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Multi-bit mapping based on constellation rotation in Quantum Noise Stream Cipher
Accepted Manuscript Multi-bit mapping based on constellation rotation in Quantum Noise Stream Cipher Kai Wang, Jie Zhang, Yajie Li, Yongli Zhao, Huibin Zhang
PII: DOI: Reference:
S0030-4018(19)30316-5 https://doi.org/10.1016/j.optcom.2019.04.024 OPTICS 24023
To appear in:
Optics Communications
Received date : 12 March 2019 Revised date : 5 April 2019 Accepted date : 6 April 2019 Please cite this article as: K. Wang, J. Zhang, Y. Li et al., Multi-bit mapping based on constellation rotation in Quantum Noise Stream Cipher, Optics Communications (2019), https://doi.org/10.1016/j.optcom.2019.04.024 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Multi-bit Mapping based on Constellation Rotation in Quantum Noise Stream Cipher KAI WANG, JIE ZHANG*, YAJIE LI, YONGLI ZHAO, AND HUIBIN ZHANG, State Key Lab of Information Photonics and Optical Communications, Beijing University of Posts and Telecommunications Beijing 100876 China *[email protected]
Keyword: Optical fiber communication, Optical security and encryption, Modulation, Orthogonal frequency-division multiplexing (OFDM).
Abstract Based on Y-00 protocol, the Quantum Noise Stream Cipher (QNSC) directly encrypts the data by employing inherent quantum noise, and thus can provide physical layer security in optical fiber communication systems. With increase of bit position of ciphertext, the effect of noise on the higher bit positions of ciphertext decreases. In order to improve the effect of noise on the higher bit positions of ciphertext to protect the plaintext, XOR operation has to be used in QNSC. However, in multi-bit data QNSC system, multi-bit data has to perform XOR operation with additional seed key to protect the multi-bit data. With increase of bit number of plaintext, many additional seed keys have to be used to perform XOR operation. Thus, it is necessary to design a new mapping method without the additional seed key to protect the multi-bit data. In this paper, we propose a multi-bit data mapping method based on constellation rotation (CR) to expand the impact of noise on the higher bit positions of ciphertext to protect data. Without using additional seed key, CR-based QNSC fully expands the random impacts of noise on the multi-bit plaintext and further enhances the security of multi-bit plaintext in QNSC scheme. The CR-based multi-bit mapping method rotates the constellation diagram of plaintext by an angle to code the plaintext alternatively in constellation diagram. Based on IM/DD-OFDM, we conduct QAM/QNSC and PSK/QNSC simulation to test the performance of CR-based multibit mapping in optical back-to-back with ASE noise. At the same time, we compare the XORbased BPSK/QNSC with CR-based BPSK/QNSC in different intensity noises. Results show that the bit error rate of multi-bit plaintext in different bit positions is close to 0.5 for multi-bit plaintext of PSK/QNSC and QAM/QNSC by using CR-based mapping. Thus, the effect of noise on the higher bit positions of plaintext is improved. Eve can’t obtain any information about plaintext. By comparing XOR-based BPSK/QNSC and CR-based BPSK/QNSC in onebit system, we find that the bit error rate is close to 0.5 in both schemes. The advantage of CRbased QNSC is able to encrypt multi-bit plaintext without extra seed key and hardly affect the transmission performance of the legitimate receiver Bob. 1. Introduction With the development of transmission technology, the demand of protecting critical data from malicious attacks is increasing, such as eavesdropping and unauthorized modification. Conventional cryptography is generally based on mathematical algorithms, such as Advanced Encryption Standard (AES). The security of conventional cryptography is related to the computing power and storage capacity of the computer. However, conventional ciphers based on mathematical algorithm will be cracked easily once quantum computer is mature in the future. To reinforce the security of data transmission in physical layer, encryption based on physical effects has been studied in optical fiber communication, such as all-optical data
encryption [1-3], optical chaos encryption [4, 5], quantum key distribution [6-8] and quantum noise stream cipher [9-11]. The all-optical data encryption can encrypt the data with high speed and low latency, but the encrypted signal is still digitized and carries all information of the original data [12]. If an eavesdropper records the encrypted data in a long time, the original data may be recovered by post-processing technique [13]. Using spread spectrum techniques to encrypt data in a broadband chaotic signal, optical chaos can enhance the robustness and privacy of the system. However, the requirement of synchronization between the transmitter and the receiver is rigorous [13]. The Quantum Key Distribution (QKD) based on Heisenberg uncertainty principle is the focus in physical layer encryption of optical fiber communication. The BB84 is a key protocol in QKD and can provide perfect security with one-time pad [14]. The optical fiber channel has various interference, such as attenuation and nonlinear effect. Thus, QKD has significantly limitations on transmission distance and distribution rate of keys when quantum signal is transmitted by optical fiber channel. In addition, the conditions for running QKD system are constrained by a lot factors, such as chromatic dispersion and fiber loss [15]. Therefore, without repeaters, QKD is hardly to be deployed in wide areas, such as the seafloor. To support long-haul secure transmission, based on Y-00 protocol, the QNSC is proposed to provide high speed and longer distance secure transmission [16, 17]. In addition, the QNSC is compatible with general optical fiber communication network. Therefore, it is feasible and practical to deploy QNSC in optical fiber communication network [18]. In QNSC system, the plaintext is modulated together with corresponding bases as amplitude/phase signal. In other word, a symbol is a representation of multiple bits, consisting of plaintext and the bases. In the basic QNSC system, one symbol carries only one-bit data, which is mapped from plaintext by XOR operation with the bases. In this case, XOR operation can fully expand the random impacts of quantum noise on the plaintext, and thus can protect the plaintext from being attacked [19-21]. However, the capacity of such basic QNSC system is very limited due to that one symbol can carry only one-bit plaintext. To improve the capacity of QNSC, the encryption method for carrying multi-bit plaintext per symbol is being investigated to achieve a balance between transmission capacity and security in QNSC [22-24]. The authors in [25,26] have proposed and demonstrated a multi-bit QAM/QNSC prototype, in which multi-bit plaintext is carried by I channel and Q channel, respectively. Hence, the ciphertext masked by quantum noise consists of multi-bit plaintext and bases in QNSC. The mapping rule of multi-bit QAM/QNSC is to place the multi-bit plaintext on the higher bit positions of ciphertext of I and Q channel, respectively. In a multi-bit QNSC system, legitimate receiver Bob shares the seed key with transmitter Alice and then only recovers the multi-bit QAM signal which is slightly affected by amplitude noise and phase noise. However, the eavesdroppers (Eve) can obtain some information of the plaintext by observing the higher bit positions of ciphertext masked by quantum amplitude noise and phase noise. In addition, the multi-bit PSK mapping scheme in a wireless system has the same problem [27]. To protect the data from Eve, additional mathematical encryptions have to be used in multi-bit QNSC. The additional mathematical encryptions can introduce random effects of noise on higher bit position of plaintext. However, additional mathematical encryptions have to use XOR operation with additional seed key against Eve. In order to solve this problem in BPSK/QNSC, many technologies have been proposed, such as Overlap Selection Keying (OSK) [28], Deliberate Signal Randomization (DSR) [9], Deliberate Error Randomization (DER) [9, 30], Irregular Mapping [31], Quantum Diffusion Mapping (QDM) [32] and so on. In the development of these randomization techniques, the BPSK based QNSC was mainly used as a reference model. Therefore, we must be careful when some of the randomization technique are applied to the model of multi-bit PSK and other modulation format such as quadrature amplitude modulation. Thus, it is necessary to consider a new mapping scheme to enhance the security ability of plaintext against eavesdroppers and fully expand the random effects of the quantum noise on plaintext in multi-bit data QNSC mapping.
