It is well known that one of the most dangerous attacks against space transmissions is jamming. In particular, space telemetry downlinks are subject to extremely tight power and bandwidth limitations and are hence vulnerable to both accidental and intentional interference. While the Consultative Committee for Space Data Systems (CCSDS) has already standardized Low-Density Parity-Check (LDPC) codes for the use in space telemetry communications, we show that their performance is significantly deteriorated by the impact of pulsed jamming. We investigate the use of spatially coupled LDPC (SC-LDPC) codes as a promising alternative for improving reliability of telemetry channels in interference-heavy environments. We perform Monte Carlo simulations to assess the sensitivity of SC-LDPC-coded deep space communications to pulsed jamming, comparing attainable error rate performance with an LDPC code recommended in the CCSDS standard for telemetry synchronization and channel coding issues. Our results show that SC-LDPC codes can achieve improved performance, in terms of reduced bit error rate, under different pulsed jamming conditions.

Cyclic simplex codes are usually deemed as impractical because of their extremely low rate and large codeword length for practical values of the code dimension. To address these limitations, researchers have resorted to punctured simplex codes, focusing on primitive polynomials and employing statistical analyses to investigate their properties. This paper delves deeper into the properties of punctured binary simplex codes, also focusing on the recently introduced family of Primitive Rate-Compatible Low-Density Parity-Check (PRC-LDPC) codes. We study the average behavior of punctured simplex codes in terms of minimum distance properties. Furthermore, our results highlight the potential of high-rate PRC-LDPC codes to reach or even surpass the performance of state-of-the-art code families. We show how to design good codes by applying puncturing and shortening operations to cyclic simplex codes, also considering complexity-constrained scenarios.

Spatially coupled low-density parity-check (SC-LDPC) codes have recently attracted a lot of attention because of their optimal asymptotic performance. Time-invariant and periodically time-varying SC-LDPC codes are particularly interesting code families, since they have good finite-length performance and allow for a very compact representation. However, one of the key challenges to make SC-LDPC codes practical consists in finding efficient techniques for their encoding. In this paper, we propose a method to construct polynomial generator matrices for time-invariant and periodically time-varying SC-LDPC codes, based on their polynomial parity-check matrix. We show that it is always possible to find a polynomial generator matrix in quasi-standard form, where the systematic part consists of a diagonal matrix with identical entries, enabling efficient and non-catastrophic encoding. We also show that it is always possible to compute an equivalent rational function generator matrix in standard form that guarantees recursive, systematic (and therefore also non-catastrophic) encoding. Finally, to illustrate the method, we introduce a family of time-invariant SC-LDPC codes characterized by a binary parity-check matrix associated to a Tanner graph with girth larger than 4 and by good asymptotic and finite length performance. These codes naturally admit a quasi-standard polynomial generator matrix that, depending on the number of control symbols per period, either can be efficiently converted into polynomial standard form or remains in quasi-standard form but is relatively sparse. Both these features enable fast and efficient encoding.

The Bit-Flipping (BF) decoder, thanks to its very low computational complexity, is widely employed in post-quantum cryptographic schemes based on Moderate Density Parity Check codes in which, ultimately, decryption boils down to syndrome decoding. In such a setting, for security concerns, one must guarantee that the Decoding Failure Rate (DFR) is negligible. Such a condition, however, is very difficult to guarantee, because simulations are of little help and the decoder performance is difficult to model theoretically. In this paper, we introduce a new version of the BF decoder, that we call BF-Max, characterized by the fact that in each iteration only one bit (the least reliable) is flipped. When the number of iterations is equal to the number of errors to be corrected, we are able to develop a theoretical characterization of the DFR that tightly matches with numerical simulations. We also show how BF-Max can be implemented efficiently, achieving low complexity and making it inherently constant time. With our modeling, we are able to accurately predict values of DFR that are remarkably lower than those estimated by applying other approaches.

In the context of space communications, as per the recommendation from the Consultative Committee for Space Data Systems regarding TeleCommand synchronization and coding, the Communications Link Transmission Unit is composed of a start sequence, coded data, and a tail sequence, which might be optional depending on the employed error correcting code. The task of detecting the tail sequence must be handled along with that of decoding the codewords containing the transmitted data, and this poses some challenges. In this paper, we propose a machine learning model for recognizing the tail sequence based on the analysis of metrics calculated during decoding, when the transmission is coded with Low-Density Parity-Check (LDPC) codes. The model is trained on data produced by an iterative decoder, commonly used in LDPC decoding, with noisy (random) codewords or the noisy tail sequence as inputs. We report the results of some preliminary experiments showing that this approach is capable of achieving very high levels of accuracy using multiple classifiers.

