPR, regardless of the modulation formats, while the differences at the probability of 10�<sup>4</sup> are about 3, 2.3, and 1.9 dB respectively for QPSK, 16-QAM, and 64-QAM. The CCDF curves in Figure 14(a) prove that, the new reconfigurable DFT does significantly reduce the PAPR of OFDM signals.

Finally, the fourth chaotic sequence {ui} is used to generate the scrambling criterion for subcarrier allocation before ZP. After an array {wi} (i=1,…,M) is obtained from {ui}, where M is the total

new order with respect to their values, for instance, from the lowest value to the highest. Then

For example, assuming an original chaotic array of {0:8, 0:3, 0:5, 1:2, 1:1}, it is swapped from the small value to the big one, and then the new array f0:3, 0:5, 0:8, 1:1, 1:2g can be obtained. In the original OFDM symbol, the first subcarrier is changed into the third position of the new OFDM symbol; similarly, the second is assigned to the first position, and so on. Following this scram-

The setup of IM/DD OFDM transmission experiment was the same as applied in the previous signal scrambling schemes; however, with the FFT points of 1024 and the number of subcarriers of 384, consequently the data rate was 13.3 Gb/s (10Gs/s · 4 · 384/1024 · 8/9). The BER measurements were shown in Figure 16, for the encrypted 16-QAM OFDM signals. The encrypted signals with DFT-S-OFDM precoding were fully recovered by the legal ONUs. Due to the effective PAPR reduction via DFT precoding, the BER performance was improved ~2 dB

OFDM transmission without DFT precoding. The power penalty was only 0.9 dB (at

illegal ONU with a wrong key, even a very small deviation of 1 · 10�15, it was not possible to decrypt the original OFDM signals. The corresponding constellations for correct and incorrect chaotic keys are plotted as insets in Figure 16, which verified the security feasibility of chaotic

To quantitatively evaluate the performance dependence on the mismatches of row or column

parameters Δm, Δn in Figure 17, where the modulation format is 16-QAM, and the received optical power is �9 dBm. In Figure 17(a), when Δm > 0:01, the BER is significantly deteriorated

) for the encrypted DFT-S-OFDM signals, if compared with the cases of original

) between the case of B2B and transmission after 20 km SSMF. Moreover, for any

Sorting of {*wi*} Permutation of OFDM symbol

0

0 } in the

}, details as shown in Figure 15.

, the BERs are shown with respect to the

…… ……

…… …………

number of subcarriers (as shown in Figure 13), the elements in {wi} are swapped into {wi

the new index order of subcarriers is rearranged as that of {wi

(BER@10�<sup>3</sup>

BER@10�<sup>3</sup>

DFT-S-OFDM signal transmission.

330 Optical Fiber and Wireless Communications

w1 w2 w3 w4 w5 …… wi …… wM

parameters Δm, Δn in the reconfigurable DFT matrix F<sup>0</sup>

w2 w3 w1 w5 w4 …… wM …… wi

Figure 15. Subcarrier allocation in OFDM symbol using digital chaotic sequences.

bling criterion, a key space of M! is created for OFDM data encryption.

Figure 17. BERs versus the index mismatches in (a) Δm (b) Δn of chaotic DFT matrix.

(BER>10�<sup>3</sup> ), thus the data cannot be correctly received, even after using forward error correction approaches. It can also be noted that, if Δm is increased further, deterioration becomes more serious, as shown in the corresponding BERs and constellations. Similarly, the BERs variation with respect to the column parameter mismatch in the reconfigurable DFT matrix in F<sup>0</sup> is plotted in Figure 17(b). From Figure 17(b), the original data can be only recovered when Δn < 0:01. The results also verify that, the rotation angles in the corresponding constellations become larger with the increase of Δn. The difference of BERs and constellations with respect to Δm, Δn again confirms that it is helpful to set two independent values of m, n in the new reconfigurable DFT matrix in F<sup>0</sup> , for the purpose of key space expansion in encryption scheme using chaotic DFT precoding.

