This paper presents a novel group secret key generation (GSKG) protocol for multi-user wireless systems with a star topology. In contrast to conventional methods relying on pairwise key derivation followed by reconciliation, the proposed approach enables the central node to directly select a group secret key uniformly at random across the full quantization space. This eliminates the need for intermediate pairwise keys, significantly reducing computational complexity, communication overhead, and latency. To ensure consistent group key reconstruction, we introduce two lightweight, non-coding reconciliation techniques: One-Sided Centering (OSC) and Quantization Pattern Alternation (QPA). Unlike coding-based reconciliation in conventional schemes, OSC and QPA allow the central node to transmit helper data – derived from adjusted measurements or shifted quantization grids – to peripheral nodes. This helper data not only aligns their channel observations with the selected quantization region, but also implicitly distributes the group key. Thus, reconciliation and distribution are seamlessly integrated into a unified protocol. The proposed GSKG framework supports both fully and partially cooperative scenarios and remains robust against unreliable or compromised nodes by selectively omitting or perturbing helper data. Furthermore, a concatenation-based group key construction method exploits existing bidirectional channel measurements to generate multiple key segments, enhancing throughput without additional channel probing. Extensive simulations validate the scheme’s performance in terms of key disagreement rate, key generation rate, computational efficiency, and robustness against eavesdropping. The results confirm that the protocol offers high scalability and security, making it a practical and efficient solution for group key establishment in IoT and other resource-constrained wireless applications.

The transfer function of a power-line connection can be modified by terminations at the end terminals between L-N or also L-PE in common three-wire home wiring, likewise at other end points. Using random terminations with, e.g., reactive loads and varying appliance connections can provide the common randomness required for physical-layer key generation. The amplitude and phase of the transfer function are noticeably modified by these variations. Hence, both can potentially be quantized for key generation. For convenient use of a linear quantizer, we opted for the phase. Dependent on the network situation and the location of the varying loads, the transfer function can be made largely independent between legitimate and eavesdropping connections. In here, we investigate the use of random capacitive or inductive loads at the L-N endpoints that are also commonly used for power-line transmission. We use a simulation program that is also modeling all transfer functions in a 3 -wire system, although the results reported here would not be influenced much by crosstalk contributions. The simulation models the connection between two end-points as a back-bone with bridge taps. We determine the transfer factors $S_{12}$ and $S_{21}$ (ideally identical) instead of the transfer functions for legitimate and critical eavesdropper placements. We include a key reconciliation option and investigate the key disagreement rates for legitimate and eavesdropping connections and provide NIST test results.

This article presents a novel low-complexity deep learning model for physical-layer secret key generation (PSKG), specifically designed to enhance wireless security in the Internet of Things (IoT). PSKG typically generates cryptographic keys by exploiting the reciprocal nature of wireless channels; however, this reciprocity is frequently compromised in time-division duplex (TDD) systems due to hardware imperfections and noise, which significantly complicate the key generation process. To effectively address these challenges, the proposed deep neural network (DNN) is developed to efficiently learn and enhance reciprocity features, even in the presence of imperfect channel state information (CSI). Furthermore, the study introduces a DNN-based PSKG method that strategically leverages the phase of channel frequency responses for key extraction. The results conclusively demonstrate that the proposed model substantially reduces the key disagreement ratio (KDR) and enhances randomness, thus providing a robust and practical solution for securing wireless communications in IoT environments.

Physical Layer Secret Key Generation (PSKG) exploits wireless channel reciprocity to establish cryptographic keys between legitimate users, providing a promising solution for securing IoT networks. However, reciprocity in time-division duplex (TDD) systems is often compromised by hardware imperfections and channel noise, posing significant challenges for reliable key generation. To address these issues, this paper proposes an autoencoder-based noise augmentation (AE-NA) model that operates in an unsupervised manner. Specifically, the proposed model integrates dynamic Gaussian noise augmentation directly into the autoencoder architecture. This approach enhances feature extraction capabilities and improves noise robustness, enabling the network to learn noise-invariant representations effectively from noisy channel estimates. Furthermore, we present a comprehensive PSKG framework that incorporates the proposed AE-NA model. Reported results demonstrate that the AE-NA-based PSKG significantly reduces the key disagreement ratio (KDR) while enhancing key randomness, thereby providing a scalable and effective solution for secure key generation in dynamic IoT environments.

