Precoded LDPC coding for physical layer security
 Kyunghoon Kwon^{1},
 Taehyun Kim^{1} and
 Jun Heo^{1}Email authorView ORCID ID profile
https://doi.org/10.1186/s1363801607617
© The Author(s) 2016
Received: 1 September 2015
Accepted: 25 October 2016
Published: 9 December 2016
Abstract
This paper examines a simple and practical security preprocessing scheme for the Gaussian wiretap channel. A security gap based error rate is used as a measure of security over the wiretap channel. In previous works, information puncturing and scrambling schemes based on lowdensity paritycheck (LDPC) codes were employed to reduce the security gap. Unlike the previous works, our goal is to improve security performance by using the precode of the feedforward (FF) structure. We demonstrate that the FF code has an advantage for the security gap compared to the perfect scrambling scheme. Furthermore, we propose the joint iterative decoding method between LDPC and FF codes to improve the reliability/security performances. The proposed joint iterative method is able to achieve outstanding performance by using the proposed scaling and correction factors based on signaltonoise ratio (SNR) evolution. The improved performances by these factors are demonstrated through the extrinsic information transfer (EXIT) chart and simulation results. Finally, the simulation results suggest that the proposed coding scheme is more effective than the conventional scrambling scheme.
Keywords
1 Introduction
For several decades, wireless communication technologies have been available that exchange information rapidly and reliably between a sender and a receiver. Owing to the continued development of communication technologies, we can today access communication networks conveniently and with transportability, whenever and wherever we wish. In conjunction with this development, a growing interest has developed in secure information transmission over wireless networks related to the specific security vulnerabilities caused by the inherent openness of wireless media. It is difficult to detect eavesdropping because anybody can acquire transmitted information over a wireless communication channel.
Shannon established communication theory in 1949 and defined the basic concept of secure communication from the informationtheoretic perspective [1]. Using Shannon’s approaches, a sender, Alice, securely transmits an information message M to a legitimate receiver, Bob, across a public channel. To be “perfectly secure", the requirement of the mutual information I(M;X)=0 must be satisfied between Alice’s information message M and the transmitted word X. From this definition, Shannon proved that Alice and Bob must share a key string to achieve perfect security. This theory was the introduction of the key distribution problem and is the basis of symmetric key cryptography defense systems for the upper layer implemented today. Present systems based on cryptography prevent the extraction of information without a secure key string when information is exposed to the eavesdropper Eve. This public key algorithm depends on the computational limit of the eavesdropper to ensure computational security. In spite of the improvements in public key algorithms, there remains a problem for security based on the assumption of Eve’s limited computational resources considering the advancement of available computing power.
An alternative technology that is not based on computational complexity, is physical layer security. Unlike the key distribution problem, physical layer security utilizes the characteristics of a communication channel and allows a legitimate receiver to decode correctly. The important difference compared to Shannon’s theory is that the eavesdropper can observe information transmitted by the sender through another channel. Physical layer security guarantees security analytically, based on information theory, regardless of the eavesdropper’s computational power. Therefore, there is no elevation of risk due to the advancement of high speed computing.
A security system based on the physical layer was introduced by Wyner in 1975 [2] and informationtheoretically secure communication was studied in [3, 4]. According to the wiretap channel model defined by Wyner, the main channel was defined between the sender, Alice, and the legitimate receiver, Bob; the wiretap channel was defined as a degraded version of the main channel. The main and wiretap channels were assumed to be discrete memoryless channels. Suppose that Alice sends Bob an sbit message M across the main channel. Alice encodes M into an nbit transmitted word X. Bob and Eve receive message X across the main and wiretap channel, respectively. Bob and Eve’s channel observations are denoted by Y and Z, respectively. Alice encodes the information for two objectives [2] as follows: (i) the error probability between the message M and Bob’s decoded message \(\hat {M}_{B}\) of the received message Y must converge to zero (with negligibly small probability of error) [reliability]. ii) no information is shared between information message M and Eve’s received message Z. For a precise expression, the formulation is articulated as the rate of mutual information \(\frac {1}{n}I(M;Z)\rightarrow 0\) when n→∞ [security]. Wyner defined that physical layer security is achieved without key distribution using forward error correction (FEC) when it corresponds to the considerations of reliability and security. Moreover, the secrecy rate is defined by the rate s/n, where s and n are the number of secret message bits and the number of bits transmitted over the channel, respectively. A detailed explanation of Wyner code could be found in [5].
where S N R _{ B,m i n } is the lowest SNR for which (a) is satisfied and S N R _{ E,m a x } is the highest SNR for which (b) holds.