In this paper we study a special family of Low-Density Parity-Check (LDPC) codes, called Primitive Rate-Compatible (PRC) LDPC codes, obtained by applying puncturing to some properly chosen simplex codes. We investigate their minimum distance properties and give insights on the design of PRC-LDPC codes for use in scenarios where constraints on hardware and software resources require maintaining a low level of complexity. We numerically assess the error rate performance of these codes in such a setting, showing that they also demonstrate good performance under belief propagation decoding, in addition to low decoding complexity.

We introduce and study a family of rate-compatible Low-Density Parity-Check (LDPC) codes. The design of these codes starts from simplex codes, defined by parity-check matrices having a simple form stemming from the coefficients of a primitive polynomial. For this reason, we call the new codes Primitive Rate-Compatible LDPC (PRC-LDPC) codes. By applying puncturing to these codes, we obtain a bit-level granularity of the code rate. We show that, in order to achieve good LDPC codes, the underlying polynomials, besides being primitive, must meet some more stringent conditions with respect to those of classical punctured simplex codes. We leverage non-modular Golomb rulers to take these new requirements into account. We characterize the minimum distance properties of PRC-LDPC codes, and study and discuss their encoding and decoding complexity. Finally, we assess the error rate performance of high rate PRC-LDPC codes under iterative decoding.

Time-invariant spatially coupled low-density parity-check (TI-SC-LDPC) codes can be obtained by unwrapping quasi-cyclic (QC) LDPC block codes. This results in a free distance that is lower bounded by the minimum distance of the underlying QC-LDPC block codes. By introducing some variability in the syndrome former matrix, time-varying (TV) SC-LDPC codes are obtained, which trade an improved error correction performance for an increased decoding memory requirement and decoding complexity. A family of codes able to combine the advantages of TI-SC-LDPC codes with those of TV-SC-LDPC codes is that of periodically time-varying (PTV) SC-LDPC codes, based on a finite and periodic variation of the syndrome former matrix. In this paper we focus on such codes, and derive new upper bounds on the free distance of PTV-SC-LDPC code ensembles as well as on specific codes. By using these bounds, we show that PTV-SC-LDPC codes can achieve important improvements in the free distance over TI-SC-LDPC codes even using a very small period of variability, which corresponds to a minimal increase in memory and complexity. We also validate the new upper bounds through numerical experiments and assess the error correction performance of the corresponding codes through Monte Carlo simulations.

The Permuted Kernel Problem (PKP) asks to find a permutation which maps an input matrix into the kernel of some given vector space. The literature exhibits several works studying its hardness in the case of the input matrix being mono-dimensional (i.e., a vector), while the multi-dimensional case has received much less attention and, de facto, only the case of a binary ambient finite field has been studied. The Subcode Equivalence Problem (SEP), instead, asks to find a permutation so that a given linear code becomes a subcode of another given code. At the best of our knowledge, no algorithm to solve the SEP has ever been proposed. In this paper we study the computational hardness of solving these problems. We first show that, despite going by different names, PKP and SEP are exactly the same problem. Then we consider the state-of-the-art solver for the mono-dimensional PKP (namely, the KMP algorithm, proposed by Koussa, Macario-Rat and Patarin), generalize it to the multi-dimensional case and analyze both the finite and the asymptotic regimes. We further propose a new algorithm, which can be thought of as a refinement of KMP. In the asymptotic regime our algorithm does not improve on KMP but, in the finite regime (and for parameters of practical interest), we achieve significant improvements, especially for the multi-dimensional version of PKP. As an evidence, we show that it is the fastest algorithm to attack several recommended instances of cryptosystems based on PKP. As a side-effect, given the mentioned equivalence between PKP and SEP, all the algorithms we analyze in this paper can be used to solve instances of the latter problem.

Cyber risk assessment is one of the top priorities of modern organizations and companies, owing to the massive amount of data they process on a daily basis and to the increasing number of successful cyber attacks. The probability of occurrence of these cyber incidents can be estimated by means of statistical tools, which exploit numerical categories to compute the probability that the organization will be breached by one or more cyber attacks. However, these approaches heavily rely on experts' estimates and/or on past data, which are not always available. In this paper we show that, by exploiting machine learning tools, cyber risk can be assessed by using some easily obtainable parameters (called maturity, complexity, attractiveness) representing the cyber posture of the organization under exam. To validate the method we propose, we apply it to three organizations in the healthcare sector having different values of maturity and complexity. The results highlight how the model can be successfully used to assign each organization a class of cyber risk, even in a crucial sector such as healthcare.