Physical Layer Secret Key Generation (PLSKG), which utilizes wireless channel reciprocity, is increasingly adopted to secure wireless communications. In Time Division Duplexing (TDD) systems, the inherent channel reciprocity facilitates the generation of cryptographic keys without requiring a key exchange process. Conversely, Frequency Division Duplexing (FDD) systems present greater challenges for key generation due to the use of distinct frequency bands for uplink and downlink, resulting in differing frequency responses. In this paper, we address this challenge by modeling the channel along with connecting cables as a two-port network and utilizing the reciprocity principles applicable to such networks within the same frequency range. For key generation, we construct the channel profile for each FDD band by utilizing the amplitude of the bidirectional scattering parameters S12and S21in two closely spaced FDD bands, thereby ensuring the required reciprocity in FDD systems. To assess the practicality of the proposed scheme, we analyze its performance across various indoor scenarios and compare it with other key generation schemes using multiple performance indicators. The comparative results confirm the feasibility and effectiveness of our scheme in terms of randomness and key disagreement ratio.

This article presents a secure spatial modulation (SM) scheme designed to protect communications within resource-constrained Internet of Things (IoT) networks. The proposed scheme leverages channel state information (CSI) and channel reciprocity in time-division duplex (TDD) communications, offering a robust security solution without demanding high computational resources. The proposed model generates two independent shuffling vectors, strategically derived from the channel’s power and phase characteristics. These vectors are utilized to permute both the signal and space constellation diagrams, disrupting eavesdropping attempts. The reported results affirm the scheme’s effectiveness. Legitimate receivers consistently achieve a low average bit error ratio (ABER), ensuring reliable information decoding. In contrast, potential eavesdroppers face significantly higher ABER, consistently around 0.5 across various signal-to-noise ratio (SNR) levels, numbers of transmit antennas, and modulation orders. Furthermore, the complexity analysis of the proposed model confirms its suitability for resource-constrained IoT devices.

Physical-layer secret key generation (PSKG) stands as a promising privacy protection technique, establishing shared encryption keys through the analysis of highly correlated wireless channel measurements. This approach relies on exploiting reciprocal channel characteristics between uplink and downlink transmissions. Nonetheless, the distinct carrier frequencies employed for uplink and downlink in frequency-division duplexing (FDD) systems pose a challenge in identifying common features. This paper presents a novel approach that exploits the inherent reciprocity between scattering parameters of passive two-port networks within same frequency ranges to overcome this obstacle. By capitalizing this reciprocity and considering closely situated FDD bands, a seamless continuity is anticipated in phase differences extracted form the corresponding S-parameters, between neighboring antennas of an antenna array from both uplink and downlink directions. This continuity, thereby ensures consistency in the generated keys from both transmission ends. Furthermore, a two-stage pre-processing method is proposed to enhance performance effectively. Additionally, the paper suggests the utilization of polynomial curve-fitting through measurement data to improve reciprocity and proposes a non-linear framework for quantizing the merging points of the two FDD bands. A statistical analysis employing multiple linear regression is provided to determine the error probability associated with the generated keys. Empirical results validate the feasibility and effectiveness of the proposed key generation scheme, affirming its attributes in terms of randomness, efficiency, key distribution uniformity, and key disagreement ratio (KDR).

Physical layer key generation is based on reciprocal channels providing common randomness, which was so far known from TDD wireless channels. This paper opens the door to wireline physical-layer security, especially focusing on power-line connections. Additionally to the known reciprocity, we now also provide randomization by terminating idle branching connections (e.g., empty sockets) with random (reactive) loads. Alternatively, unused pairs at the power-line modem's end may be terminated by random (reactive) loads. Simulation and measurement results are shown. We also indicate, how the actual key generation can be realized by quantizing a frequency range and using the position of notches of the transfer function or of transmission coefficients selecting a quantization interval and with it a binary label as a key segment. Key reconciliation can simply be realized by a publicly announced shift of the quantization grid. Applications are seen for in-home and industrial devices required to exchange data over the power-line network securely.