According to (1), the security gap should be kept as small as possible, so that the desired security is achieved with small degradation of Eve’s channel. Therefore, it is important to construct an errorcorrecting code (ECC) to reduce the security gap. As mentioned above, the main target of this paper is to keep the security gap as small as possible.
Studies on the errorcorrecting code for physical layer security have focused on lowdensity paritycheck (LDPC) codes. LDPC codes [12] have a remarkable errorcorrecting capability and a powerful analysis tool for a belief propagation (BP) decoder, [13] called density evolution (DE) [14] or the extrinsic information transfer (EXIT) chart [15]. Klinc et al. [7] proposed a securityachieving algorithm using LDPC codes with a puncturing scheme. Only parity bits are transmitted to eliminate the exposure of secret messages and the decoders recover the punctured bits using the received parity bits. Baldi proposed nonsystematic codes [16, 17] for physical layer security using a scrambling matrix inspired by the McEliece Cryptosystem [18]. This scheme causes intentional bit error propagation where transmitted bits consist of scrambled information bits. This achieves secrecy maintaining the error correction capability of FEC and the advantage of a decrease in the signal power compared with the puncturing scheme [19]. However, since the scrambling scheme produced leads to an error propagation phenomenon, an improved reliability in terms of frame error rate cannot be expected.
In this paper, we propose a feedforward (FF) precode that resolves the disadvantage of the puncturing scheme for linear block codes and addresses the advantage of a decrease in the signal power with respect to the conventional scrambling scheme. Unlike the previous scrambling scheme that uses a hard decision value for error propagation only, the proposed code has an improved reliability at a high SNR region compared to the scrambling scheme. We demonstrate that the proposed code has improved reliability performance at high SNR with a reduced security gap. The proposed system consists of an LDPC code as an inner code and an FF code as a precode (outer code). The outer code has a code rate approaching one to minimize the loss of transmitted information against the conventional scrambling scheme. By concatenating LDPC and FF codes, reliability is achieved using LDPC and security is realized using the FF code. Unlike the scrambling scheme, the FF code employs soft decision decoding to recover the secret message and has superior reliability performance compared to the scrambling scheme. The reliability performance can be improved by applying joint iterative decoding to the proposed system. The improved performance is demonstrated through the EXIT chart curves [20–22].
The outline of this paper is as follows. In Section 2, we introduce the wiretap channel model and review previous works, information puncturing, and scrambling schemes. In Section 3, the encoding and decoding procedures of the FF code are discussed and the performance is evaluated. In Section 4, the joint iterative decoding procedure is explained and the security and reliability performances of the proposed system are evaluated. Also, we approximate the factors used in this paper and analyze the performance of the proposed system using the EXIT chart curve. The conclusion is presented in Section 5.
2 Preliminaries and related works
This section discusses some background concepts and the previous works that will be used throughout the paper.
2.1 System model
where \(N_{i}^{Bob}\) and \(N_{i}^{Eve}\) are independent and identically distributed (i.i.d) zeromean Gaussian random variables of variance \({\sigma _{B}^{2}}\) and \({\sigma _{E}^{2}}\), respectively, and κ is a positive constant that models the gain advantage of the eavesdropper over the destination.
Let n _{ ch } be the number of transmitted bits over the channel, and n _{ code } denote the codeword block length of the LDPC code. Define the design rate \(R_{d}=\frac {k}{n_{ch}}\), the secret rate \(R_{s}=\frac {s}{n_{ch}}\), and the code rate \(R_{c}=\frac {k}{n_{code}}\). In general, if the number of the secret message bits s is equal to the dimension of the LDPC code k, then R _{ s }=R _{ d }. If R _{ s }<R _{ d } in [7], it may help to achieve the reduced security gap but higher power should be needed to achieve the reliability condition. Since the power saving is important in many applications, R _{ s }≈R _{ d } is preferred.