We propose a protocol to jointly achieve authentication and consensus on a blockchain network, in which endpoints are required to digitally sign some random message using fuzzy keys according to a classic fuzzy signature paradigm typical, for example, of biometric authentication. We consider classic RSA digital signatures, showing that fuzziness in the secret key translates into some noise affecting the derived signatures. The removal of such a noise provides the basis for building a blockchain consensus mechanism, which we name Proof of Fuzzy Signature (PoFS). It basically provides a special instance of Proof of Work in which the mining process corresponds to the de-noising process of RSA digital signatures derived from fuzzy keys. This way, the authentication process is delegated to a distributed network and, at the same time, requires executing the useful task of removing noise from fuzzy signatures.

In this paper we show that, when a binary primitive polynomial can be associated to a sparse Golomb ruler, the simplex code obtained by taking it as the code parity-check polynomial exhibits good distance properties and performance. We define some conditions under which the obtained codes are also Low-Density Parity-Check (LDPC) codes, and can hence be decoded through efficient iterative algorithms. We perform code puncturing, leading to a family of rate-adaptive codes, and we predict some of their structural properties in terms of minimum distance and weight distribution. We show that, in addition to having some useful properties, these codes achieve good performance in terms of error rate under LDPC decoding.

The Permuted Kernel Problem (PKP) asks to find a permutation of a given vector belonging to the kernel of a given matrix. The PKP is at the basis of PKP-DSS, a post-quantum signature scheme deriving from the identification scheme proposed by Shamir in 1989. The most efficient solver for PKP is due to a recent paper by Koussa et al. In this paper we propose an improvement of such an algorithm, which we achieve by considering an additional collision search step applied on kernel equations involving a small number of coordinates. We study the conditions for such equations to exist from a coding theory perspective, and we describe how to efficiently find them with methods borrowed from coding theory, such as information set decoding. We assess the complexity of the resulting algorithm and show that it outperforms previous approaches in several cases. We also show that, taking the new solver into account, the security level of some instances of PKP-DSS turns out to be slightly overestimated.

Time-varying spatially coupled low-density parity-check (SC-LDPC) codes with very large period are characterized by significantly better error rate performance and girth properties than their time-invariant counterparts, but the number of parameters they require to be described is usually very large and unpractical. Time-invariant SC-LDPC codes, which can be seen as periodically time-varying codes with unitary period, are represented through a small number of parameters and designed exploiting few degrees of freedom, but their error rate performance and girth properties are sub-optimal. In this paper, we show that the limits of time-invariant SC-LDPC codes can be overcome by transforming them into time-varying SC-LDPC codes with very small period. In particular, we show that periodically time-varying SC-LDPC codes with small period may exhibit significantly better girth properties than the corresponding time-invariant codes by exploiting a larger number of degrees of freedom in the code design, which however scale at most linearly with the product of the code period and the size of the considered base matrix.

We introduce a new family of linear block codes over $\mathbb{Z}_{q}$ that we name low-Lee-density parity-check (LLDPC) codes. These codes, which are embedded with the Lee metric, are characterized by a parity-check matrix whose rows and columns have low Lee weight. We propose general constructions of LLDPC codes and devise an efficient iterative decoding algorithm for them, with complexity that grows linearly with the code length. We assess the error rate performance of these codes through numerical simulations.

Public‐key cryptosystems built on quasi‐cyclic (QC) low‐density parity‐check and moderate‐density parity‐check codes are promising candidates for post‐quantum cryptography, since they are characterised by compact keys and high algorithmic efficiency. The main issue with this kind of system is represented by the fact that, since the decoding procedure is probabilistic, it may leak information about the secret key. In this work, the authors study cryptanalysis procedures that aim at recovering the secret key by exploiting this fact. They identify the phenomenon that is at the basis of these procedures and show that the QC structure plays an important role in the success of these attacks. They use a graph analogy to study the complexity of these attacks, and show that their feasibility strongly depends on the QC structure. They also devise an approach to perform full cryptanalysis by combining an information set decoding algorithm with some partial knowledge about the structure of the secret key.

Iterative decoders used for decoding low-density parity-check (LDPC) and moderate-density parity-check (MDPC) codes are not characterized by a deterministic decoding radius and their error rate performance is usually assessed through intensive Monte Carlo simulations. However, several applications, like code-based cryptography, need guaranteed low values of the error rate, which are infeasible to assess through simulations, thus requiring the development of theoretical models for the error rate of these codes. Some models of this type already exist, but become computationally intractable for parameters of practical interest. Other approaches approximate the code ensemble behaviour through assumptions, which may not hold true for a specific code. We propose a theoretical analysis of the error correction capability of LDPC and MDPC codes that allows deriving tight bounds on the error rate at the output of parallel bit-flipping decoders. Special attention is devoted to the case of codes with small girth. Single-iteration decoding is investigated through a rigorous approach, which does not require any assumption and results in a guaranteed error correction capability for any single code. We show an example of application of the new bound to the context of code-based cryptography, where guaranteed error rates are needed to achieve strong security levels.