BACKGROUND: Identification of essential genes is not only useful for our understanding of the minimal gene set required for cellular life but also aids the identification of novel drug targets in pathogens. In this work, we present a simple and effective gene essentiality prediction method using information-theoretic features that are derived exclusively from the gene sequences. RESULTS: We developed a Random Forest classifier and performed an extensive model performance evaluation among and within 15 selected bacteria. In intra-organism predictions, where training and testing sets are taken from the same organism, AUC (Area Under the Curve) scores ranging from 0.73 to 0.90, 0.84 on average, were obtained. Cross-organism predictions using 5-fold cross-validation, pairwise, leave-one-species-out, leave-one-taxon-out, and cross-taxon yielded average AUC scores of 0.88, 0.75, 0.80, 0.82, and 0.78, respectively. To further show the applicability of our method in other domains of life, we predicted the essential genes of the yeast Schizosaccharomyces pombe and obtained a similar accuracy (AUC 0.84). CONCLUSIONS: The proposed method enables a simple and reliable identification of essential genes without searching in databases for orthologs and demanding further experimental data such as network topology and gene-expression.

In this work, we investigate a physical-layer key reconciliation protocol for a reciprocal, flat fading channel between two legitimate users. We consider the scenario when the n bits of the secret key are measured independently by Alice and Bob without a transmission over the channel. Due to reciprocity, the generated keys are identical except for noise at both ends. We assume Gaussian noise and ignore non-ideal behavior of circuitry and alike. Redundancy information required to reconciliate the key is transmitted from one legitimate user to the other. LDPC codes are employed for the reconciliation procedure. The main focus of this work lies in designing the code structure through density evolution for a multi-edge-type description.

This manuscript addresses peak-to-average ratio (PAR) reduction in orthogonal frequency division multiplexing (OFDM) based multiple-input multiple-output (MIMO) systems. A new technique for PAR reduction in point-to-point scenarios is proposed. Singular value decomposition (SVD) algorithms usually sort the singular values of a MIMO channel in descending order. The last singular value and the associated eigenchannels of a MIMO-OFDM channel are often very weak. Not using them for data transmission will offer redundancy for PAR reduction. These eigenchannels are used to approximate the peaks which exceed a given target value in a least-squares fashion. This approximated exceedence model is then subtracted from the original signal in time domain for PAR reduction. It has been shown that a remarkable gain can be obtained with the proposed algorithm with a negligible increase in the average power and capacity loss.

Key reconciliation procedures are needed to correct key differences that can arise as a consequence of independent noise at the two ends of a reciprocal link. We assume a line-of-sight channel and use reconfigurable antenna elements to randomize it, such that it allows for key generation. The Linde-Buzo-Gray algorithm is employed to quantize the complex channel transfer characteristic, and adaptive guard bands, symmetric to the quantization thresholds, are further constructed. To limit the number of key errors, we ensure that only the points that fall outside the guard band interval are accepted for key generation. The steps for constructing the guard bands are presented.

This paper addresses peak-to-average ratio (PAR) reduction in orthogonal frequency division multiplexing (OFDM) based large-scale multiple-input multiple-output (MIMO) systems. We propose a new technique for PAR reduction in point-to-point scenarios. The last eigenchannels of a massive MIMO channel are often very weak. Not using them for data transmission will offer redundancy for PAR reduction. These eigenchannels are used to approximate the peaks which exceed a given target value in a least squares fashion. This approximate exceedence model is then subtracted from the original signal in time domain for PAR reduction. It has been shown through simulation results that a considerable gain can be obtained with the proposed method with marginal increase in the average power and almost negligible loss in the channel capacity.

BACKGROUND: A number of evolutionary models have been widely used for sequence alignment, phylogenetic tree reconstruction, and database searches. These models focus on how sets of independent substitutions between amino acids or codons derive one protein sequence from its ancestral sequence during evolution. In this paper, we regard the Empirical Codon Mutation (ECM) Matrix as a communication channel and compute the corresponding channel capacity. RESULTS: The channel capacity of 4.1875 bit, which is needed to preserve the information determined by the amino acid distribution, is obtained with an exponential factor of 0.26 applied to the ECM matrix. Additionally, we have obtained the optimum capacity achieving codon distribution. Compared to the biological distribution, there is an obvious difference, however, the distribution among synonymous codons is preserved. More importantly, the results show that the biological codon distribution allows for a "transmission" at a rate very close to the capacity. CONCLUSION: We computed an exponential factor for the ECM matrix that would still allow for preserving the genetic information given the redundancy that is present in the codon-to-amino acid mapping. This gives an insight how such a mutation matrix relates to the preservation of a species in an information-theoretic sense.