For the proposed multi-fold DFT-S-OFDM data encryption, the total key space can be given as follows. First, the parameters in DFT matrix (both of row and column) can be varied from 0 to 384 with the sensitivity of 0.01, which generates a key space of ~1:5 · 109 ð38, 400 · 38, 400Þ for each OFDM symbol. Second, the subcarrier allocation generates a key space of 384!. Finally, the chaotic TS for OFDM symbol time synchronization enlarges the key space by a factor of ~10.

#### 4.2.2. Chaotic Walsh-Hadamard transform (WHT)

Chaotic WHT is an alternative encryption technique using signal precoding, which provides lower computational complexity if compared with DFT precoding. The encryption scheme based on a reconfigurable chaotic WHT precoding matrix is shown in Figure 18 [21], where the modified version of reconfigurable WHT matrix is also determined by digital chaos.

Theoretically, it can be proven that the permutation on the standard WHT matrix does not degrade the precoding properties. Since the WHT matrix is an orthogonal matrix whose elements are constrained from the set {�1, 1},

$$\mathbf{H} \mathbf{H}^{\mathsf{T}} = \mathbf{H}^{\mathsf{T}} \mathbf{H} = \mathbf{n} \mathbf{I} \tag{21}$$

Assuming Q<sup>n</sup> · <sup>n</sup> and R<sup>n</sup> · <sup>n</sup> are the row and column permutation matrices on WHT matrix H<sup>n</sup> · <sup>n</sup> respectively, then the chaotic row/column permutated WHT matrix becomes P<sup>n</sup> · <sup>n</sup>.

$$\mathbf{P} = \mathbf{Q} \mathbf{H} \mathbf{R}, \quad \mathbf{P}^{\mathrm{T}} = \mathbf{R}^{\mathrm{T}} \mathbf{H}^{\mathrm{T}} \mathbf{Q}^{\mathrm{T}} \tag{22}$$

$$\mathbf{P}\mathbf{P}^{\mathsf{T}} = \mathbf{Q}\mathbf{H}(\mathbf{R}\mathbf{R}^{\mathsf{T}})\mathbf{H}^{\mathsf{T}}\mathbf{Q}^{\mathsf{T}} = \mathbf{Q}\mathbf{H}\mathbf{H}\mathbf{H}^{\mathsf{T}}\mathbf{Q}^{\mathsf{T}} = \mathbf{Q}\mathbf{n}\mathbf{I}\mathbf{Q}^{\mathsf{T}} = \mathbf{n}\mathbf{I} \tag{23}$$

To specify the PAPR reduction capability of chaotic WHT matrix, the same analysis based on the autocorrelation coefficient can be applied as for chaotic DFT.

As plotted in Figure 19(a), the original 16-QAM OFDM symbols are applied and compared with the standard and chaotic WHT. The autocorrelation function is reduced in sidelobe if either chaotic or standard WHT is employed. An enlarged sidelobe for a specific zone is inserted in Figure 19(a) for clear view. Moreover, the sidelobe shape is exactly the same for both the standard and chaotic WHT precoders.

Furthermore, the CCDF curves are calculated for the cases of chaotic WHT, standard WHT, and un-precoded original OFDM signals, as shown in Figure 19(b). The PAPR is effectively reduced by ~1 dB by either the standard or chaotic WHT precoder, compared to the un-precoded OFDM signals. In addition, the CCDFs are exactly the same for the standard and chaotic WHT, which again verifies the above theoretical analysis.

Therefore, row/column permutated WHT can be still applied as a precoder not only to reduce the PAPR but also to enhance the physical-layer security of OFDM data, since the correct

Figure 18. Schematic diagram of OFDM encryption precoded with chaotic WHT.

4.2.2. Chaotic Walsh-Hadamard transform (WHT)

are constrained from the set {�1, 1},

332 Optical Fiber and Wireless Communications

Chaotic WHT is an alternative encryption technique using signal precoding, which provides lower computational complexity if compared with DFT precoding. The encryption scheme based on a reconfigurable chaotic WHT precoding matrix is shown in Figure 18 [21], where the modified version of reconfigurable WHT matrix is also determined by digital chaos.