The error correction performance of low-density parity-check codes under iterative message-passing decoding is degraded by the presence of certain harmful objects existing in their Tanner graph representation. Depending on the context, such harmful objects are known as stopping sets, trapping sets, absorbing sets, or pseudocodewords. In this paper, we propose a general procedure, based on edge spreading, that enables the design of good quasi-cyclic spatially coupled low-density parity-check codes. These codes are derived from quasi-cyclic low-density parity-check (QC-LDPC) block codes and possess a significantly reduced multiplicity of harmful objects with respect to the original QC-LDPC block codes. The proposed procedure relies on a novel algorithm that greedily spans the search space of potential candidates to reduce the multiplicity of the target harmful objects. The effectiveness of the method is validated via examples and numerical computer simulations.

Differently from bounded-distance decoders used for algebraic codes, iterative decoders used for low-density parity-check (LDPC) codes are not characterized by a deterministic decoding radius. Therefore, the error rates of LDPC-coded transmissions are usually estimated heuristically through simulations. This is adequate for many applications like wireless communications, where a frame error rate (FER) in the order of 10-6or higher is usually targeted. However, lower values of FER can barely be assessed through simulations, and this limits the use of LDPC codes in applications requiring a lower FER, like optical communications and code-based cryptography. In this paper we introduce and study a version of the classic bit flipping (BF) decoder for which we are able to devise and develop a theoretical characterization of the FER. In addition, we consider a two-iteration hard-decision decoder for LDPC codes derived from BF, and discuss its error rate performance. Our results are validated through numerical simulations.

Time-varying spatially-coupled low-density parity-check convolutional codes (SC-LDPC-CCs) exhibit excellent features, but their representation requires a very large number of parameters. On the other hand, the description of time-invariant SC-LDPC-CCs is very convenient and their error rate performance, though usually worse, is often satisfactory. In this paper we investigate the girth properties of these codes, showing that the time-invariant ones have some weaknesses, which can be compensated by introducing a small periodicity in the code. By considering periodically time-varying codes, we achieve considerable improvements in the girth properties using few more degrees of freedom with respect to the time-invariant case.

We consider a one-time digital signature scheme recently proposed by Persichetti and show that a successful key recovery attack can be mounted with limited complexity. The attack we propose exploits a single signature intercepted by the attacker, and relies on a statistical analysis performed over such a signature, followed by information set decoding. We assess the attack complexity and show that a full recovery of the secret key can be performed with a work factor that is far below the claimed security level. The efficiency of the attack is motivated by the sparsity of the signature, which leads to a significant information leakage about the secret key.

In this study, the authors elaborate on a recently proposed variant of the public‐key McEliece and Niederreiter cryptosystems using generalised Reed–Solomon (GRS) codes as private codes. The use of these codes brings known advantages in terms of public key size, but particular care is needed in the choice of parameters not to endanger the system security. In fact, the considered system exploits a strong disguising technique of the private code within the public code. However, it has recently been pointed out that some new attacks exist which may threaten some instances of such a system, therefore the choice of parameters needs to consider some further constraints compared to the original version. After outlining these constraints, the authors propose a new modification of the system achieving greater flexibility in the parameter choice. Moreover, the new system exhibits a lower complexity than the original GRS code‐based system. Its very competitive features such as key size and encryption rate are highlighted with respect to classic systems.

Cyber risk assessment requires defined and objective methodologies; otherwise, its results cannot be considered reliable. The lack of quantitative data can be dangerous: if the assessment is entirely qualitative, subjectivity will loom large in the process. Too much subjectivity in the risk assessment process can weaken the credibility of the assessment results and compromise risk management programs. On the other hand, obtaining a sufficiently large amount of quantitative data allowing reliable extrapolations and previsions is often hard or even unfeasible. In this paper, we propose and study a quantitative methodology to assess a potential annualized economic loss risk of a company. In particular, our approach only relies on aggregated empirical data, which can be obtained from several sources. We also describe how the method can be applied to real companies, in order to customize the initial data and obtain reliable and specific risk assessments.

In this paper, we provide an analysis of the performance of concatenation of polar codes with outer cyclic redundancy check (CRC) codes, separated by an interleaver, in the short and moderate block length regimes. The analysis addresses maximum likelihood decoding as a proxy to the code performance under successive cancellation list decoding. The analysis is carried out by introducing the concatenated polar code (CPC) ensembles, whose distance properties can be analyzed (for sufficiently short block lengths) by means of the uniform interleaver approach. At moderate block lengths, we resort to the Monte Carlo simulations. Results show that if the inner polar code possesses a low minimum distance and the outer CRC code has a sufficiently large amount of redundancy, then the choice of the outer code generator polynomial and the interleaver may yield to a large variability in the performance of the resulting CPC.