A Middleton Class-A (MCA) model is one of the most accurate statistical-physical models for narrowband impulse noise. The previous studies show that time diversity can efficiently be used to reduce the impact of MCA noise. The optimum combiner in such noise consists of a nonlinear preprocessor followed by a conventional combiner. Since an MCA noise process consists of an infinite number of noise states, there is no closed-form solution of the optimum nonlinearity. In this paper, we adopt a two-term model for the MCA process, which is further approximated to a simpler noise model. Therefore, we introduce a closed-form approximation of the optimum nonlinearity in the presence of real-valued MCA noise. In fading channels, we use a complex extension of an MCA model. We show how the nonlinearity operation maintains the diversity advantage in such a noise model.

The Middleton Class-A (MCA) model is one of the most accepted models for narrow-band impulsive interference superimposed to additive white Gaussian noise (AWGN). The MCA density consists of a weighted linear combination of infinite Gaussian densities, which leads to a non-tractable form of the optimum detector. To reduce the receiver complexity, one can start with a two-term approximation of the MCA model, which has only two noise states (Gaussian and impulsive state). Our objective is to introduce a simple method to estimate the noise state at the receiver and accordingly, reduce the complexity of the optimum detector. Furthermore, we show for the first time how the decision boundaries of binary signals in MCA noise should look like. In this context, we provide a new analysis of the behavior of many suboptimum detectors such as a linear detector, a locally optimum detector (LOD), and a clipping detector. Based on this analysis, we insert a new clipping threshold for the clipping detector, which significantly improves the bit-error rate performance.

A Middleton Class-A (MCA) density is well known to model impulsive interference. The statistical-physical extension of this model for multiple receive antennas is currently limited to two antennas. An algebraic extension of the univariate MCA model leads to a multivariate MCA distribution, which can be used for an arbitrary number of receive antennas. Since recent studies show a significant level of noise correlation in several wireless systems, we develop MIMO receivers for Rayleigh fading channels in the presence of spatially correlated MCA interference. We derive an upper bound pairwise error probability (PEP) for orthogonal space time block codes (OSTBCs). We show that the performance improvement of OSTBCs is highly dependent on the impulse noise environment and it becomes minor as the number of transmit and receive antennas increases. In the design of MIMO receivers, the maximum likelihood (ML) detection has a high computational complexity. Since the MCA model can be seen as a multivariate Gaussian distribution conditioned on the knowledge of noise state, we introduce a simple approach to estimate the state of noise at the receiver, which subsequently reduces the complexity of the ML decision rule.

We propose a scheme to improve system performance in Discrete MultiTone systems, which combines the joint differential and common mode processing at the receiver side with erasure marking. This is achieved by using the strong correlation between differential and common mode in the case of a few strong interferers to obtain an estimate of the impulse noise or RFI, which is used for error localization before decoding. When side information is available, a Reed Solomon (RS) decoder is able to correct twice as many erasures than errors.

In MIMO transmission, channel state information (CSI) is crucial for achieving channel adaptation. However, the inaccuracy of CSI may induce severe interferences. Hereto, limitations of linear equalizers to combat severe interference and noise enhancements necessitate the need for investigating non-linear schemes. Thus, we propose a modified successive interference cancellation (SIC) technique based on the well-known V-BLAST non-linear spatial equalizer. First, we implement a linear pre-processing filter in order to pre-sort the eigenchannels at the transmitter. This simplifies the complexity of non-linear equalization significantly by reducing the effort needed for sorting at the receiver. To protect the strong eigenbeams against errors and minimize the SIC error propagation, an unequal-error protection (UEP) bit-loading algorithm is used. A comparison to an MMSE linear equalization shows that our design operates at a lower symbol-error ratio (SER) with almost identical complexity.

The Middleton Class-A (MCA) model is one of the most widely applied models for narrow-band impulsive interference superimposed to additive white Gaussian noise (AWGN). The MCA noise process consists of an infinite number of Gaussian-distributed noise states with different variances. As a result, the optimum detector has irreducible form. Here, our analysis is based on a two-state model, where we further approximate it to a single noise state. Therefore, a log-function reduces the likelihood ratio test (LRT) to a closed-form expression. Since the low-pass equivalent of the noise process can be expressed by in-phase and quadrature (IQ) components. We derive the nonlinear decision rules when the IQ components of noise are independent and identically distributed (i.i.d.). Furthermore, for jointly distributed IQ noise components, we show that the conventional coherent detector over a fading channel with Gaussian noise is still optimum for impulse noise.

