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Polar codes have garnered a lot of attention from the scientific community, owing to their low-complexity implementation and provable capacity achieving capability. They have been standardized to be used for encoding information on the control channels in 5G wireless networks due to their robustness for short codeword lengths. The conventional approach to generate polar codes is to recursively use 2×2 kernels and polarize channel capacities. This approach however, has a limitation of only having the ability to generate codewords of length Norig=2n form. In order to mitigate this limitation, multiple techniques have been developed, e.g., polarization kernels of larger sizes, multi-kernel polar codes, and downsizing techniques like puncturing or shortening. However, the availability of so many design options and parameters, in turn makes the choice of design parameters quite challenging. In this paper, the authors propose a novel polar code construction technique called Adaptive Segmented Aggregation which generates polar codewords of any arbitrary codeword length. This approach involves dividing the entire codeword into smaller segments that can be independently encoded and decoded, thereby aggregated for channel processing. Additionally a rate assignment methodology has been derived for the proposed technique, that is tuned to the design requirement.
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The ever-growing threats in cybersecurity growing with the rapid development of quantum computing, necessitates the development of robust and quantum-resistant cryptographic systems. This paper introduces a novel cryptosystem, Public Key Cryptosystem based on Systematic Polar Encoding (PKC-SPE), based on the combination of systematic polar encoding and public-key cryptographic principles. The Systematic Polar Encoding (SPE), derived from the well-established field of polar codes, serves as the foundation for this proposed cryptographic scheme. Here, we have used MATLAB Software to introduce and implement the PKC-SPE Cryptosystem. The paper examines key generation, encryption, and decryption algorithms, providing insights into the adaptability and efficiency of systematic polar encoding in public-key cryptography. We assess the efficiency of the PKC-SPE Cryptosystem in three aspects: key size, computational complexity, and system implementation timings. In addition, we compare the PKC-SPE Cryptosystem with PKC-PC cryptosystem and find that it has reduced key sizes ( P r = 0.8436 kbytes). The results obtained through simulations validate the effectiveness of the proposed cryptosystem and highlighting its potential for integration into real-world communication systems. Thus, in the paradigm shift to quantum computing, the PKC-SPE cryptosystem emerges as a promising candidate to secure digital communication in the quantum computing era.
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Over binary-input memoryless symmetric (BMS) channels, the performance of polar codes under successive cancellation list (SCL) decoding can approach maximum likelihood (ML) algorithm when the list size L is greater than or equal to 2MF, where MF, known as mixing factor of code, represents the number of information bits before the last frozen bit. Recently, Yao et al. showed the upper bound of the mixing factor of decreasing monomial codes with length n=2m and rate R≤12 when m is an odd number; moreover, this bound is reachable. Herein, we obtain an achievable upper bound in the case of an even number. Further, we propose a new decoding hard-decision rule beyond the last frozen bit of polar codes under BMS channels.
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Channel coding is a fundamental procedure in wireless telecommunication systems and has a strong impact on the data transmission quality. This effect becomes more important when the transmission must be characterised by low latency and low bit error rate, as in the case of vehicle-to-everything (V2X) services. Thus, V2X services must use powerful and efficient coding schemes. In this paper, we thoroughly examine the performance of the most important channel coding schemes in V2X services. More specifically, the impact of use of 4th-Generation Long-Term Evolution (4G-LTE) turbo codes, 5th-Generation New Radio (5G-NR) polar codes and low-density parity-check codes (LDPC) in V2X communication systems is researched. For this purpose, we employ stochastic propagation models that simulate the cases of line of sight (LOS), non-line of sight (NLOS) and line of sight with vehicle blockage (NLOSv) communication. Different communication scenarios are investigated in urban and highway environments using the 3rd-Generation Partnership Project (3GPP) parameters for the stochastic models. Based on these propagation models, we investigate the performance of the communication channels in terms of bit error rate (BER) and frame error rate (FER) performance for different levels of signal to noise ratio (SNR) for all the aforementioned coding schemes and three small V2X-compatible data frames. Our analysis shows that turbo-based coding schemes have superior BER and FER performance than 5G coding schemes for the vast majority of the considered simulation scenarios. This fact, combined with the low-complexity requirements of turbo schemes for small data frames, makes them more suitable for small-frame 5G V2X services.
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CRC-aided successive cancellation list (CA-SCL) decoding is a powerful algorithm that dramatically improves the error performance of polar codes. Path selection is a major issue that affects the decoding latency of SCL decoders. Generally, path selection is implemented using a metric sorter, which causes its latency to increase as the list grows. In this paper, intelligent path selection (IPS) is proposed as an alternative to the traditional metric sorter. First, we found that in the path selection, only the most reliable paths need to be selected, and it is not necessary to completely sort all paths. Second, based on a neural network model, an intelligent path selection scheme is proposed, including a fully connected network construction, a threshold and a post-processing unit. Simulation results show that the proposed path-selection method can achieve comparable performance gain to the existing methods under SCL/CA-SCL decoding. Compared with the conventional methods, IPS has lower latency for medium and large list sizes. For the proposed hardware structure, IPS's time complexity is O(klog2(L)) where k is the number of hidden layers of the network and L is the list size.