In this paper we study recent reaction attacks against QC-LDPC and QC-MDPC code-based cryptosystems, which allow an opponent to recover the private parity-check matrix through its distance spectrum by observing a sufficiently high number of decryption failures. We consider a special class of codes, known as monomial codes, to form private keys with the desirable property of having a unique and complete distance spectrum. We verify that for these codes the problem of recovering the secret key from the distance spectrum is equivalent to that of finding cliques in a graph, and use this equivalence to prove that current reaction attacks are not applicable when codes of this type are used in the McEliece cryptosystem.

Low decoding latency and complexity are two important requirements of channel codes used in many applications, like machine-to-machine communications. In this paper, we show how these requirements can be fulfilled by using some special quasi-cyclic low-density parity-check block codes and spatially coupled low-density parity-check convolutional codes that we denote as compact. They are defined by parity-check matrices designed according to a recent approach based on sequentially multiplied columns. This method allows obtaining codes with girth up to 12. Many numerical examples of practical codes are provided.

In this paper, we deal with time-invariant spatially coupled low-density parity-check convolutional codes (SC-LDPC-CCs). Classic design approaches usually start from quasi-cyclic low-density parity-check block codes and exploit suitable unwrapping procedures to obtain SC-LDPC-CCs. We show that the direct design of the SC-LDPC-CCs syndrome former matrix or, equivalently, the symbolic parity-check matrix, leads to codes with smaller syndrome former constraint lengths with respect to the best solutions available in the literature. We provide theoretical lower bounds on the syndrome former constraint length for the most relevant families of SC-LDPC-CCs, under constraints on the minimum length of cycles in their Tanner graphs. We also propose new code design techniques that approach or achieve such theoretical limits.

The writing operation of multi-level flash memories can suffer from voltage overshoots, which can be generally modeled as asymmetric errors of limited magnitude. Using suitable error correcting codes, these kinds of errors can be corrected. In particular, q-ary non-linear codes of length 2 are equivalent to packings of the plane modulo q with quasi-crosses. The design procedures for a number of such packings are presented.

Transmissions over the wiretap channel have been studied for a long time from the information theory standpoint. This has allowed to assess the secrecy performance against eavesdropping while ensuring reliable transmission towards the legitimate receiver. However, most previous studies rely on a number of assumptions which are far from practical wireless communications, like infinite length codewords, random coding, discrete channels or continuous channels with Gaussian signaling. In this paper, we show how the level of security at the physical layer can be assessed from the information theoretic standpoint while taking into account the constraints of practical transmissions over realistic wireless wiretap channels, i.e., by considering practical codes with finite length, discrete modulation formats and continuous channels with fading. For this purpose, we consider the notion of mutual information security, which is provably equivalent to semantic security. Our target is to show that classical and already implemented coding and modulation schemes can be used to achieve some level of security at the physical layer, opposed to approaches resorting to completely new designs tailored to secure transmissions. To corroborate this thesis, we consider some coding and modulation schemes compliant with the IEEE 802.16e (WiMax) standard and show how they can be used to achieve some given security level.

Distributed storage systems and caching systems are becoming widespread, and this motivates the increasing interest on assessing their achievable performance in terms of reliability for legitimate users and security against malicious users. While the assessment of reliability takes benefit of the availability of well established metrics and tools, assessing security is more challenging. The classical cryptographic approach aims at estimating the computational effort for an attacker to break the system, and ensuring that it is far above any feasible amount. This has the limitation of depending on attack algorithms and advances in computing power. The information-theoretic approach instead exploits capacity measures to achieve unconditional security against attackers, but often does not provide practical recipes to reach such a condition. We propose a mixed cryptographic/information-theoretic approach with a twofold goal: estimating the levels of information-theoretic security and defining a practical scheme able to achieve them. In order to find optimal choices of the parameters of the proposed scheme, we exploit an effective probabilistic model checker, which allows us to overcome several limitations of more conventional methods.

In this paper, we study of the performance of the concatenation of a short polar code with an outer binary linear block code from a distance spectrum viewpoint. Our analysis targets the case where an outer cyclic code is employed together with an inner systematic polar code. A concatenated code ensemble is defined placing an interleaver at the input of the polar encoder. The introduced ensemble allows deriving bounds on the achievable error rates under maximum likelihood decoding, by applying the union bound to the (expurgated) average weight enumerators. The analysis suggests the need of careful optimization of the outer code, to attain low error floors. We also investigate the performance of a number of randomly chosen interleavers, with the aim to discuss the dispersion around the ensemble.