From our earlier works, we know that in the case of analog codes, a Turbo-like iterative decoding can be nicely illustrated as iterative projections onto super codes that correspond to parts of the parity check matrix. So-called LCD (linear code with complementary dual) codes are recognized as a counterpart in finite fields for the orthogonal case, where two iterative projections lead to the final solution. A method for decomposing an arbitrary LCD code C into two super LCD codes C1and C2such that decoding by iteratively projecting the received vector onto C1and C2results in the same decoding solution as directly projecting the vector onto the original code space C. This is not necessarily a maximum-likelihood solution opposite to the analog case. A bound on the probability of finding the nearest codeword is provided.

The inlay approach for the Broadband Aeronautical Multicarrier Communications (B-AMC) system is exposed to severe interference from the adjacent channels of the distance measuring equipment (DME) system. Here, we propose a simple technique to mitigate DME interference and subsequently enable transmission in the spectral gaps of the DME channels. The proposed technique uses a precoding at the transmitter based on employing lattice signal sets in order to modify the shape of the DME signal spectrum. Hereto, a simple clipping method is applied to the received subcarriers to mitigate the impact of the DME interference. Simulations show that the proposed subcarrier clipping technique can considerably reduce the effect of the DME signal by choosing appropriate clipping thresholds. They have been selected to maximize the signal-to interference-and-noise ratio (SINR) after the clipping operation. It has also been confirmed by simulations that the proposed method offers a significantly better performance when compared to current mitigation techniques.

A multi-edge framework for unequal error protecting (UEP) LT codes is derived by distinguishing between the edges connected to each protection class. Under the framework introduced, two existing techniques for the design of unequal error protecting LT codes can be evaluated and explained in a unified way. Furthermore, a simple and flexible design technique is proposed for UEP LT codes with good performance.

Irregular low-density parity-check (LDPC) codes are particularly well-suited for transmission schemes that require unequal error protection (UEP) of the transmitted data due to the different connection degrees of its variable nodes. However, this UEP capability is strongly dependent on the connection profile among the protection classes. This paper applies a multi-edge type analysis of LDPC codes for optimizing such a connection profile according to the performance requirements of each protection class. This allows the construction of UEP-LDPC codes where the difference between the performance of the protection classes can be adjusted and with an UEP capability that does not vanish as the number of decoding iterations grows.

Unequal Error Protection (UEP) is the key to future multi-layer and scalable data and video transmission. This paper presents a novel bit-loading and channel adaptation technique to realize UEP properties in the physical transport using greedy and sub-optimal bit-loading algorithms. In this case, we exploited the power minimization approach developed by the greedy algorithm to allow for an arbitrary number of classes, arbitrary SNR margins between the classes, and arbitrary numbers of bits per class. We achieved a similar results using a fast sub-optimum power minimization bit-loading algorithm based on Campello's algorithm. Our results show the suitability of this technique to realize UEP for a single user case using orthogonal frequency division multiplexing (OFDM). Additionally, we extended the results to demonstrate the performance of allocating resources to two users with different quality of service (QoS), each of which requires data of different priorities.

This paper presents the design of multilevel coding schemes optimized for the transport of multimedia data. The key feature is unequal error protection which allows for more protection for header information and essential data, accepting worse performance for less important payload. The approach is based on the information theoretical description of mutual information in multilevel coding schemes. We discuss the suitability and UEP capability of standard as well as non-uniform signal constellations. We investigate the flexibility of this method regarding the design freedom and verify the approach by an image transmission application.

We investigate the sum-product decoding on graphs of analog compound codes and show that the iterative decoding can be completely analyzed by tracing the mean vector at each iteration. A novel geometric analysis is proposed to visualize the iterative decoding process in the Euclidean space. Based on this geometric analysis, we propose to decompose the analog compound codes into several orthogonal constituent code spaces to achieve the fastest convergence speed. Simulations are given to verify our conclusions.

This paper describes the turbo decoding of hybrid concatenated codes with interleavers and presents a new analysis of the decoding process including information processing and convergence. From this analysis, the decoding can be scheduled with respect to optimization issues like computational complexity and convergence behavior

The common-mode (CM) signal in wireline transmission systems has proven to provide valuable information exploited for mitigating narrowband noise at the receive side. We focus on the case of broadband noise. Treating the CM signal as an additional receive signal, we investigate the capacity of the copper cable channel for different levels of coordination among the users. We introduce a channel model which includes the common-mode paths and derive a suitable form of the channel capacity formula. CM crosstalk measurement results, essential for evaluation of the channel capacity, are presented. Using the measurement data, exemplary results of capacity gain achievable by CM-aided data transmission over the copper cable are shown.