In this paper, we propose a multi-bit mapping method based on constellation rotation (CR) to satisfy the multi-bit data encryption in QNSC. CR-based mapping is a symbol encryption method which encrypts multi-bit data symbol by using one-bit key. According to the bases, the CR-based mapping encrypts the multi-bit plaintext by rotating the constellation diagram of plaintext with an angle. The CR-based mapping method can expand the random impacts of quantum noise on different bit positions of plaintext, so as to improve the security of plaintext in multi-bit QNSC scheme. In the proposed CR-based multi-bit mapping, the rotation angle of plaintext constellation diagram is determined by the bases. Thus, the legitimate receiver Bob can precisely recover the plaintext according to the angle with the shared bases in CR-based multi-bit mapping. Simulation results show that Eve’s bit error rate of plaintext in different bit positions is close to 0.5. It means that the effect of noise on different bit position of plaintext is the largest and balanced. Eve can’t obtain any information about plaintext. For XOR-based BPSK/QNSC and CR-based BPSK/QNSC, they have the same performance. However, the advantage of CR-based QNSC is able to encrypt multi-bit plaintext and hardly affect the transmission performance of the legitimate receiver Bob. The rest of this paper is organized as follows. In Section II, we review and analyze the multi-bit mapping QAM/QNSC scheme. In Section III, we introduce the CR-based multi-bit mapping method for PSK/QNSC and QAM/QNSC in detail. In Section IV, a QNSC based on IM/DD-OFDM simulation is built for different intensity noises and bases. In Section V, the security and transmission performance of CR-based multi-bit mapping is evaluated in different parameters. Meanwhile, the performance of XOR-based BPSK/QNSC and CR-based BPSK/QNSC is compared. Section VI concludes this work. 2. The Secure Issue in Multi-bit Mapping
Fig.1. The constellation diagram of 16QAM/QNSC.
The essence of QNSC is to maximize the masked signal space by mapping rule and inherent quantum noise to increase the difficulty of Eve’s breaking the system. Figure 1 shows the constellation diagram of a 16QAM/QNSC, where 16QAM (2-bit for I and Q, respectively) data is encrypted by using 16 bases (2-bit for I and Q, respectively). The mapping rule is (I, Q)encrypted = (I data + BI , Q data + BQ ) , where Idata and Qdata are plaintext message. The BI and BQ are bases produced by Pseudo Random Number Generator (PRNG). According to the mapping rule, 4-bit information is hidden in the constellation diagram of 16*16 encrypted symbols. For the QAM/QNSC decryption process, legitimate receiver Bob only recovers the Idata and Qdata by detecting 16QAM signal masked by amplitude noise and phase noise. However, Eve has to detect the 256QAM signal with quantum noise. According to the mapping rule of multi-bit QAM/QNSC scheme, we observe that the plaintext of Idata and Qdata in a nearby area are the same in Fig. 1. It allows Eve to access 2-bit Idata and Qdata data by
obtaining the most significant bit (MSB) and second significance bit (SSB) of the ciphertext for I and Q, even though Eve don’t know the bases. Furthermore, we also analyze Eve’s bit error rate at different bit positions in a symbol for I channel in Fig. 2. For I and Q channel, the noise is independent and approximated Gaussian distribution. Thus, the bit error rate of ciphertext in I channel is discussed. Here, from the top to the bottom, the bit position is numbered from 1 to 11. The Eve don’t know any information of seed key and plaintext. Thus, Eve can only obtain ciphertext by optical fiber channel. Compared with bases and plaintext in transmitter, the bit error rate of Eve can be calculated. The effect of noise on bit position decreases as bit position increases in Fig. 2 (a) and (b). Compared Fig. 2 (a) and Fig. 2 (b), effect of noise on bit position increases with number of bases. Thus, increasing number of bases can improve the security of QNSC system. The bit error rate is close to 0.5. This means that Eve can’t access the bit information with effect of noise. As Fig. 2 shows, the bit error rate is relatively low at the higher bit positions of ciphertext in QAM/QNSC. It means that the quantum noise has little influence on the higher bit positions of ciphertext. In other words, the security of plaintext reduces when there are more bits of plaintext in QAM/QNSC. Thus, it is essential to propose new method to fully expand the random effects of quantum noise on plaintext in QNSC without extra seed key.
(a) 4QAM/QNSC with 11-bit bases (b) 4QAM/QNSC with 8-bit bases Fig.2. Bit error rate of ciphertext in 4QAM/QNSC for different bit positions
3. Multi-bit Mapping based on Constellation Rotation In order to address the problem of multi-bit mapping in QNSC, we propose the CR-based multibit mapping to expand the random effects of quantum noise on plaintext. The CR-based multibit mapping rotates constellation diagram of plaintext with an angle θ to make plaintext alternate distribution. Meanwhile, the alternate distribution of plaintext expands the random effects of quantum noise in QNSC. Then, we will introduce the CR-based multi-bit mapping for PSK and QAM in detail. 3.1 multi-bit mapping based on constellation rotation (CR) for QAM/QNSC
Firstly, n-bit Idata and Qdata generate the analogue plaintext I AP and Q AP between 0~2n-1, respectively. For example, the analogue plaintext I AP and Q AP are 2 and 5 when
Idata and Qdata are 010 and 101 (3-bit for I and Q). Meanwhile, bases m-bit BI and BQ generate the analogue bases BI,AP and BQ,AP between 0~2m-1, respectively. The analogue plaintext with the constellation rotation (CR) I AP,CR and Q AP,CR are given by the following formulas
I AP, CR real[( I AP + iQ AP )exp(iθ )] = = Q AP , CR imag[( I AP + iQ AP )exp(iθ )] The angle θ in Eqn. (1) is defined as follows
(1)
BI,AP = θ B= I,AP π =BQ,AP π , if BQ,AP π θ =( BQ,AP − BI,AP ) , if BQ,AP ≠ BI,AP 2
(2)
According to the Eqns. (1) and (2), the constellation diagram of multi-bit plaintext is rotated with an angle θ by employing CR-based mapping in QAM/QNSC scheme. The rotation angle θ is determined by bases in CR-based mapping. If BQ, AP = BI , AP , the constellation diagram of plaintext is rotated by a counterclockwise angle BQ, AP π , and if
BQ, AP > BI , AP , the
constellation diagram of plaintext is rotated by a counterclockwise angle | BQ, AP - BI, AP |π . 2
Otherwise, the constellation diagram of plaintext is rotated by a clockwise angle π | BQ, AP - BI, AP | . Therefore, we obtain new n-bit data Idata,CR and Q data,CR by using CR2
based mapping. The ciphertext with CR Iencrypted,CR and Qencrypted,CR are produced by mapping
(I data ,CR + BI , Q data ,CR + BQ ) . According to CR encryption and mapping rule, rule (I, Q)encrypted = the QAM constellation diagram of multi-bit plaintext moves and rotates an angle θ with a different bases at the same time. Fig. 3 shows constellation diagram of 16QAM/QNSC with CR. Here, the number of bases are 64 (3-bit for I and Q, respectively). The Fig. 3(a) is a constellation diagram of 16QAM/QNSC. It is notable that plaintext of adjacent symbol in constellation diagram of 16QAM/QNSC with CR is different for I and Q channel in Fig. 3(b). When plaintexts are 00 and 00 for I and Q channel, respectively and bases are 001 and 001 for I and Q channel, respectively, the ciphertexts are 00001 and 00001 for I and Q channel, respectively. However, when plaintexts are 00 and 00 for I and Q channel, respectively and bases are 001 and 001 for I and Q channel, respectively, the constellation of 16QAM is rotated by a counterclockwise angle π and ciphertexts are 11001 and 11001 by using the CR-based mapping. When plaintexts are 00 and 00 for I and Q channel, respectively and bases are 000 and 001 for I and Q channel, respectively, the ciphertexts are 00000 and 00001 for I and Q channel, respectively. When plaintexts are 00 and 00 for I and Q channel, respectively and bases are 00000 and 00001 for I and Q channel, respectively, the ciphertexts are 00001 and 11001 by using the CR-based mapping. Thus, the noise has more influence on the higher bit positions of ciphertext by using CR encryption. In addition, the Eve can’t recover message by accessing the higher bit positions of ciphertext when signal is masked by inherent quantum noise. For the decryption process of QAM/QNSC with CR at the receiver, legitimate receiver n n Bob knows the shared key with transmitter Alice and only makes 2 * 2 QAM level decision to recover the QAM signal. Then, according to the bases, Bob rotates the constellation diagram an angle -θ to recover plaintext (negative represents opposite direction). The additional key of encryption and decryption procedures is not required for CR-based QAM/QNSC systems. Thus, CR-based mapping enhances the security of plaintext against Eve in QAM/QNSC, while keeping the simplicity of Alice and receiver Bob to perform encryption and decryption procedures.
Fig.3. (a) constellation diagram of 16QAM/QNSC; (b) constellation diagram of 16QAM/QNSC with constellation rotation.