We present a method for estimating the undetected error rate when a cyclic redundancy check (CRC) is performed on the output of the decoder of short low-density parity-check (LDPC) codes. This system is of interest for telecommand links, where new LDPC codes have been designed for updating the current standard. We show that these new LDPC codes combined with CRC are adequate for complying with the stringent requirements of this kind of transmissions in terms of error detection.

We propose to use real-valued errors instead of classical bit flipping intentional errors in the McEliece cryptosystem based on moderate-density parity-check (MDPC) codes. This allows to exploit the error correcting capability of these codes to the utmost, by using soft-decision iterative decoding algorithms instead of hard-decision bit flipping decoders. However, soft reliability values resulting from the use of real-valued noise can also be exploited by attackers. We devise new attack procedures aimed at this, and compute the relevant work factors and security levels. We show that, for a fixed security level, these new systems achieve the shortest public key sizes ever reached, with a reduction up to 25% with respect to previous proposals.

We consider a system where an agent (Alice) aims at transmitting a message to a second agent (Bob) over a set of parallel channels, while keeping it secret from a third agent (Eve) by using physical layer security techniques. We assume that Alice perfectly knows the set of channels with respect to Bob, but she has only a statistical knowledge of the channels with respect to Eve. We derive bounds on the achievable outage secrecy rates, by considering coding either within each channel or across all parallel channels. Transmit power is adapted to the channel conditions, with a constraint on the average power over the whole transmission. We also focus on the maximum cumulative outage secrecy rate that can be achieved. Moreover, in order to assess the performance in a real life scenario, we consider the use of practical error correcting codes. We extend the definitions of security gap and equivocation rate, previously applied to the single additive white Gaussian noise channel, to Rayleigh distributed parallel channels, on the basis of the error rate targets and the outage probability. Bounds on these metrics are also derived, considering the statistics of the parallel channels. Numerical results are provided, that confirm the feasibility of the considered physical layer security techniques.

We consider the framework of the McEliece cryptosystem based on low-density parity-check (LDPC) codes, which is a promising post-quantum alternative to classical public key cryptosystems. The use of LDPC codes in this context allows to achieve good security levels with very compact keys, which is an important advantage over the classical McEliece cryptosystem based on Goppa codes. However, only regular LDPC codes have been considered up to now, while some further improvement can be achieved by using irregular LDPC codes, which are known to achieve better error correction performance than regular LDPC codes. This is shown in this paper, for the first time at our knowledge. The possible use of irregular transformation matrices is also investigated, which further increases the efficiency of the system, especially in regard to the public key size.

In this paper, we carry out a practical assessment of the performance of finite-length LDPC codes over the wiretap channel with fast Rayleigh fading. Classical metrics for physical layer security, like the secrecy capacity, are based on information theoretic arguments, and provide the ultimate security bounds for these schemes. However, it is difficult to design practical schemes, using some specific finite-length code, able to approach such a performance. Then we use a more practical metric, based on the error probability, which allows assessing the performance achieved in terms of both reliability and security over the fast Rayleigh fading wire-tap channel.

In this paper, we provide a tight estimation of the security gap for the wiretap channel with fast Rayleigh fading. The error rate has already been used as a practical physical layer security metric, and the security gap has been defined to exploit such a metric over the wiretap channel with additive white Gaussian noise. We study two different approaches for estimating the security gap also over a fast fading wiretap channel. The first approach is quite conservative, and tends to overestimate the security gap. The second approach instead allows to perform a more realistic evaluation, and to highlight the gain achievable through information scrambling and coding, even when both the legitimate receiver and the eavesdropper channels are known only in statistical terms.

We investigate and compare different options for updating the error correcting code currently used in space mission telecommand links. Taking as a reference the solutions recently emerged as the most promising ones, based on Low-Density Parity-Check codes, we explore the behavior of alternative schemes, based on parallel concatenated turbo codes and soft-decision decoded BCH codes. Our analysis shows that these further options can offer similar or even better performance.

In the context of public key cryptography, the McEliece cryptosystem represents a very smart solution based on the hardness of the decoding problem, which is believed to be able to resist the advent of quantum computers. Despite this, the original McEliece cryptosystem based on Goppa codes, has encountered limited interest in practical applications, partly because of some constraints imposed by this very special class of codes. The authors have recently introduced a variant of the McEliece cryptosystem including low‐density parity‐check codes, that are state‐of‐the‐art codes, now used in many telecommunication standards and applications. In this study, the authors discuss the possible use of a bit‐flipping decoder in this context, which gives a significant advantage in terms of complexity. The authors also provide theoretical arguments and practical tools for estimating the trade‐off between security and complexity, in such a way to give a simple procedure for the system design.