Digital subscriber line (DSL) technology for high-speed data transmission over the local-loop telephone twisted pairs was advanced in the labs of several research institutions during the late 1980’s and early 1990’s. Researchers began to develop and experiment with new transmission ideas and to realize high-speed modems in prototype form. As technology matured and standards evolved, these modems came to be known collectively as “xDSL” modems. The ISDN basic-rate interface, which was developed during the early 1980’s and operated at a “mere” 160 kb/s, today is often cited as being the archetype of this technology.

This paper proposes a suitable method for simulating impulses with appropriate amplitude, spectral, and inter-arrival characteristics. The statistics used to develop the parameters of this model are based on statistics derived from observations of impulse noise on the telephone networks of British Telecom (BT) and Deutsche Telekom (DT). This paper initially reviews the former DT approach to impulse noise generation for testing digital subscriber line systems, so called xDSL systems. Some problems are highlighted and an alternative technique is suggested that is capable of generating impulses with both appropriate amplitude an spectral characteristics.

An adaptation algorithm for determining the time-domain equalizer coefficients is described that maximizes the total channel capacity for all carriers of a multitone (discrete multitone) transmission. It takes into account the crosstalk noise environment and the interblock interference as a common disturbance. Furthermore, the leakage effect of the discrete Fourier transform (fast Fourier transform) is considered, too. Including this into the algorithm for the equalizer coefficients leads to a notable improvement in the signal-to-noise ratio, especially at lower frequencies for a typical asymmetrical digital subscriber line application.

A bound for the possible reduction of the peak-to-average ratio (PAR) dependent on the rate as well as possible practical procedures are presented. The idea of trellis shaping, originally used to minimize average transmit power in single-carrier systems, is applied to the problem of PAR reduction in multicarrier transmission. Its impact, as a function of code rate, as well as design practicability is considered using metrics in both the time and frequency domains.

This paper presents the performance of various coding schemes for the asymmetrical digital subscriber line (ADSL) in an impulse-noise environment. Impulse noise is considered to be one of the most damaging impairments in the ADSL, in which compressed video signals are delivered to residential customers. The impulse noise used in this study was measured and collected in German telephone networks. Based on this measurement and the corresponding statistical modeling, a simulation model for impulse noise is proposed and its properties are outlined. The coding schemes considered here utilize burst-error correcting Reed-Solomon codes and/or random error correcting trellis codes as well as symbol interleaving between the two codes. It has been found through computer simulations that a proper concatenation of the two codes could increase the immunity against impulse noise compared to an uncoded scheme. Specifically, a concatenated code, using a 2-dimensional 8-state trellis code and a 4-error-correcting Reed-Solomon code with an interleaving depth of 18 symbols, was able to eliminate all the errors caused by the impulse noise used in the study. It has also been found that the trellis codes are not very effective against impulse noise, unless they are used in conjunction with Reed-Solomon codes and a proper symbol interleaving. Performance results of other coding configurations using Reed-Solomon codes with different error-correcting capabilities are also presented. In addition, we also show the performance results when simple array codes are used instead of the Reed-Solomon codes

Based on V.A. Zinoviev's (1981) generalized concatenated codes, conditions for the construction of 90 degrees phase-invariant QAM are derived. Furthermore, a proposal for the necessary differential encoding/decoding is made. The conditions for phase invariance are specialized for the case of Reed-Muller codes as outer codes of the generalized concatenation.>

The Berlekamp-Massey algorithm (BMA) which solves special Toeplitz systems of linear equations is extended to an algorithm for inverting Toeplitz matrices. The original BMA itself leads to one row of the inverse of the corresponding Toeplitz matrix. The other rows are derived using the same operations that are central to the original BMA. Two alternatives for the extended BMA are presented: the first includes the usual BMA without any changes, and the second simplifies the structure by some modifications also in the original BMA part. Both versions follow a tree-like structure. If the branches of the tree are implemented in parallel, the time demand would be nearly the same as for the usual BMA. In contrast to other Toeplitz algorithms, only slight modifications must be incorporated to handle singular submatrices.>