3.2 multi-bit mapping based on constellation rotation (CR) for PSK/QNSC
The rotation angle θ of CR-based multi-bit mapping is related to constellation shape. Thus, the process of PSK/QNSC with CR is different with that in QAM/QNSC. The n-bit data X generates the analogue plaintext X AP . According to MPSK mapping rule, the In-phase analogue plaintext I AP and Quadrature-phase analogue plaintext Q AP for I and Q channel are given by the following formulas 2π I AP = cos( n X AP ) 2 2 Q = sin( π X ) AP AP 2n
(3)
The m-bit bases B produces analogue bases BAP between 0~2m-1. Thus, In-phase analogue plaintext I AP ,CR and Quadrature-phase analogue plaintext Q AP ,CR with CR are given as follows
= I AP ,CR real[( I AP + Q AP i )exp(i θ )] = Q AP ,CR imag[( I AP + Q AP i )exp(i θ )]
(4)
The Eqn. (4) is the same with Eqn. (1) for QNSC. However, the angle θ in Eqn. (4) is determined by the following formulas. 2π θ =(BAP + 1)2n +1 , if BAP is odd θ = − B 2π , if B is even AP AP 2n +1
(5)
According to Eqns. (4) and (5), we get the multi-bit plaintext XCR after CR encryption. Then, = ( X CR + B) , we generate the M-bit ciphertext with CR CCR by mapping rule CCR，encrypted ( M =(m+n) ). The ciphertext with CR CCR is modulated by IQ modulator as MPSK signal. The negative sign represents clockwise rotation of constellation diagram of plaintext and positive sign is just the opposite in Eqn. (5). The rotation angle θ is decided by bases and bit number of plaintext in PSK/QNSC. This process of CR-based multi-bit mapping in PSK/QNSC is different from that in QAM/QNSC. The plaintext’s constellation diagram not only rotates
according to the different bases but also rotates itself by CR encryption. The plaintext of adjacent symbol is different by using CR encryption in PSK/QNSC. Consequently, CR-based multi-bit mapping can encrypt the whole symbol of QAM and PSK by employing the different bases. The CR-based multi-bit mapping can expand the random effects of noise on the multi-bit plaintext in QNSC. The bases not only expands the data decision space for Bob but also encrypts the data for Eve by using CR in multi-bit QNSC system. The CR-based encryption can be applied to different modulation format such as QAM and PSK. However, the weakness of CR-based multi-bit mapping is that it cannot balance the impact of noise on different bit position of bases. Therefore, the security of key doesn’t increase by using CR-based mapping. 4. Simulation Setup The QNSC encrypts data by expanding the signal space and quantum noise. However, it leads to the complexity of channel equalization algorithms and carrier recovery algorithms in QNSC. The OFDM facilitates the channel equalization and improves spectrum utilization [32]. In this section, the QNSC simulation system is built by using IM/DD-OFDM in optical back-to-back condition. Compared with quantum noise, ASE noise has more influence on the signal. Besides, ASE noise is unavoidable when Eve detects the signal by accessing optical fiber. A QNSC system is mainly affected by amplified spontaneous emission (ASE) noise of amplifier in this paper. In order to analyze security and transmission performance of CR-based QNSC mapping in different intensity noises, bit error rate is discussed in different OSNR. The OSNR is calculated as follows. Ps − power (6) OSNR = Pn − power In Eqn. (6), Ps − power refers to the optical power value of the signal within the range of signal bandwidth and Pn − power refers to the ASE noise power within 0.1nm bandwidth. According to different OSNR, different ASE noise powers are added to analyze security performance of CR-based mapping. Fig. 4 shows the system components of QNSC with OFDM. The OFDM uses intensity modulator and directly detection. Firstly, a serial to parallel (S/P) conversion is used to separate the stream bit by bit. Then Pseudo Random Binary Sequence (PRBS) with different seed keys to generate data and bases. According to different modulation formats, including to PSK and QAM, the data is performed QNSC mapping with bases to produce PSK/QNSC or QAM/QNSC signal. The QNSC module represents the PSK/QNSC or QAM/QNSC mapping in Fig. 4. The encrypted data stream is performed Hermitian symmetry and then 64-point IFFT to modulate each symbol with a subcarrier to generate a real signal. QNSC symbols are mapped to 50 subcarriers, both DC subcarrier and Nyquist subcarrier carry null data, and the other 14 subcarriers are set to zero to reduce the aliasing impact. Then training symbols are added for synchronization and channel estimation purposes. In order to prevent Inter-Symbol Interference (ISI), the Cyclic Prefix (CP) is added and length of CP is 1/4 length of OFDM symbols. The Fig. 4(a) shows the electrical spectrum of OFDM signal. Then, the signal sampled by the DAC with 14-bit resolution and 10Gsymbol/s sample rate is used to modulate the optical field by using a Mach–Zehndner modulator (MZM). The OFDM signal 3.9 GHz, the frequency spacing is 156.25MHz. The optical source bandwidth is 10 × 25 / 64 = is a CW laser with a central wavelength of 1550nm to produce optical signal. The extinction ratio of the laser is infinite. Besides, transmitted power of laser is set as 1mW. The encrypted optical signal masked by ASE noise can be transmitted in SSMF optical channel. At the receiver, the encrypted signal is detected by a photodiode and then sampled at two samples per symbol by analog-digital converter (ADC) with 14-bit resolution. This is followed by synchronization, which finds the symbol window correctly to apply the FFT accurately. Next, a S/P conversion is done to divide the signal correctly for the FFT and CP is removed.
The estimation of the channel is performed to calculate the channel multiplication coefficients, then each subcarrier channel is equalized to compensate for phase and amplitude distortion due to the optical and electrical paths. The data processing is the same for Bob and Eve. The legal receiver Bob shares the seed key with transmitter Alice and can detect signal accurately with little effect of noise. To evaluate the safety performance of CR-based QNSC, a perfect assumption that Eve could get nearly full copy of signal without being detected is made in this paper. The biggest difference between Bob and Eve is that Eve does not know the information of the seed key and plaintext. This is a ciphertext-only attack scenario for Eve. Thus, Eve can only use M-ary signal decision to recover data with effect of noise. In order to analyze the impact of CR-based QNSC, we discuss the bit error rate for Eve and Bob. Fig. 4 (b) and (c) are encrypted and decrypted QAM signal for Eve and Bob when OSNR is 25dB. The bit error rate of Eve’s plaintext in different positions is calculated for CR-based QNSC and QNSC in different intensity noises, modulation formats and the number of bases. The XOR-based onebit mapping can fully expand the random effects of quantum noise on one-bit plaintext. Thus, the XOR-based BPSK/QNSC and CR-based BPSK/QNSC are compared in different intensity noises to analyze the performance of CR-based BPSK/QNSC.
Fig. 4 The schematic model of QAM/QNSC with IM/DD-OFDM.
5. Results and Analysis 5.1 PSK/QNSC and QAM/QNSC with constellation rotation for Eve
(a) 4QAM/QNSC without CR (b) 4QAM/QNSC with CR Fig. 5 The bit error rate in 4QAM/QNSC with different bases for I channel
Figure 5 shows bit error rate of I channel with different OSNR for 4QAM/QNSC and 4QAM/QNSC with CR. Here, 4QAM (1-bit for I and Q, respectively) signal is encrypted by using 256*256 (8-bit for I and Q, respectively), 512*512 (9-bit for I and Q, respectively), and 1024*1024 (10-bit for I and Q, respectively). We only consider bit error rate of I channel in Fig. 5, because I channel and Q channel are equivalent in QAM/QNSC. The bit error rate of 4QAM/QNSC decreases with noise of system when bit number of bases is fixed in Fig. 5(a). However, the bit error rate of 4QAM/QNSC with CR fluctuates between 0.498 and 0.5 when noise of system decreases in Fig. 5(b). The bit error rate in different bit number of bases is very
close for 4QAM/QNSC and 4QAM/QNSC with CR in Fig. 5 (a) and (b). This means that the noise has a greater effect on bit error rate than the bases for QNSC system. The larger the OSNR, the less noise affects the QAM/QNSC signal. Thus, the quantum noise plays a decisive role in the safety of the QAM/QNSC system. Compare Fig. 5(a) with Fig. 5(b), the CR-based multibit mapping expands the random effects of noise on the different bit positions of plaintext by using bases to encode plaintext in QAM/QNSC system. Figure 6 shows bit error rate of 16QAM/QNSC in different bit positions of plaintext with different OSNR for I channel. The MSB and SSB of 16QAM/QNSC are considered in Fig. 6 for I channel. The 16QAM (2-bit for I and Q, respectively) data is encrypted by using different number of bases in Fig. 6. The bit error rate of MSB of plaintext decreases with noise of system when the number of bases is fixed in Fig. 6(a). The bit error rate of SSB of plaintext decreases with noise of system when the number of bases is fixed in Fig. 6(b). In Fig .6(c), the bit error rate of MSB of plaintext fluctuates between 0.497 and 0.5 even through noise of system decreases. The bit error rate of SSB of plaintext fluctuates between 0.498 and 0.5 even through noise of system decreases in Fig. 6(d). Compare Fig. 6(a) with Fig. 6(b), the higher bit positions of plaintext, the smaller the bit error rate. This means that Eve can obtain more information of plaintext when bit number of plaintext increases. According to Fig. 6(c) and Fig. 6(d), the bit error rate of MSB and SSB in plaintext is close to 0.5 by using CR-based mapping for QAM/QNSC. Thus, the CR encryption has good security performance of protecting plaintext for multi-bit plaintext in QAM/QNSC. According to Fig. 5 and Fig. 6, the bit error rate in different bit positions of plaintext is close to 0.5 by using CR when plaintext has multi-bit in QAM/QNSC.