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Polar codes are closer to the Shannon limit with lower complexity in coding and decoding. As traditional decoding techniques suffer from high latency and low throughput, with the development of deep learning technology, some deep learning-based decoding methods have been proposed to solve these problems. Usually, the deep neural network is treated as a black box and learns to map the polar codes with noise to the original information code directly. In fact, it is difficult for the network to distinguish between valid and interfering information, which leads to limited BER performance. In this paper, a deep residual network based on information refinement (DIR-NET) is proposed for decoding polar-coded short packets. The proposed method works to fully distinguish the effective and interference information in the codewords, thus obtaining a lower bit error rate. To achieve this goal, we design a two-stage decoding network, including a denoising subnetwork and decoding subnetwork. This structure can further improve the accuracy of the decoding method. Furthermore, we construct the whole network solely on the basis of the attention mechanism. It has a stronger information extraction ability than the traditional neural network structure. Benefiting from cascaded attention modules, information can be filtered and refined step-by-step, thus obtaining a low bit error rate. The simulation results show that DIR-Net outperforms existing decoding methods in terms of BER performance under both AWGN channels and flat fading channels.
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In the successive cancellation (SC) list decoding, the tree pruning operation retains the L best paths with respect to a metric at every decoding step. However, the correct path might be among the L worst paths due to the imposed penalties. In this case, the correct path is pruned and the decoding process fails. shifted pruning (SP) scheme can recover the correct path by additional decoding attempts when decoding fails, in which the pruning window is shifted by κ≤L paths over certain bit positions. A special case of the shifted pruning scheme where κ=L is known as SCL-flip decoding, which was independently proposed in 2019. In this work, a new metric that performs better in particular for medium and long codes is proposed, and nested shift-pruning schemes are suggested for improving the average complexity.
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Polar codes are a relatively new family of linear block codes which have garnered a lot of attention from the scientific community, owing to their low-complexity implementation and provably capacity achieving capability. They have been proposed to be used for encoding information on the control channels in 5G wireless networks due to their robustness for short codeword lengths. The basic approach introduced by Arikan can only be used to generate polar codes of length N=2n, ∀n∈N. To overcome this limitation, polarization kernels of size larger than 2×2 (like 3×3, 4×4, and so on), have already been proposed in the literature. Additionally, kernels of different sizes can also be combined together to generate multi-kernel polar codes, further improving the flexibility of codeword lengths. These techniques undoubtedly improve the usability of polar codes for various practical implementations. However, with the availability of so many design options and parameters, designing polar codes that are optimally tuned to specific underlying system requirements becomes extremely challenging, since a variation in system parameters can result in a different choice of polarization kernel. This necessitates a structured design technique for optimal polarization circuits. We developed the DTS-parameter to quantify the best rate-matched polar codes. Thereafter, we developed and formalized a recursive technique to design polarization kernels of higher order from component smaller order. A scaled version of the DTS-parameter, namely SDTS-parameter (denoted by the symbol ζ in this article) was used for the analytical assessment of this construction technique and validated for single-kernel polar codes. In this paper, we aim to extend the analysis of the aforementioned SDTS parameter for multi-kernel polar codes and validate their applicability in this domain as well.
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Reconciliation is an essential procedure for continuous-variable quantum key distribution (CV-QKD). As the most commonly used reconciliation protocol in short-distance CV-QKD, the slice error correction (SEC) allows a system to distill more than 1 bit from each pulse. However, the quantization efficiency is greatly affected by the noisy channel with a low signal-to-noise ratio (SNR), which usually limits the secure distance to about 30 km. In this paper, an improved SEC protocol, named Rotated-SEC (RSEC), is proposed through performing a random orthogonal rotation on the raw data before quantization, and deducing a new estimator for the quantized sequences. Moreover, the RSEC protocol is implemented with polar codes. The experimental results show that the proposed protocol can reach up to a quantization efficiency of about 99%, and maintain at around 96% even at the relatively low SNRs (0.5,1), which theoretically extends the secure distance to about 45 km. When implemented with the polar codes with a block length of 16 Mb, the RSEC achieved a reconciliation efficiency of above 95%, which outperforms all previous SEC schemes. In terms of finite-size effects, we achieved a secret key rate of 7.83×10-3 bits/pulse at a distance of 33.93 km (the corresponding SNR value is 1). These results indicate that the proposed protocol significantly improves the performance of SEC and is a competitive reconciliation scheme for the CV-QKD system.