Theoretically, it can be proven that the permutation on the standard WHT matrix does not degrade the precoding properties. Since the WHT matrix is an orthogonal matrix whose elements

Assuming Q<sup>n</sup> · <sup>n</sup> and R<sup>n</sup> · <sup>n</sup> are the row and column permutation matrices on WHT matrix H<sup>n</sup> · <sup>n</sup>

To specify the PAPR reduction capability of chaotic WHT matrix, the same analysis based on

As plotted in Figure 19(a), the original 16-QAM OFDM symbols are applied and compared with the standard and chaotic WHT. The autocorrelation function is reduced in sidelobe if either chaotic or standard WHT is employed. An enlarged sidelobe for a specific zone is inserted in Figure 19(a) for clear view. Moreover, the sidelobe shape is exactly the same for

Furthermore, the CCDF curves are calculated for the cases of chaotic WHT, standard WHT, and un-precoded original OFDM signals, as shown in Figure 19(b). The PAPR is effectively reduced by ~1 dB by either the standard or chaotic WHT precoder, compared to the un-precoded OFDM signals. In addition, the CCDFs are exactly the same for the standard and chaotic WHT, which

Therefore, row/column permutated WHT can be still applied as a precoder not only to reduce the PAPR but also to enhance the physical-layer security of OFDM data, since the correct

> . . .

Subcarrier

al ol cation . . .

**Key Chaotic system**

**Column permutations**

Chaotic WHT matrix

respectively, then the chaotic row/column permutated WHT matrix becomes P<sup>n</sup> · <sup>n</sup>.

the autocorrelation coefficient can be applied as for chaotic DFT.

both the standard and chaotic WHT precoders.

again verifies the above theoretical analysis.

PRBS

S/P

QAM

Figure 18. Schematic diagram of OFDM encryption precoded with chaotic WHT.

Mapping

. . .

. . .

**Row permutations**

**X**

HHT¼HTH <sup>¼</sup> nI <sup>ð</sup>21<sup>Þ</sup>

<sup>P</sup> <sup>¼</sup> QHR, PT¼RTHTQT <sup>ð</sup>22<sup>Þ</sup>

Add CP **Synchronization**

**TS**

P/S

. . .

IFFT

**Y Z**

PPT<sup>¼</sup> QHðRRTÞHTQT<sup>¼</sup> QHIHTQT<sup>¼</sup> QnIQ<sup>T</sup><sup>¼</sup> nI <sup>ð</sup>23<sup>Þ</sup>

Figure 19. PAPR reduction precoded by the standard/chaotic WHT, (a) normalized aperiodic autocorrelation function; (b) CCDF of PAPR.

original OFDM data can be recovered only after obtaining the correct row/column sequences to reconstruct the chaotic WHT matrix through legal ONUs.

In the proposed chaotic WHT scheme, a 3D hyper Chen's chaos is implemented to generate the digital chaotic sequences for data encryption [22]:

$$\begin{cases} \dot{\mathbf{x}} = -a\mathbf{x} + y\\ \dot{\mathbf{y}} = (c - a)\mathbf{x} + cy + \mathbf{x}z\\ \dot{z} = bz + xy \end{cases} \tag{24}$$

where a, b, and c are constants. Generally, with a=35, b=3, c∈[20, 28.4], it enters into chaotic zones. To generate the permutation vectors Dx, Dy, and Dz, each of the chaotic sequences {x}, {y}, and {z} is post-processed as follows.

First, we assume that the length of chaotic sequences {x} and {y} is M, where M is the order of WHT matrix, and then

$$D\_x = \text{sort}\{\mathbf{x}\}, \quad D\_y = \text{sort}\{\mathbf{y}\}, \quad D\_z = \text{sort}\{\mathbf{z}\} \tag{25}$$

where sort function returns an index vector, according to the ascending order of the values in the chaotic sequences.