(a) Bit error rate of MSB without CR
(b) Bit error rate of SSB without CR
(c) Bit error rate of MSB with CR (d) Bit error rate of SSB with CR Fig. 6 The bit error rate in 16QAM/QNSC for different bases for I channel
Figure 7 shows bit error rate of 16PSK/QNSC in different bit positions with different OSNR. Here, 16PSK (4 bits) data is encrypted by using 128 bases (7 bits). The TSB and FSB represent third and fourth significant bit of 16PSK. With the increase of OSNR, the bit error rate decreases when bit position is fixed in Fig 7(a). Meanwhile, the bit error rate of plaintext increases with bit position when OSNR is fixed in Fig. 7(a). Thus, Eve can obtain more information when bit position of plaintext is the lower in multi-bit PSK/QNSC. The bit error rate in different bit positions of plaintext fluctuates between 0.498 and 0.5 even through OSNR increases in Fig. 7(b). Compare Fig. 7(a) with Fig. 7(b), the effect of quantum noise on different bit positions of plaintext is equivalent by using CR-based mapping for PSK/QNSC. Thus, CR-
based mapping has good security of protecting plaintext for multi-bit plaintext in PSK/QNSC. According to Fig. 6 and Fig. 7, CR-based mapping can encrypt multi-bit plaintext in different intensity noises and be applied to both MPSK/QNSC and MQAM/QNSC.
(a) 16PSK/QNSC without CR (b) 16PSK/QNSC with CR Fig. 7 Bit error rate of plaintext in 16PSK/QNSC for different bit positions for (a)~(b).
With the different intensity noises, we compare the difference between BPSK/QNSC with XOR and BPSK/QNSC with CR in Fig. 8(a) and Fig. 8(b). Here, BPSK (1-bit) data is encrypted by using 128 bases (7-bit) for BPSK/QNSC, XOR-based BPSK/QNSC, and CR-based BPSK/QNSC. The BPSK/QNSC encrypts the BPSK signal by placing the data in the highest position of ciphertext. Thus, the encryption process of BPSK/QNSC is the same with one channel of QAM/QNSC. However, the XOR-based BPSK/QNSC first performs XOR operation with the lowest bit position of bases and then does the same process with BPSK/QNSC. In addition, The BPSK/QNSC, XOR-based BPSK/QNSC, and CR-based BPSK/QNSC are simulated under the same conditions by IM/DD OFDM system in Fig. 8(a) and Fig. 8(b). The bit error rate of BPSK/QNSC decreases with the noise in Fig. 8(a). With the OSNR increasing, the bit error rate of BPSK/QNSC is close to 0.0298 for Eve in Fig. 8(a). This means that quantum noise can encrypt plaintext in BPSK/QNSC. However, the bit error rate of XOR-based BPSK/QNSC and CR-based BPSK/QNSC is close to 0.5 when the noise decreases in Fig. 8(b). Thus, XOR-based BPSK/QNSC and CR-based BPSK/QNSC can expand the random impacts of quantum noise on the plaintext. The XOR-based BPSK/QNSC and CRbased BPSK/QNSC are superior to BPSK/QNSC in terms of protective plaintext.
(a) The BPSK/QNSC, BPSK/QNSC with XOR and (b) The BPSK/QNSC with XOR and BPSK/QNSC BPSK/QNSC with CR with XOR Fig. 8 BER vs. OSNR in BPSK/QNSC, BPSK/QNSC with CR and BPSK/QNSC with XOR
5.2 QAM/QNSC and PSK/QNSC with constellation rotation for Bob
We show BER of Bob with different OSNR for different modulation formats in Fig. 9. Here, the BPSK, 4PSK and 8PSK signal are encrypted by using 128 bases in Fig. 9(a) and (b). Meanwhile, 4QAM (1-bit for I and Q, respectively) and 16QAM (2-bit for I and Q, respectively) signals are encrypted by using 256*256 bases (8-bit for I and Q, respectively) in Fig. 9(c) and (d). Reed Solomon (255,239) code is used for error correction to improve the transmission performance in Fig. 9. The difference of Bob’s BER between PSK/QNSC and PSK/QNSC with CR is small for different intensity noises in Fig. 9(a) and (b). Meanwhile, the difference of Bob’s BER between QAM/QNSC and QAM/QNSC with CR is small for different OSNR in Fig. 9(c) and (d). The transmission performance of system can be improved by about 4dB by
using RS code for QNSC and QNSC with CR. Thus, the QNSC with CR has little influence on Bob’s BER for different OSNR and modulation formats.
(a) BER in PSK/QNSC
(b) BER in PSK/QNSC with FEC
(c) BER in 4QAM/QNSC (d) BER in 16QAM/QNSC Fig. 9 BER in QNSC for different OSNR
Figure 10 shows the BER of Bob with different bit number of bases for different OSNR. The 16QAM (2 bits for I and Q, respectively) and 8PSK (3 bits) signals are encrypted by using different number of bases in Fig. 10(a) and (b). Reed Solomon (255,239) code is used for error correction to improve the transmission performance in Fig. 10. The difference of Bob’s BER between 16QAM/QNSC and 16QAM/QNSC with CR is small when bit number of bases is fixed in Fig. 10(a). The BER of Bob slightly changes when bit number of bases increases in Fig. 10(a). In addition, the difference of Bob’s BER between 8PSK/QNSC and 8PSK/QNSC with CR is small with different bit number of bases in Fig. 10(b). The different number of bases has little influence on BER of Bob in Fig. 10(b). Thus, according to Fig. 9 and Fig. 10, CRbased mapping hardly affects the Bob’s BER, even though noise and the number of bases change.
(a) BER in 16QAM/QNSC with FEC (b) BER in 8PSK/QNSC with FEC Fig. 10 BER in QNSC for different bit number of bases.
The CR-based multi-bit mapping makes plaintext of adjacent symbol different in the region masked by quantum noise. Thus, it can enhance security ability of the multi-bit plaintext in QNSC. Meanwhile, the bit error rate of multi-bit plaintext is close to 0.5 for different intensity noises and modulation formats in QNSC for Eve. Thus, CR-based QNSC expands the random effects of noise on different bit positions of plaintext. The CR-based multi-bit mapping doesn’t need additional keys. In addition, the encryption and decryption operation of CR-based multi-
bit mapping are simple for Alice and Bob. However, the CR-based QNSC can’t improve the security of key.
6. Conclusion This paper proposes a multi-bit mapping method based on constellation rotation (CR) in Quantum Noise Stream Cipher. The CR-based mapping can encrypt multi-bit data symbol by using one-bit key. According to different bases to rotate constellation diagram of the plaintext, CR-based multi-bit mapping makes plaintext of adjacent symbol different to expand the random effects of quantum noise on the plaintext in QNSC system. And the CR-based multibit mapping can be applied to multi-bit plaintext for PSK/QNSC and QAM/QNSC. We setup a simulation of QNSC with OFDM in optical back-to-back, and evaluate the CR-based mapping in terms of the bit error rate for different intensity noises, modulation formats and the number of bases. The bit error rate of plaintext in different bit positions is close to 0.5 by using CRbased mapping for Eve in QNSC system, even though the noise is small and the number of bases is less for QAM/QNSC and PSK/QNSC. Thus, CR-based QNSC expands the random effects of noise on multi-bit plaintext to enhance the security of multi-bit plaintext against Eve in QNSC system. The Eve can’t obtain any information about plaintext. In addition, we compare XOR-based BPSK/QNSC with CR-based BPSK/QNSC in this paper. The CR-based BPSK/QNSC can achieve the same performance as XOR-based BPSK/QNSC for one-bit plaintext. However, the advantage of CR-based multi-bit mapping is able to encrypt multi-bit plaintext and hardly affect the transmission performance of the legitimate receiver Bob for QAM/QNSC and PSK/QNSC. However, the CR-based mapping can’t improve the effect of noise on running key. This is a weakness of CR-based multi-bit mapping. The next work is to improve the scheme to increase the impact of noise on bases.