Product LDPC codes take advantage of LDPC decoding algorithms and the high minimum distance of product codes. We propose to add suitable interleavers to improve the waterfall performance of LDPC decoding. Interleaving also reduces the number of low weight codewords, that gives a further advantage in the error floor region.

This paper examines the use of nonsystematic channel codes to obtain secure transmissions over the additive white Gaussian noise wire-tap channel. Unlike the previous approaches, we propose to implement nonsystematic coded transmission by scrambling the information bits, and characterize the bit error rate of scrambled transmissions through theoretical arguments and numerical simulations. We have focused on some examples of Bose-Chaudhuri-Hocquenghem and low-density parity-check codes to estimate the security gap, which we have used as a measure of physical layer security, in addition to the bit error rate. Based on a number of numerical examples, we found that such a transmission technique can outperform alternative solutions. In fact, when an eavesdropper (Eve) has a worse channel than the authorized user (Bob), the security gap required to reach a given level of security is very small. The amount of degradation of Eve's channel with respect to Bob's that is needed to achieve sufficient security can be further reduced by implementing scrambling and descrambling operations on blocks of frames, rather than on single frames. While Eve's channel has a quality equal to or better than that of Bob's channel, we have shown that the use of a hybrid automatic repeat-request protocol with authentication still allows achieving a sufficient level of security. Finally, the secrecy performance of some practical schemes has also been measured in terms of the equivocation rate about the message at the eavesdropper and compared with that of ideal codes.

Binary linear [n,k] codes that are proper for error detection are known for many combinations ofnandk. For the remaining combinations, existence of proper codes is conjectured. In this paper, a particular class of [n,k] codes is studied in detail. In particular, it is shown that these codes are proper for many combinations ofnandkwhich were previously unsettled.

In this paper we extend the analysis of the performance of the APSK modulation, often limited to the AWGN channel, by considering its application in wireless tactical scenarios for land mobile systems. First, the implications of reduced PAPR on the total degradation are explored, taking into account the non-linear effects due to HPAs and the need to use adaptive pre-distortion. Then, the bit error rate performance is assessed by simulation, for some typical multipath scenarios with decision feedback equalization, also including the presence of turbo channel coding.

We present an exact analysis of the correlation coefficient, in the frequency domain, between the real and imaginary parts of the impulse noise squared norms for OFDM systems. We analyze a scenario whereMGaussian randomly positioned impulses interfere the OFDM signal, with and without the presence of a background thermal noise. Variability ofMis taken into account through the Bernoulli-Gauss model. We demonstrate that rather high correlation values occur with not negligible probabilities under certain operation conditions. This has an impact on the design of impulse noise suppression techniques as well as on the error rate performance.

We present a new class of sparse and easily invertible circulant matrices that can have a sparse inverse though not being permutation matrices. Their study is useful in the design of quasi-cyclic low-density generator matrix codes that are able to join the inner structure of quasi-cyclic codes with sparse generator matrices, so limiting the number of elementary operations needed for encoding. Circulant matrices of the proposed class permit to hit both targets without resorting to identity or permutation matrices that may penalize the code minimum distance and often cause significant error floors.

This paper is a first study on the usage of non-systematic codes based on scrambling matrices for physical layer security. The chance of implementing transmission security at the physical layer is known since many years, but it is now gaining an increasing interest due to its several possible applications. It has been shown that channel coding techniques can be effectively exploited for designing physical layer security schemes, in such a way that an unauthorized receiver, experiencing a channel different from that of the authorized receiver, is not able to gather any information. Recently, it has been proposed to exploit puncturing techniques in order to reduce the security gap between the authorized and unauthorized channels. In this paper, we show that the security gap can be further reduced by using non-systematic codes, able to scramble information bits within the transmitted codeword.

Computation of the undetected error probability for error detecting codes over the Z-channel is an important issue, explored only in part in previous literature. In this paper, Varshamov-Tenengol'ts (VT) codes are considered. First, an exact formula for the probability of undetected errors is given. It can be explicitly computed for small code lengths (up to approximately 25). Next, some lower bounds that can be explicitly computed up to almost twice this length are studied. A comparison to the Hamming codes is given. It is further shown that heuristic arguments give a very good approximation that can easily be computed even for large lengths. Finally, Monte Carlo methods are used to estimate performance for long code lengths.

In this paper, a new variant of the McEliece cryptosystem, based on quasi-cyclic low-density parity-check (QC-LDPC) codes, is studied. In principle, such codes can substitute Goppa codes, originally used by McEliece; their adoption, however, is subject to cryptanalytic evaluation to ensure sufficient system robustness. The authors conclude that some families of QC-LDPC codes, based on circulant permutation matrices, are inapplicable in this context, due to security issues, whilst other codes, based on the "difference families" approach, can be able to ensure a good level of security against intrusions, even if very large lengths are needed.