2.2 Punctured and scrambled code for Gaussian wiretap channel
It is possible to recover the secret message with correct decoding. However, if decoding fails, an error propagation phenomenon is observed due to the density of the descrambling matrix S ^{−1} in the rightside term of the above equation. In [17], perfect scrambling is denoted by a descrambling matrix with row and column weight >1 and a density close to 0.5. Thus, perfect scrambling with one (or more) error(s) causes an error rate around 0.5 in the final decoded message. Since the BER of Eve is very close to 0.5 (if errors are randomly distributed), it would be difficult to extract much information about the message. In terms of the gain of signal power, Baldi et al. showed that the puncturing scheme has worse error correcting performance than the scrambling scheme with respect to systematic LDPC coding. This is because the puncturing scheme increases the code rate and has a negative impact on the code minimum distance which is reduced [23, 24]. However, the scrambling scheme can only provide an error propagation effect, not error correction. The use of the scrambling scheme without FEC (as unitary rate coding, section 3A in [17]) guarantees security performance on average, though it does not provide improved reliability.
3 Feedforward precode for physical layer security
The proposed coding scheme employs the simplest convolutional encoding with one tail bit to protect the secret message for an improved reliability performance, and the decoding complexity of the proposed scheme is higher due to soft decision decoding (BCJR algorithm).
3.1 Encoding
The proposed code with density 0.5 guarantees the requirement of perfect scrambling, and achieves the limit of security performance when n goes to infinity. In contrast to the conventional scrambling scheme based on a nonsingular random matrix, the FF code consists of the straightforward structures of the encoder and decoder.
3.2 Decoding
While the scrambling scheme only has error propagation capability, the proposed FF code, with increased minimum Hamming distance (d _{ min }=2) using redundant bit (tail bit) and coding gain using the BCJR algorithm, has a noticeable performance gain in the high SNR region. In the low SNR region, this code demonstrates a BER of 0.5. Security as defined in this paper is achieved. Moreover, this code has an improved performance of about 0.4 dB compared to the uncoded system at the BER of 10^{−7}, owing to the BCJR decoding algorithm. Compared with the conventional scrambling scheme, the proposed code has a performance improvement of approximately 1.4 dB at the BER of 10^{−7}.
If information from other symbols with low reliability is incorrect, errors accumulate for the entire code sequence, which cause error propagation. Unlike channel errors, the error positions after FF decoding (or descrambling) are not exactly i.i.d. Moreover, the operation of the FF code employs the correlation effect between consecutive symbols and each symbol is dependent on other symbols. Therefore, we cannot state that this system has a perfect secrecy even though Eve’s BER is equal to 0.5. This does not ensure the maximum entropy for Eve, since the error positions are not i.i.d.
Security gap performances with uncoded BPSK, perfect scrambling and FF code over the AWGN channel
Code  S N R _{ E,m a x }[ d B]  S N R _{ B,m i n }[ d B]  S _{ g } [dB] 

uncoded BPSK  −14.94  9.59  24.53 
Perf. scramb.  5.15  11.44  6.29 
FF coded  4.25  9.8  5.55 
3.3 Complexity

Forward/backward recursion: let t be the number of states of the FF code, n be the number of the length of a trellis, respectively. From the Fig. 7, each state has two outgoing branches. For each state, (2t) multiplication operations and t addition operation are needed. Therefore, for a trellis with length n, a total of (2t n) multiplication operations and (t n) addition operations are required. Likewise, the operations required to backward recursion are also equal to forward recursion.

Branch metric (probability): to compute the branch metric on the probability domain, (2t) branch metrics are needed since there are t states and each state has two outgoing branches. For each branch, two multiplications are required. Therefore, a total of (4t n) multiplications are needed for a trellis length n.

LLR computation: the numerator (denominator) of LLR computation is the total sum of the probability of branch metric corresponding to 0 (1). Since the precode has two states and two outgoing branches per each state, there are four branch metrics of probability domain. Among the metrics, two branch metrics are corresponded to the probability of 0. For each numerator and denominator, (t−1) addition operations are needed. Then, 1 logarithm operation and 1 division operation are needed to compute LLR. In total, 2(t−1)n addition, n logarithm, and n division operations are needed.
To compute the perfect scrambling scheme (randomly generated), 1×n hard decision vector and n×n descrambling matrix are needed. For the 1st decoded (descrambled) bit, n multiplication operations and n−1 addition operations are needed. In total, n ^{2} multiplication and n(n−1) addition operations are needed to obtain the descrambled message.