7. Funding This work is supported by the joint Foundation of Ministry of Education of China and Department of Equipment Pre-research, and NSFC (Grant No.: 61831003).
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PII: DOI: Reference:
S0030-4018(19)30316-5 https://doi.org/10.1016/j.optcom.2019.04.024 OPTICS 24023
To appear in:
Optics Communications
Received date : 12 March 2019 Revised date : 5 April 2019 Accepted date : 6 April 2019 Please cite this article as: K. Wang, J. Zhang, Y. Li et al., Multi-bit mapping based on constellation rotation in Quantum Noise Stream Cipher, Optics Communications (2019), https://doi.org/10.1016/j.optcom.2019.04.024 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Multi-bit Mapping based on Constellation Rotation in Quantum Noise Stream Cipher KAI WANG, JIE ZHANG*, YAJIE LI, YONGLI ZHAO, AND HUIBIN ZHANG, State Key Lab of Information Photonics and Optical Communications, Beijing University of Posts and Telecommunications Beijing 100876 China *[email protected]
Keyword: Optical fiber communication, Optical security and encryption, Modulation, Orthogonal frequency-division multiplexing (OFDM).
Abstract Based on Y-00 protocol, the Quantum Noise Stream Cipher (QNSC) directly encrypts the data by employing inherent quantum noise, and thus can provide physical layer security in optical fiber communication systems. With increase of bit position of ciphertext, the effect of noise on the higher bit positions of ciphertext decreases. In order to improve the effect of noise on the higher bit positions of ciphertext to protect the plaintext, XOR operation has to be used in QNSC. However, in multi-bit data QNSC system, multi-bit data has to perform XOR operation with additional seed key to protect the multi-bit data. With increase of bit number of plaintext, many additional seed keys have to be used to perform XOR operation. Thus, it is necessary to design a new mapping method without the additional seed key to protect the multi-bit data. In this paper, we propose a multi-bit data mapping method based on constellation rotation (CR) to expand the impact of noise on the higher bit positions of ciphertext to protect data. Without using additional seed key, CR-based QNSC fully expands the random impacts of noise on the multi-bit plaintext and further enhances the security of multi-bit plaintext in QNSC scheme. The CR-based multi-bit mapping method rotates the constellation diagram of plaintext by an angle to code the plaintext alternatively in constellation diagram. Based on IM/DD-OFDM, we conduct QAM/QNSC and PSK/QNSC simulation to test the performance of CR-based multibit mapping in optical back-to-back with ASE noise. At the same time, we compare the XORbased BPSK/QNSC with CR-based BPSK/QNSC in different intensity noises. Results show that the bit error rate of multi-bit plaintext in different bit positions is close to 0.5 for multi-bit plaintext of PSK/QNSC and QAM/QNSC by using CR-based mapping. Thus, the effect of noise on the higher bit positions of plaintext is improved. Eve can’t obtain any information about plaintext. By comparing XOR-based BPSK/QNSC and CR-based BPSK/QNSC in onebit system, we find that the bit error rate is close to 0.5 in both schemes. The advantage of CRbased QNSC is able to encrypt multi-bit plaintext without extra seed key and hardly affect the transmission performance of the legitimate receiver Bob. 1. Introduction With the development of transmission technology, the demand of protecting critical data from malicious attacks is increasing, such as eavesdropping and unauthorized modification. Conventional cryptography is generally based on mathematical algorithms, such as Advanced Encryption Standard (AES). The security of conventional cryptography is related to the computing power and storage capacity of the computer. However, conventional ciphers based on mathematical algorithm will be cracked easily once quantum computer is mature in the future. To reinforce the security of data transmission in physical layer, encryption based on physical effects has been studied in optical fiber communication, such as all-optical data
encryption [1-3], optical chaos encryption [4, 5], quantum key distribution [6-8] and quantum noise stream cipher [9-11]. The all-optical data encryption can encrypt the data with high speed and low latency, but the encrypted signal is still digitized and carries all information of the original data [12]. If an eavesdropper records the encrypted data in a long time, the original data may be recovered by post-processing technique [13]. Using spread spectrum techniques to encrypt data in a broadband chaotic signal, optical chaos can enhance the robustness and privacy of the system. However, the requirement of synchronization between the transmitter and the receiver is rigorous [13]. The Quantum Key Distribution (QKD) based on Heisenberg uncertainty principle is the focus in physical layer encryption of optical fiber communication. The BB84 is a key protocol in QKD and can provide perfect security with one-time pad [14]. The optical fiber channel has various interference, such as attenuation and nonlinear effect. Thus, QKD has significantly limitations on transmission distance and distribution rate of keys when quantum signal is transmitted by optical fiber channel. In addition, the conditions for running QKD system are constrained by a lot factors, such as chromatic dispersion and fiber loss [15]. Therefore, without repeaters, QKD is hardly to be deployed in wide areas, such as the seafloor. To support long-haul secure transmission, based on Y-00 protocol, the QNSC is proposed to provide high speed and longer distance secure transmission [16, 17]. In addition, the QNSC is compatible with general optical fiber communication network. Therefore, it is feasible and practical to deploy QNSC in optical fiber communication network [18]. In QNSC system, the plaintext is modulated together with corresponding bases as amplitude/phase signal. In other word, a symbol is a representation of multiple bits, consisting of plaintext and the bases. In the basic QNSC system, one symbol carries only one-bit data, which is mapped from plaintext by XOR operation with the bases. In this case, XOR operation can fully expand the random impacts of quantum noise on the plaintext, and thus can protect the plaintext from being attacked [19-21]. However, the capacity of such basic QNSC system is very limited due to that one symbol can carry only one-bit plaintext. To improve the capacity of QNSC, the encryption method for carrying multi-bit plaintext per symbol is being investigated to achieve a balance between transmission capacity and security in QNSC [22-24]. The authors in [25,26] have proposed and demonstrated a multi-bit QAM/QNSC prototype, in which multi-bit plaintext is carried by I channel and Q channel, respectively. Hence, the ciphertext masked by quantum noise consists of multi-bit plaintext and bases in QNSC. The mapping rule of multi-bit QAM/QNSC is to place the multi-bit plaintext on the higher bit positions of ciphertext of I and Q channel, respectively. In a multi-bit QNSC system, legitimate receiver Bob shares the seed key with transmitter Alice and then only recovers the multi-bit QAM signal which is slightly affected by amplitude noise and phase noise. However, the eavesdroppers (Eve) can obtain some information of the plaintext by observing the higher bit positions of ciphertext masked by quantum amplitude noise and phase noise. In addition, the multi-bit PSK mapping scheme in a wireless system has the same problem [27]. To protect the data from Eve, additional mathematical encryptions have to be used in multi-bit QNSC. The additional mathematical encryptions can introduce random effects of noise on higher bit position of plaintext. However, additional mathematical encryptions have to use XOR operation with additional seed key against Eve. In order to solve this problem in BPSK/QNSC, many technologies have been proposed, such as Overlap Selection Keying (OSK) [28], Deliberate Signal Randomization (DSR) [9], Deliberate Error Randomization (DER) [9, 30], Irregular Mapping [31], Quantum Diffusion Mapping (QDM) [32] and so on. In the development of these randomization techniques, the BPSK based QNSC was mainly used as a reference model. Therefore, we must be careful when some of the randomization technique are applied to the model of multi-bit PSK and other modulation format such as quadrature amplitude modulation. Thus, it is necessary to consider a new mapping scheme to enhance the security ability of plaintext against eavesdroppers and fully expand the random effects of the quantum noise on plaintext in multi-bit data QNSC mapping.
In this paper, we propose a multi-bit mapping method based on constellation rotation (CR) to satisfy the multi-bit data encryption in QNSC. CR-based mapping is a symbol encryption method which encrypts multi-bit data symbol by using one-bit key. According to the bases, the CR-based mapping encrypts the multi-bit plaintext by rotating the constellation diagram of plaintext with an angle. The CR-based mapping method can expand the random impacts of quantum noise on different bit positions of plaintext, so as to improve the security of plaintext in multi-bit QNSC scheme. In the proposed CR-based multi-bit mapping, the rotation angle of plaintext constellation diagram is determined by the bases. Thus, the legitimate receiver Bob can precisely recover the plaintext according to the angle with the shared bases in CR-based multi-bit mapping. Simulation results show that Eve’s bit error rate of plaintext in different bit positions is close to 0.5. It means that the effect of noise on different bit position of plaintext is the largest and balanced. Eve can’t obtain any information about plaintext. For XOR-based BPSK/QNSC and CR-based BPSK/QNSC, they have the same performance. However, the advantage of CR-based QNSC is able to encrypt multi-bit plaintext and hardly affect the transmission performance of the legitimate receiver Bob. The rest of this paper is organized as follows. In Section II, we review and analyze the multi-bit mapping QAM/QNSC scheme. In Section III, we introduce the CR-based multi-bit mapping method for PSK/QNSC and QAM/QNSC in detail. In Section IV, a QNSC based on IM/DD-OFDM simulation is built for different intensity noises and bases. In Section V, the security and transmission performance of CR-based multi-bit mapping is evaluated in different parameters. Meanwhile, the performance of XOR-based BPSK/QNSC and CR-based BPSK/QNSC is compared. Section VI concludes this work. 2. The Secure Issue in Multi-bit Mapping
Fig.1. The constellation diagram of 16QAM/QNSC.