The procedure of the proposed WHT chaotic encryption is given as follows. The two permutation vectors Dx and Dy, are used to permute the row and column indexes of the standard WHT matrix, respectively. Assuming the standard WHT matrix H<sup>ð</sup>0<sup>Þ</sup> <sup>M</sup> · <sup>M</sup> is

$$\mathbf{H}\_{M \times M}^{(0)} = [\mathbf{r}\_1^T, \mathbf{r}\_2^T, \dots, \mathbf{r}\_M^T] \tag{26}$$

where r1, r2,…, rM are the row vectors. After row permutation, it becomes

$$\mathbf{H}\_{M \times M}^{(1)} = [\mathbf{r}\_{\mathrm{Dx}(1)}^T \mathbf{r}\_{\mathrm{Dx}} \mathbf{r}\_{(2)} \cdots \mathbf{r}\_{\mathrm{Dx}} \mathbf{r}\_{(M)}] = [\mathbf{c}\_1, \mathbf{c}\_2 \cdots \mathbf{c}\_M] \tag{27}$$

Similarly, the columns are permutated via vector Dy , and the corresponding chaotic WHT matrix after both row and column permutations becomes

$$\mathbf{H}^{(2)}\_{\text{M}\times\text{M}} = [\mathbf{c}\_{\text{D}\_{\text{y}}(1)\prime} \mathbf{c}\_{\text{D}\_{\text{y}}(2)\prime} \cdots \mathbf{c}\_{\text{D}\_{\text{y}}(M)}] = [\mathbf{v}\_{1}^{T}, \mathbf{v}\_{2}^{T}, \cdots, \mathbf{v}\_{M}^{T}] \tag{28}$$

The chaotic sequence Dz is used to generate the chaotic TS for OFDM symbol synchronization, which is composed of random {�1, 1} and defined as [23]

$$\text{TS} = \{ (\text{mod}(D\_{z,i}, \mathcal{D}) - 0.5) \times \mathcal{D} \}\tag{29}$$

Since the hyper-chaotic sequences have high sensitivity to the initial values (~1 · 10�15) and present exceptional random behavior, the initial values are served as the security key between OLT and ONUs. A secure channel has to be employed to transfer the keys between OLT and ONUs, for instance, using quantum key distribution (QKD), which is out of the scope of this chapter and will not be given in detail here. Figure 20 shows the sensitivity of the row/column permutation indexes of chaotic WHT matrix with respect to a tiny change in the initial conditions.

Figure 21 shows the experimental setup of the encrypted transmission of OFDM signals using chaotic WHT precoding. The IM/DD was also applied, and the details were described in Section 4.1. At OLT, there were 256 points in IFFT, among which 64 subcarriers were used for OFDM data. The effective data rate was 8.9 Gb/s (10 Gs/s · 4 · 64/256/(1+1/8)). The signal waveform, electric and optical spectra are plotted as insets in Figure 21.

The transmission performance was evaluated for OFDM signals precoded by chaotic WHT via BER measurements of 128 OFDM symbols, for the cases of B2B and transmission after 20 km

Figure 20. Chaotic permutation index versus a slight change in the initial values.

Figure 21. Experimental setup of encrypted optical OFDM transmission using chaotic WHT.

SSMF, as shown in Figure 22. The BER was improved by ~1 dB (BER@10<sup>3</sup> ) in receiver sensitivity for chaotic WHT, compared with the un-precoded OFDM signals. This improvement can be mainly attributed to the precoding gain from chaotic WHT, since BER was improved also for the case of B2B, which verified the effective PAPR reduction in OFDM signals through chaotic WHT precoding. For transmission after 20 km SSMF, the effect of fiber dispersion is negligible due to the narrow spectrum of OFDM signals.

Hð2<sup>Þ</sup>

334 Optical Fiber and Wireless Communications

which is composed of random {�1, 1} and defined as [23]

of chaotic WHT matrix with respect to a tiny change in the initial conditions.

waveform, electric and optical spectra are plotted as insets in Figure 21.