The types and numbers of operations needed to implement the perfect scrambling (randomly generated), FF soft decoding (BCJR), and FF hard decoding (as perfect scrambling)
Operations  Perfect scrambling  FF soft  FF hard decoding 

(random matrix)  (BCJR)  (perfect scrambling)  
Addition  n(n−1)  2(2t−1)n  n−1 
Multiplication  n ^{2}  8t n  
Division  n  
Logarithm  n 
From Table 2, it is possible to incorrectly evaluate that the perfect scrambling scheme (random matrix) has more complexity than the FF soft decoding, since it only provides the types and numbers of operations for real value computation. In terms of the hardware implementation, the perfect scrambling only uses binary operations (modulo2 operations); however, BCJR algorithm of FF soft decoding requires the operations of the real values and it needs more cost per one operation than the perfect scrambling. For those reasons, it is difficult to precisely compare the algorithms with the data in Table 2. Therefore, the matrix \(G_{FF}^{1}\) is used as perfect scrambling for a fair comparison in this paper.
4 Joint iterative decoding for improved reliability
where L ^{0}(C _{2}) is the input LLR values of the FF decoder at the first iteration.
where α and β are the correction and scaling factors, respectively.
4.1 Computing the correction and scaling factors via Monte Carlo simulation
Since we assume that allzero codeword modulated into x=+1=[+1,+1,⋯,+1] by BPSK {+1,−1} is transmitted, the leftside of (17) must be larger than zero for the next iteration without errors.
4.2 Extrinsic information transfer (EXIT) chart analysis
The EXIT chart [20–22] is a useful analysis tool of the iterative decoding system. EXIT charts indicate mutual information exchange between the extrinsic information of two constituent codes. In most cases, the output LLR messages of these codes can be assumed to follow the Gaussian distribution. The extrinsic information between the constituent codes can then be sequentially used to process the computation. In this paper, the information from the channel (intrinsic information) and the output knowledge from the previous iteration (extrinsic information) can be used as the input of the current iteration, and the output of the current iteration can be used as the input of the next iteration. We use LDPC code and FF code as two constituent codes and assume that their input and output LLR are approximated by the Gaussian distribution.
4.3 Simulation results
In the previous subsections, it is suggested that the proposed scheme needs the correction and scaling factors in MTN for joint iterative decoding, and the joint iterative decoding of the proposed system is evaluated through EXIT chart curves. In this subsection, we evaluate the proposed system through BER and security gap performance.
As noted in Fig. 10, the values of α and β are sensitive functions of the signaltonoise ratio. It may cause the entire system to become very complex and lead to performance loss when both values are wrongly evaluated. Therefore, the simulation results for the fixed values are also presented to avoid the impact of wrong evaluation. The fixed values are selected in the SNR region having the gain of the joint iterative decoding compared to the perfect scrambling. The fixed values selected at low SNR region (≤1.5 dB) may cause critical error propagation at high SNR. In addition, the fixed values selected at high SNR region (≥2.8 dB) render it difficult to achieve the joint iterative decoding gain since the values are too small. Based on these rules, the values are selected as α=0.07 and β=0.18, which are optimized at 2.4 dB. The reasons for these values are as follows: (i) the JID at high SNR region has some of iterative decoding gain compared to the perfect scrambled LDPC code. For use of the fixed values, S N R _{ B,m i n } should possibly be kept as small as S N R _{ B,m i n } for use of the optimized values. That is, despite the use of the fixed values, outstanding reliability performance should be achieved at high SNR region; ii) error propagation effect is properly maintained at low SNR region by using these values. That is, the S N R _{ E,m a x } for use of the fixed values should be kept as close to the S N R _{ E,m a x } as possible for use of the optimized values. These optimized values shown in Fig. 12 are evaluated for the best security performance at each SNR. Although the fixed values (α=0.07, β=0.18) are experimentally selected to show the prevention of the wrong evaluation impact as an example, the security performance will differ following the values that are selected. In summary, these values (α=0.07, β=0.18) are relevantly selected by considering error propagation at low SNR and error correction at high SNR.
Security gap performances with systematic LDPC, perfect scrambled LDPC, LDPCFF serially concatenated (SC) and LDPCFF JID (opt./fix.) over the AWGN channel
Code  S N R _{ E,m a x }[ d B]  S N R _{ B,m i n }[ d B]  S _{ g } [dB] 

Syst.  −11.87  2.835  14.705 
Perf. scramb.  0.685  3.49  2.805 
SC  0.67  3.285  2.615 
JID (opt.)  0.72  2.98  2.26 
JID (fix.)  0.62  2.99  2.37 
In conclusion, the SISO decoder of the FF code provides a performance improvement over the scrambled scheme and we can achieve a better performance improvement using the proposed JID scheme. Furthermore, the proposed system using the fixed factors has similar security/reliability performances to that using the optimized factors and can still achieve the performance improvement over the scrambled scheme. From the figure, we can also observe that the security gap advantage vanishes as Eve’s BER tends toward the ideal value of 0.5, hereafter \(P_{e,min}^{E}\geq 0.45\). However, since \(P_{e,min}^{E}\geq 0.4\) is sufficiently significant for a practical system, physical layer security as defined in this paper can be achieved.