The essence of QNSC is to maximize the masked signal space by mapping rule and inherent quantum noise to increase the difficulty of Eve’s breaking the system. Figure 1 shows the constellation diagram of a 16QAM/QNSC, where 16QAM (2-bit for I and Q, respectively) data is encrypted by using 16 bases (2-bit for I and Q, respectively). The mapping rule is (I, Q)encrypted = (I data + BI , Q data + BQ ) , where Idata and Qdata are plaintext message. The BI and BQ are bases produced by Pseudo Random Number Generator (PRNG). According to the mapping rule, 4-bit information is hidden in the constellation diagram of 16*16 encrypted symbols. For the QAM/QNSC decryption process, legitimate receiver Bob only recovers the Idata and Qdata by detecting 16QAM signal masked by amplitude noise and phase noise. However, Eve has to detect the 256QAM signal with quantum noise. According to the mapping rule of multi-bit QAM/QNSC scheme, we observe that the plaintext of Idata and Qdata in a nearby area are the same in Fig. 1. It allows Eve to access 2-bit Idata and Qdata data by
obtaining the most significant bit (MSB) and second significance bit (SSB) of the ciphertext for I and Q, even though Eve don’t know the bases. Furthermore, we also analyze Eve’s bit error rate at different bit positions in a symbol for I channel in Fig. 2. For I and Q channel, the noise is independent and approximated Gaussian distribution. Thus, the bit error rate of ciphertext in I channel is discussed. Here, from the top to the bottom, the bit position is numbered from 1 to 11. The Eve don’t know any information of seed key and plaintext. Thus, Eve can only obtain ciphertext by optical fiber channel. Compared with bases and plaintext in transmitter, the bit error rate of Eve can be calculated. The effect of noise on bit position decreases as bit position increases in Fig. 2 (a) and (b). Compared Fig. 2 (a) and Fig. 2 (b), effect of noise on bit position increases with number of bases. Thus, increasing number of bases can improve the security of QNSC system. The bit error rate is close to 0.5. This means that Eve can’t access the bit information with effect of noise. As Fig. 2 shows, the bit error rate is relatively low at the higher bit positions of ciphertext in QAM/QNSC. It means that the quantum noise has little influence on the higher bit positions of ciphertext. In other words, the security of plaintext reduces when there are more bits of plaintext in QAM/QNSC. Thus, it is essential to propose new method to fully expand the random effects of quantum noise on plaintext in QNSC without extra seed key.
(a) 4QAM/QNSC with 11-bit bases (b) 4QAM/QNSC with 8-bit bases Fig.2. Bit error rate of ciphertext in 4QAM/QNSC for different bit positions
3. Multi-bit Mapping based on Constellation Rotation In order to address the problem of multi-bit mapping in QNSC, we propose the CR-based multibit mapping to expand the random effects of quantum noise on plaintext. The CR-based multibit mapping rotates constellation diagram of plaintext with an angle θ to make plaintext alternate distribution. Meanwhile, the alternate distribution of plaintext expands the random effects of quantum noise in QNSC. Then, we will introduce the CR-based multi-bit mapping for PSK and QAM in detail. 3.1 multi-bit mapping based on constellation rotation (CR) for QAM/QNSC
Firstly, n-bit Idata and Qdata generate the analogue plaintext I AP and Q AP between 0~2n-1, respectively. For example, the analogue plaintext I AP and Q AP are 2 and 5 when
Idata and Qdata are 010 and 101 (3-bit for I and Q). Meanwhile, bases m-bit BI and BQ generate the analogue bases BI,AP and BQ,AP between 0~2m-1, respectively. The analogue plaintext with the constellation rotation (CR) I AP,CR and Q AP,CR are given by the following formulas
I AP, CR real[( I AP + iQ AP )exp(iθ )] = = Q AP , CR imag[( I AP + iQ AP )exp(iθ )] The angle θ in Eqn. (1) is defined as follows
(1)
BI,AP = θ B= I,AP π =BQ,AP π , if BQ,AP π θ =( BQ,AP − BI,AP ) , if BQ,AP ≠ BI,AP 2
(2)
According to the Eqns. (1) and (2), the constellation diagram of multi-bit plaintext is rotated with an angle θ by employing CR-based mapping in QAM/QNSC scheme. The rotation angle θ is determined by bases in CR-based mapping. If BQ, AP = BI , AP , the constellation diagram of plaintext is rotated by a counterclockwise angle BQ, AP π , and if
BQ, AP > BI , AP , the
constellation diagram of plaintext is rotated by a counterclockwise angle | BQ, AP - BI, AP |π . 2
Otherwise, the constellation diagram of plaintext is rotated by a clockwise angle π | BQ, AP - BI, AP | . Therefore, we obtain new n-bit data Idata,CR and Q data,CR by using CR2
based mapping. The ciphertext with CR Iencrypted,CR and Qencrypted,CR are produced by mapping
(I data ,CR + BI , Q data ,CR + BQ ) . According to CR encryption and mapping rule, rule (I, Q)encrypted = the QAM constellation diagram of multi-bit plaintext moves and rotates an angle θ with a different bases at the same time. Fig. 3 shows constellation diagram of 16QAM/QNSC with CR. Here, the number of bases are 64 (3-bit for I and Q, respectively). The Fig. 3(a) is a constellation diagram of 16QAM/QNSC. It is notable that plaintext of adjacent symbol in constellation diagram of 16QAM/QNSC with CR is different for I and Q channel in Fig. 3(b). When plaintexts are 00 and 00 for I and Q channel, respectively and bases are 001 and 001 for I and Q channel, respectively, the ciphertexts are 00001 and 00001 for I and Q channel, respectively. However, when plaintexts are 00 and 00 for I and Q channel, respectively and bases are 001 and 001 for I and Q channel, respectively, the constellation of 16QAM is rotated by a counterclockwise angle π and ciphertexts are 11001 and 11001 by using the CR-based mapping. When plaintexts are 00 and 00 for I and Q channel, respectively and bases are 000 and 001 for I and Q channel, respectively, the ciphertexts are 00000 and 00001 for I and Q channel, respectively. When plaintexts are 00 and 00 for I and Q channel, respectively and bases are 00000 and 00001 for I and Q channel, respectively, the ciphertexts are 00001 and 11001 by using the CR-based mapping. Thus, the noise has more influence on the higher bit positions of ciphertext by using CR encryption. In addition, the Eve can’t recover message by accessing the higher bit positions of ciphertext when signal is masked by inherent quantum noise. For the decryption process of QAM/QNSC with CR at the receiver, legitimate receiver n n Bob knows the shared key with transmitter Alice and only makes 2 * 2 QAM level decision to recover the QAM signal. Then, according to the bases, Bob rotates the constellation diagram an angle -θ to recover plaintext (negative represents opposite direction). The additional key of encryption and decryption procedures is not required for CR-based QAM/QNSC systems. Thus, CR-based mapping enhances the security of plaintext against Eve in QAM/QNSC, while keeping the simplicity of Alice and receiver Bob to perform encryption and decryption procedures.
Fig.3. (a) constellation diagram of 16QAM/QNSC; (b) constellation diagram of 16QAM/QNSC with constellation rotation.