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There exists a natural trade-off in public key encryption (PKE) schemes based on ring learning with errors (RLWE), namely: we would like a wider error distribution to increase the security, but it comes at the cost of an increased decryption failure rate (DFR). A straightforward solution to this problem is the error-correcting code, which is commonly used in communication systems and already appears in some RLWE-based proposals. However, applying error-correcting codes to those cryptographic schemes is far from simply installing an add-on. Firstly, the residue error term derived by decryption has correlated coefficients, whereas most prevalent error-correcting codes with remarkable error tolerance assume the channel noise to be independent and memoryless. This explains why only simple error-correcting methods are used in existing RLWE-based PKE schemes. Secondly, the residue error term has correlated coefficients leaving accurate DFR estimation challenging even for uncoded plaintext. It can be found in the literature that a tighter DFR estimation can effectively create a DFR margin. Thirdly, most error-correcting codes are not well designed for safety considerations, e.g., syndrome decoding has a nonconstant time nature. A code good at error correcting might be weak under a variety of attacks. In this work, we propose a polar coding scheme for RLWE-based PKE. A relaxed "independence" assumption is used to derive an uncorrelated residue noise term, and a wireless communication strategy, outage, is used to construct polar codes. Furthermore, some knowledge about the residue noise is exploited to improve the decoding performance. With the parameterization of NewHope Round 2, the proposed scheme creates a considerable DRF margin, which gives a competitive security improvement compared to state-of-the-art benchmarks. Specifically, the security is improved by 28.8%, while a DFR of 2-149 is achieved a for code rate pf 0.25, n=1024,q= 12,289, and binomial parameter k=55. Moreover, polar encoding and decoding have a quasilinear complexity O(Nlog2N) and intrinsically support constant-time implementations.
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Polar code has been adopted as the control channel coding scheme for the fifth generation (5G), and the performance of short polar codes is receiving intensive attention. The successive cancellation flipping (SC flipping) algorithm suffers a significant performance loss in short block lengths. To address this issue, we propose a double long short-term memory (DLSTM) neural network to locate the first error bit. To enhance the prediction accuracy of the DLSTM network, all frozen bits are clipped in the output layer. Then, Gaussian approximation is applied to measure the channel reliability and rank the flipping set to choose the least reliable position for multi-bit flipping. To be robust under different codewords, padding and masking strategies aid the network architecture to be compatible with multiple block lengths. Numerical results indicate that the error-correction performance of the proposed algorithm is competitive with that of the CA-SCL algorithm. It has better performance than the machine learning-based multi-bit flipping SC (ML-MSCF) decoder and the dynamic SC flipping (DSCF) decoder for short polar codes.
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Polar coding gives rise to the first explicit family of codes that provably achieve capacity with efficient encoding and decoding for a wide range of channels. However, its performance at short blocklengths under standard successive cancellation decoding is far from optimal. A well-known way to improve the performance of polar codes at short blocklengths is CRC precoding followed by successive-cancellation list decoding. This approach, along with various refinements thereof, has largely remained the state of the art in polar coding since it was introduced in 2011. Recently, Arikan presented a new polar coding scheme, which he called polarization-adjusted convolutional (PAC) codes. At short blocklengths, such codes offer a dramatic improvement in performance as compared to CRC-aided list decoding of conventional polar codes. PAC codes are based primarily upon the following main ideas: replacing CRC codes with convolutional precoding (under appropriate rate profiling) and replacing list decoding by sequential decoding. One of our primary goals in this paper is to answer the following question: is sequential decoding essential for the superior performance of PAC codes? We show that similar performance can be achieved using list decoding when the list size L is moderately large (say, L⩾128). List decoding has distinct advantages over sequential decoding in certain scenarios, such as low-SNR regimes or situations where the worst-case complexity/latency is the primary constraint. Another objective is to provide some insights into the remarkable performance of PAC codes. We first observe that both sequential decoding and list decoding of PAC codes closely match ML decoding thereof. We then estimate the number of low weight codewords in PAC codes, and use these estimates to approximate the union bound on their performance. These results indicate that PAC codes are superior to both polar codes and Reed-Muller codes. We also consider random time-varying convolutional precoding for PAC codes, and observe that this scheme achieves the same superior performance with constraint length as low as ν=2.