Although the proposed JID scheme has the advantage of enhanced reliability/security performances, this is achieved from an extra complexity/decoding delay. Basically, extra decoding complexity is needed since the FF decoding procedure is performed l _{ o } times for JID. If more JID is demanded, the extra complexity will be increased.
4.4 Randomness measurement
The proposed precode includes operations between consecutive symbols. From these operations, the proposed precode has a correlation effect and is able to employ soft decision decoding. Due to the correlation effect and soft decision decoding, Eve’s decoding performance is better than that of the perfect scrambling scheme. However, the security gap between Bob and Eve is maintained since both performances are equally enhanced. The proposed scheme may have a negative impact on the randomness of the produced sequence since the FF code is highly structured. For this reason, the entire distribution of errors should be analyzed since Eve’s average error rate does not guarantee the randomness of the decoded sequence.
From Fig. 3, Eve’s received Z decodes LDPC decoded message \(\hat {M}_{E}=M+\mathbf {E}\) by using the BP decoder, where M and E are LDPC codeword and the error vector after LDPC decoding, respectively. In addition, through the FF decoder, Eve’s FF decoder outputs the FF decoded message \(\hat {U}_{E}=U+\mathbf {e}\), where U and e are the information message and error vector after FF decoding, respectively. For the erroneous frame, let \({e_{i}^{l}}\) be the number of errors for the ith position in the lth erroneous frame, l _{ max } be the total number of erroneous frames, t _{ i } be the total number of errors at the ith position, \(t_{i}=\sum _{l=1}^{l_{max}}{e_{i}^{l}}\), and T be the total number of errors, \(T=\sum _{i=1}^{n}t_{i}\), where n is the length of the information bits. Therefore, the error expectation value (or bit error probability) at the ith position for an erroneous frame is \({P_{m}^{i}}=t_{i}/l_{max}\). Therefore, \({P_{m}^{i}}\) is the bit error probability for each position when e≠0.
5 Conclusions
In this paper, we have examined securityprocessing schemes for physical layer security. We proposed a serially concatenated system that consists of an outer code and conventional FEC as an inner code. Compared with previous works relating to channel coding for physical layer security, the puncturing scheme for LDPC code (or linear block code) has a weakness in that it should be required higher signal power to achieve reliability than scrambling scheme. The disadvantages of the scrambling scheme as unitary rate coding are that it is only capable of reducing the security gap and it does not provide the error correction capability. The proposed security scheme adopts the FF code using a SISO decoding procedure (BCJR algorithm). We demonstrated that the proposed scheme is capable of performing error correction and error propagation simultaneously. Simulation results confirm that the FF code using a BCJR algorithm has an improved reliability performance and reduced security gap.
Furthermore, we proposed a joint iterative decoding algorithm between the FF code and conventional LDPC to improve the reliability performance through the bit and frame error rate with a correction factor α and scaling factor β obtained by using Monte Carlo simulation. These factors are the functions of the signaltonoise ratio. In the case of the proposed JID using these factors, our best results indicate reliability/securitygap performance improvements of 0.51 and 0.545 dB, respectively. The reliability performance of the proposed JID scheme using these factors is observed to be only 0.145 dB away from the systematic LDPC code. Despite the use of fixed factors to avoid the impact of wrong evaluation, our results indicate reliability/securitygap performance improvements to perfect scrambled LDPC of 0.5 and 0.435 dB, respectively. It is demonstrated that error floor phenomenon of LDPC code in the high SNR region can be reduced when using joint iterative decoding with the proper α and β; thus, a reduced security gap can be achieved. This is analyzed via the EXIT chart curve. In future works, equivocation rate analysis of the proposed scheme will be performed with α and β for an informationtheoretic approach.
6 Endnote
Declarations
Acknowledgements
This work was supported by the ICT R&D program of MSIP/IITP [1711028311, Reliable cryptosystem standards and core technology development for secure quantum key distribution network].
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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