3.2 multi-bit mapping based on constellation rotation (CR) for PSK/QNSC
The rotation angle θ of CR-based multi-bit mapping is related to constellation shape. Thus, the process of PSK/QNSC with CR is different with that in QAM/QNSC. The n-bit data X generates the analogue plaintext X AP . According to MPSK mapping rule, the In-phase analogue plaintext I AP and Quadrature-phase analogue plaintext Q AP for I and Q channel are given by the following formulas 2π I AP = cos( n X AP ) 2 2 Q = sin( π X ) AP AP 2n
(3)
The m-bit bases B produces analogue bases BAP between 0~2m-1. Thus, In-phase analogue plaintext I AP ,CR and Quadrature-phase analogue plaintext Q AP ,CR with CR are given as follows
= I AP ,CR real[( I AP + Q AP i )exp(i θ )] = Q AP ,CR imag[( I AP + Q AP i )exp(i θ )]
(4)
The Eqn. (4) is the same with Eqn. (1) for QNSC. However, the angle θ in Eqn. (4) is determined by the following formulas. 2π θ =(BAP + 1)2n +1 , if BAP is odd θ = − B 2π , if B is even AP AP 2n +1
(5)
According to Eqns. (4) and (5), we get the multi-bit plaintext XCR after CR encryption. Then, = ( X CR + B) , we generate the M-bit ciphertext with CR CCR by mapping rule CCR，encrypted ( M =(m+n) ). The ciphertext with CR CCR is modulated by IQ modulator as MPSK signal. The negative sign represents clockwise rotation of constellation diagram of plaintext and positive sign is just the opposite in Eqn. (5). The rotation angle θ is decided by bases and bit number of plaintext in PSK/QNSC. This process of CR-based multi-bit mapping in PSK/QNSC is different from that in QAM/QNSC. The plaintext’s constellation diagram not only rotates
according to the different bases but also rotates itself by CR encryption. The plaintext of adjacent symbol is different by using CR encryption in PSK/QNSC. Consequently, CR-based multi-bit mapping can encrypt the whole symbol of QAM and PSK by employing the different bases. The CR-based multi-bit mapping can expand the random effects of noise on the multi-bit plaintext in QNSC. The bases not only expands the data decision space for Bob but also encrypts the data for Eve by using CR in multi-bit QNSC system. The CR-based encryption can be applied to different modulation format such as QAM and PSK. However, the weakness of CR-based multi-bit mapping is that it cannot balance the impact of noise on different bit position of bases. Therefore, the security of key doesn’t increase by using CR-based mapping. 4. Simulation Setup The QNSC encrypts data by expanding the signal space and quantum noise. However, it leads to the complexity of channel equalization algorithms and carrier recovery algorithms in QNSC. The OFDM facilitates the channel equalization and improves spectrum utilization [32]. In this section, the QNSC simulation system is built by using IM/DD-OFDM in optical back-to-back condition. Compared with quantum noise, ASE noise has more influence on the signal. Besides, ASE noise is unavoidable when Eve detects the signal by accessing optical fiber. A QNSC system is mainly affected by amplified spontaneous emission (ASE) noise of amplifier in this paper. In order to analyze security and transmission performance of CR-based QNSC mapping in different intensity noises, bit error rate is discussed in different OSNR. The OSNR is calculated as follows. Ps − power (6) OSNR = Pn − power In Eqn. (6), Ps − power refers to the optical power value of the signal within the range of signal bandwidth and Pn − power refers to the ASE noise power within 0.1nm bandwidth. According to different OSNR, different ASE noise powers are added to analyze security performance of CR-based mapping. Fig. 4 shows the system components of QNSC with OFDM. The OFDM uses intensity modulator and directly detection. Firstly, a serial to parallel (S/P) conversion is used to separate the stream bit by bit. Then Pseudo Random Binary Sequence (PRBS) with different seed keys to generate data and bases. According to different modulation formats, including to PSK and QAM, the data is performed QNSC mapping with bases to produce PSK/QNSC or QAM/QNSC signal. The QNSC module represents the PSK/QNSC or QAM/QNSC mapping in Fig. 4. The encrypted data stream is performed Hermitian symmetry and then 64-point IFFT to modulate each symbol with a subcarrier to generate a real signal. QNSC symbols are mapped to 50 subcarriers, both DC subcarrier and Nyquist subcarrier carry null data, and the other 14 subcarriers are set to zero to reduce the aliasing impact. Then training symbols are added for synchronization and channel estimation purposes. In order to prevent Inter-Symbol Interference (ISI), the Cyclic Prefix (CP) is added and length of CP is 1/4 length of OFDM symbols. The Fig. 4(a) shows the electrical spectrum of OFDM signal. Then, the signal sampled by the DAC with 14-bit resolution and 10Gsymbol/s sample rate is used to modulate the optical field by using a Mach–Zehndner modulator (MZM). The OFDM signal 3.9 GHz, the frequency spacing is 156.25MHz. The optical source bandwidth is 10 × 25 / 64 = is a CW laser with a central wavelength of 1550nm to produce optical signal. The extinction ratio of the laser is infinite. Besides, transmitted power of laser is set as 1mW. The encrypted optical signal masked by ASE noise can be transmitted in SSMF optical channel. At the receiver, the encrypted signal is detected by a photodiode and then sampled at two samples per symbol by analog-digital converter (ADC) with 14-bit resolution. This is followed by synchronization, which finds the symbol window correctly to apply the FFT accurately. Next, a S/P conversion is done to divide the signal correctly for the FFT and CP is removed.
The estimation of the channel is performed to calculate the channel multiplication coefficients, then each subcarrier channel is equalized to compensate for phase and amplitude distortion due to the optical and electrical paths. The data processing is the same for Bob and Eve. The legal receiver Bob shares the seed key with transmitter Alice and can detect signal accurately with little effect of noise. To evaluate the safety performance of CR-based QNSC, a perfect assumption that Eve could get nearly full copy of signal without being detected is made in this paper. The biggest difference between Bob and Eve is that Eve does not know the information of the seed key and plaintext. This is a ciphertext-only attack scenario for Eve. Thus, Eve can only use M-ary signal decision to recover data with effect of noise. In order to analyze the impact of CR-based QNSC, we discuss the bit error rate for Eve and Bob. Fig. 4 (b) and (c) are encrypted and decrypted QAM signal for Eve and Bob when OSNR is 25dB. The bit error rate of Eve’s plaintext in different positions is calculated for CR-based QNSC and QNSC in different intensity noises, modulation formats and the number of bases. The XOR-based onebit mapping can fully expand the random effects of quantum noise on one-bit plaintext. Thus, the XOR-based BPSK/QNSC and CR-based BPSK/QNSC are compared in different intensity noises to analyze the performance of CR-based BPSK/QNSC.
Fig. 4 The schematic model of QAM/QNSC with IM/DD-OFDM.
5. Results and Analysis 5.1 PSK/QNSC and QAM/QNSC with constellation rotation for Eve
(a) 4QAM/QNSC without CR (b) 4QAM/QNSC with CR Fig. 5 The bit error rate in 4QAM/QNSC with different bases for I channel
Figure 5 shows bit error rate of I channel with different OSNR for 4QAM/QNSC and 4QAM/QNSC with CR. Here, 4QAM (1-bit for I and Q, respectively) signal is encrypted by using 256*256 (8-bit for I and Q, respectively), 512*512 (9-bit for I and Q, respectively), and 1024*1024 (10-bit for I and Q, respectively). We only consider bit error rate of I channel in Fig. 5, because I channel and Q channel are equivalent in QAM/QNSC. The bit error rate of 4QAM/QNSC decreases with noise of system when bit number of bases is fixed in Fig. 5(a). However, the bit error rate of 4QAM/QNSC with CR fluctuates between 0.498 and 0.5 when noise of system decreases in Fig. 5(b). The bit error rate in different bit number of bases is very
close for 4QAM/QNSC and 4QAM/QNSC with CR in Fig. 5 (a) and (b). This means that the noise has a greater effect on bit error rate than the bases for QNSC system. The larger the OSNR, the less noise affects the QAM/QNSC signal. Thus, the quantum noise plays a decisive role in the safety of the QAM/QNSC system. Compare Fig. 5(a) with Fig. 5(b), the CR-based multibit mapping expands the random effects of noise on the different bit positions of plaintext by using bases to encode plaintext in QAM/QNSC system. Figure 6 shows bit error rate of 16QAM/QNSC in different bit positions of plaintext with different OSNR for I channel. The MSB and SSB of 16QAM/QNSC are considered in Fig. 6 for I channel. The 16QAM (2-bit for I and Q, respectively) data is encrypted by using different number of bases in Fig. 6. The bit error rate of MSB of plaintext decreases with noise of system when the number of bases is fixed in Fig. 6(a). The bit error rate of SSB of plaintext decreases with noise of system when the number of bases is fixed in Fig. 6(b). In Fig .6(c), the bit error rate of MSB of plaintext fluctuates between 0.497 and 0.5 even through noise of system decreases. The bit error rate of SSB of plaintext fluctuates between 0.498 and 0.5 even through noise of system decreases in Fig. 6(d). Compare Fig. 6(a) with Fig. 6(b), the higher bit positions of plaintext, the smaller the bit error rate. This means that Eve can obtain more information of plaintext when bit number of plaintext increases. According to Fig. 6(c) and Fig. 6(d), the bit error rate of MSB and SSB in plaintext is close to 0.5 by using CR-based mapping for QAM/QNSC. Thus, the CR encryption has good security performance of protecting plaintext for multi-bit plaintext in QAM/QNSC. According to Fig. 5 and Fig. 6, the bit error rate in different bit positions of plaintext is close to 0.5 by using CR when plaintext has multi-bit in QAM/QNSC.