**OLT**

**PRBS Keys**

**-30 -50 -70 Power (dB)**

**Electrical spectrum**

**0 2 1 3 4**

Figure 20. Chaotic permutation index versus a slight change in the initial values.

**Offline DSP**

**5 Frequency (GHz)**

> **Encrypted OFDM data**

Figure 21. Experimental setup of encrypted optical OFDM transmission using chaotic WHT.

**MZM**

**20Km SMF 1:n CW Laser PD**

**0 -40 -80 Power (dBm)**

**Optical spectrum**

**Wavelength (nm) 1549.5**

**1550 1550.5**

**10 GSa/s AWG**

**Signal waveform**

**Time (µs) 0 25 50**

**0 10 20 30 40 50 -1 -0.5 0 0.5 1 0 1 -1 Amplitude (V)**

**Permutation Index**

<sup>M</sup> · <sup>M</sup> ¼ ½cDyð1Þ, cDyð2Þ, <sup>⋯</sup>, cDyð<sup>M</sup>Þ�¼½vT

The chaotic sequence Dz is used to generate the chaotic TS for OFDM symbol synchronization,

Since the hyper-chaotic sequences have high sensitivity to the initial values (~1 · 10�15) and present exceptional random behavior, the initial values are served as the security key between OLT and ONUs. A secure channel has to be employed to transfer the keys between OLT and ONUs, for instance, using quantum key distribution (QKD), which is out of the scope of this chapter and will not be given in detail here. Figure 20 shows the sensitivity of the row/column permutation indexes

Figure 21 shows the experimental setup of the encrypted transmission of OFDM signals using chaotic WHT precoding. The IM/DD was also applied, and the details were described in Section 4.1. At OLT, there were 256 points in IFFT, among which 64 subcarriers were used for OFDM data. The effective data rate was 8.9 Gb/s (10 Gs/s · 4 · 64/256/(1+1/8)). The signal

The transmission performance was evaluated for OFDM signals precoded by chaotic WHT via BER measurements of 128 OFDM symbols, for the cases of B2B and transmission after 20 km

**0 10 20 30 40 50 60**

**Row/Column Index**

<sup>1</sup> , vT <sup>2</sup> , ⋯, vT

TS ¼ {ðmodðDz,i, 2Þ � 0:5Þ · 2} ð29Þ

 **x0=1.528121243912453 x0=1.528121243912454**

> **20 GSa/s Oscilloscope**

**Illegal ONU**

**ONU1 ONU2**

**:** .

> **Offline DSP**

<sup>M</sup>� ð28Þ

The robustness of the multi-fold data encryption is evaluated via the total key space created by chaotic WHT, since only legal ONUs can generate the correct chaotic WHT matrix to recover the original OFDM data. Assuming that the dimension of chaotic WHT matrix is 64 · 64, it creates a total key space of 64! · 64! (~10178). In addition, the chaotic TS for OFDM symbol synchronization further increases the key space by a factor of ~10. As a result, a total key space of ~10179 is achieved in chaotic WHT encryption scheme.

The computational complexities the chaotic encryption schemes with PAPR reduction are listed in Table 1. The computational complexity is reduced significantly in WHT, since it does not require complex multiplication if compared with DFT, SLM, and PTS, because the elements in WHT matrix are constrained to the value set of {1,1}. Meanwhile, WHT has the same

Figure 22. BER measurements of the OFDM transmission with and without chaotic WHT.


B is the number of candidate signals in SLM, V is the number of sub-blocks, and W is the number of phase factors.

Table 1. Computation complexity and sideband information of chaotic encryption schemes with PAPR reduction.

addition complexity as DFT. In addition, if compared with PTL or SLM schemes, simpler optical OFDM transmission structure and higher spectral efficiency can be expected for the WHT and DFT precoding schemes, since no additional sideband information needs to be transmitted.