Computation of the undetected error probability for error correcting codes over the Z-channel is an important issue, explored only in part in previous literature. In this paper we consider the case of Varshamov-Tenengol'ts codes, by presenting some analytical, numerical, and heuristic methods for unveiling this additional feature.

We adopt a class of quasi-cyclic low-density parity-check codes that allow to overcome the main limitations of the original McEliece cryptosystem based on Goppa codes, that are large key size and low transmission rate. The codes are designed by using a new algorithm based on "random difference families" that permits to construct very large sets of equivalent codes. An extensive cryptanalysis is developed to verify the security level achievable through a selected choice of the system parameters. While previous versions of the McEliece cryptosystem based on LDPC codes are vulnerable to the considered attacks, a new scheme is proposed that ensures satisfactory system robustness with reduced key size and increased transmission rate. Moreover, it is established that the new cryptosystem can be fast enough to justify its adoption as an alternative to widespread solutions, like RSA.

Future space missions will put severe constraints on communication links in terms of data rates, bandwidth occupancy, complexity, and performance. The requirements imposed by the new missions and their consequences on channel code design are presented in the first part of the paper. All relevant issues, including code rates, frame lengths, modulation formats, performance metrics, complexity, and others, are discussed. In the second part of the paper, long erasure correcting codes are presented and their properties explained. These codes operate at the upper layers of the space link protocol and constitute an attractive new frontier for zero packet loss in future space communications.

In this paper, we study a new class of Low Density Parity Check codes, designed for satellite communications, and achieving both high spectral efficiency and large coding gain. Given a constant-envelope 8-PSK constellation, a generic linear binary code does not lead to a geometrically uniform Euclidean-space code. Low Density Parity Check codes over Z_8, the group of integers modulo 8, are considered. When applied to 8-PSK, they generate geometrically uniform Euclidean space codes: distance profile and word error probability are the same for each transmitted codeword. These properties highly simplify both design and analysis. Some insights on code design, encoding, decoding, and simulated performance are provided.

We study product codes based on extended Hamming codes. We focus on their performance at low error rates, which are important for wireless multimedia applications. We present the basis and a complete set of techniques which allows one to analytically evaluate this performance without resorting to extremely long simulations. We present new theoretical results concerning the popular approximation where the bit error rate is nearly equal to the frame error rate times the ratio of the minimum distance to the codeword length. We prove that: 1) binary codes with a transitive automorphism group satisfy this approximation with equality; and 2) extended Hamming product codes belong to this class. Closed-form expressions for their dominant multiplicity values are derived. Analytical curves are plotted, discussed, and validated by comparison with iterative decoding. This analytical approach is then extended to both shortened and punctured codes, which are important for practical design. The first case is solved by applying the extended MacWilliams identity to the dual codes. For punctured codes, we present a new analytical approach for estimating their average performance using a "random" puncturer.

A new encryption system is presented for compressed video signals. It employs the twofish algorithm, recently proposed for standardization purposes, as the core cipher but, through simple modifications, it permits to achieve the very high security levels promised by the standard with reduced overhead and processing time. So the new method is particularly adapted for bandwidth limited applications operating in real time. For the sake of clarity, the proposal is tested on H.263+ coded signals, but it can be easily adapted to other formats, of MPEG-x or H.26x type. Performance evaluation is done off-line, by using a simulator, but also implemented on-line in a practical environment. A comparison is made with the previous systems.

Turbo codes have excellent performance at low and medium signal-to-noise ratios (SNR) very close to the Shannon limit, and are at the basis of their success. However, a turbo code performance curve can change its slope at high SNR if the code free distance is small. This "error floor" phenomenon is not acceptable for applications requiring very low values of bit error rates. A knowledge of the free distance and its multiplicity allows one to analytically estimate the error floor. An algorithm for computing the turbo code free distance, based on the notion of constrained subcodes, is described. Some considerations on the free distance distribution of turbo codes with growing interleaver length are also provided.

A turbo code has been included in the new Consultative Committee for Space Data Systems (CCSDS) channel coding standard for space telemetry. Many future missions with critical link budgets will benefit from its large coding gain. In this paper, the properties of this turbo code are analyzed for symbol synchronization recovery, where the transition density is essential. Key parameters like transition/symbol probability and run-length distribution are studied and compared against practical requirements. Critical working conditions are considered separately and discussed. It is shown that, thanks to the interleaver action, turbo-encoded sequences have very good properties in terms of randomness.