(a) Bit error rate of MSB without CR
(b) Bit error rate of SSB without CR
(c) Bit error rate of MSB with CR (d) Bit error rate of SSB with CR Fig. 6 The bit error rate in 16QAM/QNSC for different bases for I channel
Figure 7 shows bit error rate of 16PSK/QNSC in different bit positions with different OSNR. Here, 16PSK (4 bits) data is encrypted by using 128 bases (7 bits). The TSB and FSB represent third and fourth significant bit of 16PSK. With the increase of OSNR, the bit error rate decreases when bit position is fixed in Fig 7(a). Meanwhile, the bit error rate of plaintext increases with bit position when OSNR is fixed in Fig. 7(a). Thus, Eve can obtain more information when bit position of plaintext is the lower in multi-bit PSK/QNSC. The bit error rate in different bit positions of plaintext fluctuates between 0.498 and 0.5 even through OSNR increases in Fig. 7(b). Compare Fig. 7(a) with Fig. 7(b), the effect of quantum noise on different bit positions of plaintext is equivalent by using CR-based mapping for PSK/QNSC. Thus, CR-
based mapping has good security of protecting plaintext for multi-bit plaintext in PSK/QNSC. According to Fig. 6 and Fig. 7, CR-based mapping can encrypt multi-bit plaintext in different intensity noises and be applied to both MPSK/QNSC and MQAM/QNSC.
(a) 16PSK/QNSC without CR (b) 16PSK/QNSC with CR Fig. 7 Bit error rate of plaintext in 16PSK/QNSC for different bit positions for (a)~(b).
With the different intensity noises, we compare the difference between BPSK/QNSC with XOR and BPSK/QNSC with CR in Fig. 8(a) and Fig. 8(b). Here, BPSK (1-bit) data is encrypted by using 128 bases (7-bit) for BPSK/QNSC, XOR-based BPSK/QNSC, and CR-based BPSK/QNSC. The BPSK/QNSC encrypts the BPSK signal by placing the data in the highest position of ciphertext. Thus, the encryption process of BPSK/QNSC is the same with one channel of QAM/QNSC. However, the XOR-based BPSK/QNSC first performs XOR operation with the lowest bit position of bases and then does the same process with BPSK/QNSC. In addition, The BPSK/QNSC, XOR-based BPSK/QNSC, and CR-based BPSK/QNSC are simulated under the same conditions by IM/DD OFDM system in Fig. 8(a) and Fig. 8(b). The bit error rate of BPSK/QNSC decreases with the noise in Fig. 8(a). With the OSNR increasing, the bit error rate of BPSK/QNSC is close to 0.0298 for Eve in Fig. 8(a). This means that quantum noise can encrypt plaintext in BPSK/QNSC. However, the bit error rate of XOR-based BPSK/QNSC and CR-based BPSK/QNSC is close to 0.5 when the noise decreases in Fig. 8(b). Thus, XOR-based BPSK/QNSC and CR-based BPSK/QNSC can expand the random impacts of quantum noise on the plaintext. The XOR-based BPSK/QNSC and CRbased BPSK/QNSC are superior to BPSK/QNSC in terms of protective plaintext.
(a) The BPSK/QNSC, BPSK/QNSC with XOR and (b) The BPSK/QNSC with XOR and BPSK/QNSC BPSK/QNSC with CR with XOR Fig. 8 BER vs. OSNR in BPSK/QNSC, BPSK/QNSC with CR and BPSK/QNSC with XOR
5.2 QAM/QNSC and PSK/QNSC with constellation rotation for Bob
We show BER of Bob with different OSNR for different modulation formats in Fig. 9. Here, the BPSK, 4PSK and 8PSK signal are encrypted by using 128 bases in Fig. 9(a) and (b). Meanwhile, 4QAM (1-bit for I and Q, respectively) and 16QAM (2-bit for I and Q, respectively) signals are encrypted by using 256*256 bases (8-bit for I and Q, respectively) in Fig. 9(c) and (d). Reed Solomon (255,239) code is used for error correction to improve the transmission performance in Fig. 9. The difference of Bob’s BER between PSK/QNSC and PSK/QNSC with CR is small for different intensity noises in Fig. 9(a) and (b). Meanwhile, the difference of Bob’s BER between QAM/QNSC and QAM/QNSC with CR is small for different OSNR in Fig. 9(c) and (d). The transmission performance of system can be improved by about 4dB by
using RS code for QNSC and QNSC with CR. Thus, the QNSC with CR has little influence on Bob’s BER for different OSNR and modulation formats.
(a) BER in PSK/QNSC
(b) BER in PSK/QNSC with FEC
(c) BER in 4QAM/QNSC (d) BER in 16QAM/QNSC Fig. 9 BER in QNSC for different OSNR
Figure 10 shows the BER of Bob with different bit number of bases for different OSNR. The 16QAM (2 bits for I and Q, respectively) and 8PSK (3 bits) signals are encrypted by using different number of bases in Fig. 10(a) and (b). Reed Solomon (255,239) code is used for error correction to improve the transmission performance in Fig. 10. The difference of Bob’s BER between 16QAM/QNSC and 16QAM/QNSC with CR is small when bit number of bases is fixed in Fig. 10(a). The BER of Bob slightly changes when bit number of bases increases in Fig. 10(a). In addition, the difference of Bob’s BER between 8PSK/QNSC and 8PSK/QNSC with CR is small with different bit number of bases in Fig. 10(b). The different number of bases has little influence on BER of Bob in Fig. 10(b). Thus, according to Fig. 9 and Fig. 10, CRbased mapping hardly affects the Bob’s BER, even though noise and the number of bases change.
(a) BER in 16QAM/QNSC with FEC (b) BER in 8PSK/QNSC with FEC Fig. 10 BER in QNSC for different bit number of bases.
The CR-based multi-bit mapping makes plaintext of adjacent symbol different in the region masked by quantum noise. Thus, it can enhance security ability of the multi-bit plaintext in QNSC. Meanwhile, the bit error rate of multi-bit plaintext is close to 0.5 for different intensity noises and modulation formats in QNSC for Eve. Thus, CR-based QNSC expands the random effects of noise on different bit positions of plaintext. The CR-based multi-bit mapping doesn’t need additional keys. In addition, the encryption and decryption operation of CR-based multi-
bit mapping are simple for Alice and Bob. However, the CR-based QNSC can’t improve the security of key.
6. Conclusion This paper proposes a multi-bit mapping method based on constellation rotation (CR) in Quantum Noise Stream Cipher. The CR-based mapping can encrypt multi-bit data symbol by using one-bit key. According to different bases to rotate constellation diagram of the plaintext, CR-based multi-bit mapping makes plaintext of adjacent symbol different to expand the random effects of quantum noise on the plaintext in QNSC system. And the CR-based multibit mapping can be applied to multi-bit plaintext for PSK/QNSC and QAM/QNSC. We setup a simulation of QNSC with OFDM in optical back-to-back, and evaluate the CR-based mapping in terms of the bit error rate for different intensity noises, modulation formats and the number of bases. The bit error rate of plaintext in different bit positions is close to 0.5 by using CRbased mapping for Eve in QNSC system, even though the noise is small and the number of bases is less for QAM/QNSC and PSK/QNSC. Thus, CR-based QNSC expands the random effects of noise on multi-bit plaintext to enhance the security of multi-bit plaintext against Eve in QNSC system. The Eve can’t obtain any information about plaintext. In addition, we compare XOR-based BPSK/QNSC with CR-based BPSK/QNSC in this paper. The CR-based BPSK/QNSC can achieve the same performance as XOR-based BPSK/QNSC for one-bit plaintext. However, the advantage of CR-based multi-bit mapping is able to encrypt multi-bit plaintext and hardly affect the transmission performance of the legitimate receiver Bob for QAM/QNSC and PSK/QNSC. However, the CR-based mapping can’t improve the effect of noise on running key. This is a weakness of CR-based multi-bit mapping. The next work is to improve the scheme to increase the impact of noise on bases.
7. Funding This work is supported by the joint Foundation of Ministry of Education of China and Department of Equipment Pre-research, and NSFC (Grant No.: 61831003).
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