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With the emergence of wireless networks, cooperation for secrecy is recognized as an attractive way to establish secure communications. Departing from cryptographic techniques, secrecy can be provided by exploiting the wireless channel characteristics; that is, some error-correcting codes besides reliability have been shown to achieve information-theoretic security. In this paper, we propose a polar-coding-based technique for the primitive relay wiretap channel and show that this technique is suitable to provide information-theoretic security. Specifically, we integrate at the relay an additional functionality, which allows it to smartly decide whether it will cooperate or not based on the decoding detector result. In the case of cooperation, the relay operates in a decode-and-forward mode and assists the communication by transmitting a complementary message to the destination in order to correctly decode the initial source's message. Otherwise, the communication is completed with direct transmission from source to the destination. Finally, we first prove that the proposed encoding scheme achieves weak secrecy, then, in order to overcome the obstacle of misaligned bits, we implement a double-chaining construction, which achieves strong secrecy.
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A traditional successive cancellation (SC) decoding algorithm produces error propagation in the decoding process. In order to improve the SC decoding performance, it is important to solve the error propagation. In this paper, we propose a new algorithm combining reinforcement learning and SC flip (SCF) decoding of polar codes, which is called a Q-learning-assisted SCF (QLSCF) decoding algorithm. The proposed QLSCF decoding algorithm uses reinforcement learning technology to select candidate bits for the SC flipping decoding. We establish a reinforcement learning model for selecting candidate bits, and the agent selects candidate bits to decode the information sequence. In our scheme, the decoding delay caused by the metric ordering can be removed during the decoding process. Simulation results demonstrate that the decoding delay of the proposed algorithm is reduced compared with the SCF decoding algorithm, based on critical set without loss of performance.
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Polarization adjusted convolutional (PAC) codes are a class of codes that combine channel polarization with convolutional coding. PAC codes are of interest for their high performance. This paper presents a systematic encoding and shortening method for PAC codes. Systematic encoding is important for lowering the bit-error rate (BER) of PAC codes. Shortening is important for adjusting the block length of PAC codes. It is shown that systematic encoding and shortening of PAC codes can be carried out in a unified framework.
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A polar coding scheme is proposed for the Wiretap Broadcast Channel with two legitimate receivers and one eavesdropper. We consider a model in which the transmitter wishes to send the same private (non-confidential) message and the same confidential message reliably to two different legitimate receivers, and the confidential message must also be (strongly) secured from the eavesdropper. The coding scheme aims to use the optimal rate of randomness and does not make any assumption regarding the symmetry or degradedness of the channel. This paper extends previous work on polar codes for the wiretap channel by proposing a new chaining construction that allows to reliably and securely send the same confidential message to two different receivers. This construction introduces new dependencies between the random variables involved in the coding scheme that need to be considered in the secrecy analysis.
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Neural network decoders (NNDs) for rate-compatible polar codes are studied in this paper. We consider a family of rate-compatible polar codes which are constructed from a single polar coding sequence as defined by 5G new radios. We propose a transfer learning technique for training multiple NNDs of the rate-compatible polar codes utilizing their inclusion property. The trained NND for a low rate code is taken as the initial state of NND training for the next smallest rate code. The proposed method provides quicker training as compared to separate learning of the NNDs according to numerical results. We additionally show that an underfitting problem of NND training due to low model complexity can be solved by transfer learning techniques.
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The generator matrices of polar codes and Reed-Muller codes are submatrices of the Kronecker product of a lower-triangular binary square matrix. For polar codes, the submatrix is generated by selecting rows according to their Bhattacharyya parameter, which is related to the error probability of sequential decoding. For Reed-Muller codes, the submatrix is generated by selecting rows according to their Hamming weight. In this work, we investigate the properties of the index sets selecting those rows, in the limit as the blocklength tends to infinity. We compute the Lebesgue measure and the Hausdorff dimension of these sets. We furthermore show that these sets are finely structured and self-similar in a well-defined sense, i.e., they have properties that are common to fractals.
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Asymptotic secrecy-capacity achieving polar coding schemes are proposed for the memoryless degraded broadcast channel under different reliability and secrecy requirements: layered decoding or layered secrecy. In these settings, the transmitter wishes to send multiple messages to a set of legitimate receivers keeping them masked from a set of eavesdroppers. The layered decoding structure requires receivers with better channel quality to reliably decode more messages, while the layered secrecy structure requires eavesdroppers with worse channel quality to be kept ignorant of more messages. Practical constructions for the proposed polar coding schemes are discussed and their performance evaluated by means of simulations.
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This paper proposes a distributed joint source-channel coding (DJSCC) scheme using polar-like codes. In the proposed scheme, each distributed source encodes source message with a quasi-uniform systematic polar code (QSPC) or a punctured QSPC, and only transmits parity bits over its independent channel. These systematic codes play the role of both source compression and error protection. For the infinite code-length, we show that the proposed scheme approaches the information-theoretical limit by the technique of joint source-channel polarization with side information. For the finite code-length, the simulation results verify that the proposed scheme outperforms the distributed separate source-channel coding (DSSCC) scheme using polar codes and the DJSCC scheme using classic systematic polar codes.