arXiv subjects

Hannes Bartz

Publications and source records attributed to Hannes Bartz.

At least 19 recordsLinked to original sources

arXiv2026

Scientific Rigor and Human Warmth: Remembering Vladimir Sidorenko (1949-2025)

During the Foundations of Future Communication Systems (FFCS) conference in Braunschweig, a dedicated memorial session was held in honor of Dr. Vladimir (Volodya) Sidorenko (1949-2025). The session, chaired by Minglai Cai, brought together colleagues, collaborators, and former students to commemorate his scientific achievements and his exceptional human qualities. This report summarizes the biographical tribute, the personal recollections shared by speakers, and the broader impact of Volodya's work in coding theory, cryptography, telecommunications, and quantum error correction. Beyond his more than 150 publications and substantial technical contributions, the session highlighted his intellectual rigor, mentorship, humor, generosity, and lasting influence on the international research community.

cs.IT↗

arXiv2025

Fast Decoding of Interleaved Linearized Reed-Solomon Codes and Variants

We construct $s$-interleaved linearized Reed--Solomon (ILRS) codes and variants and propose efficient decoding schemes that can correct errors beyond the unique decoding radius in the sum-rank metric. The proposed interpolation-based scheme for ILRS codes can be used as a list decoder or as a probabilistic unique decoder that corrects errors of sum-rank up to $t\leq\frac{s}{s+1}(n-k)$, where $s$ is the interleaving order, $n$ the length and $k$ the dimension of the code. Upper bounds on the list size and the decoding failure probability are given where the latter is based on a novel Loidreau--Overbeck-like decoder for ILRS codes. We show how the proposed decoding schemes can be used to decode errors beyond the unique decoding radius in the skew metric by using an isometry between the sum-rank metric and the skew metric. We generalize fast minimal approximant basis interpolation techniques to obtain efficient decoding schemes for ILRS codes (and variants) with subquadratic complexity in the code length. Up to our knowledge, the presented decoding schemes are the first being able to correct errors beyond the unique decoding region in the sum-rank and skew metric. The performance of the proposed decoding schemes and the tightness of the upper bound on the decoding failure probability are validated via Monte Carlo simulations.

cs.IT↗

arXiv2025

On the Security of a Code-Based PIR Scheme

Private Information Retrieval (PIR) schemes allow clients to retrieve files from a database without disclosing the requested file's identity to the server. In the pursuit of post-quantum security, most recent PIR schemes rely on hard lattice problems. In contrast, the so called CB-cPIR scheme stands out as a pioneering effort to base PIR schemes on hard problems in coding theory, thereby contributing significantly to the diversification of security foundations. However, our research reveals a critical vulnerability in CB-cPIR, substantially diminishing its security levels. Moreover, a comparative analysis with state-of-the-art PIR schemes shows that CB-cPIR's advantages are reduced, making it less competitive in terms of the communication cost. Nevertheless, our findings highlight the importance of continued research into code-based PIR schemes, as they have the potential to provide a valuable alternative to lattice-based approaches.

cs.CR↗

arXiv2024

Syndrome-Based Error-Erasure Decoding of Interleaved Linearized Reed-Solomon Codes

Linearized Reed--Solomon (LRS) codes are sum-rank-metric codes that generalize both Reed--Solomon and Gabidulin codes. We study vertically and horizontally interleaved LRS (VILRS and HILRS) codes whose codewords consist of a fixed number of stacked or concatenated codewords of a chosen LRS code. Our unified presentation of results for horizontal and vertical interleaving is novel and simplifies the recognition of resembling patterns. This paper's main results are syndrome-based decoders for both VILRS and HILRS codes. We first consider an error-only setting and then present more general error-erasure decoders, which can handle full errors, row erasures, and column erasures simultaneously. Here, an erasure means that parts of the row space or the column space of the error are already known before decoding. We incorporate this knowledge directly into Berlekamp--Massey-like key equations and thus decode all error types jointly. The presented error-only and error-erasure decoders have an average complexity in $O(sn^2)$ and $\widetilde{O}(sn^2)$ in most scenarios, where $s$ is the interleaving order and $n$ denotes the length of the component code. Errors of sum-rank weight $τ=t_{\mathcal{F}}+t_{\mathcal{R}}+t_{\mathcal{C}}$ consist of $t_{\mathcal{F}}$ full errors, $t_{\mathcal{R}}$ row erasures, and $t_{\mathcal{C}}$ column erasures. Their successful decoding can be guaranteed for $t_{\mathcal{F}}\leq\tfrac{1}{2}(n-k-t_{\mathcal{R}}-t_{\mathcal{C}})$, where $n$ and $k$ represent the length and the dimension of the component LRS code. Moreover, probabilistic decoding beyond the unique-decoding radius is possible with high probability when $t_{\mathcal{F}}\leq\tfrac{s}{s+1}(n-k-t_{\mathcal{R}}-t_{\mathcal{C}})$ holds for interleaving order $s$. We give an upper bound on the failure probability for probabilistic unique decoding and showcase its tightness via Monte Carlo simulations.

cs.IT↗

arXiv2024

Support-Guessing Decoding Algorithms in the Sum-Rank Metric

The sum-rank metric generalizes the Hamming and rank metric by partitioning vectors into blocks and defining the total weight as the sum of the rank weights of these blocks, based on their matrix representation. In this work, we explore support-guessing algorithms for decoding sum-rank-metric codes. Support-guessing involves randomly selecting candidate supports and attempting to decode the error under the assumption that it is confined to these supports. While previous works have focused on worst-case scenarios, we analyze the average case and derive an optimal support-guessing distribution in the asymptotic regime. We show that this distribution also performs well for finite code lengths. Our analysis provides exact complexity estimates for unique decoding scenarios and establishes tighter bounds beyond the unique decoding radius. Additionally, we introduce a randomized decoding algorithm for Linearized Reed--Solomon (LRS) codes. This algorithm extends decoding capabilities beyond the unique decoding radius by leveraging an efficient error-and-erasure decoder. Instead of requiring the entire error support to be confined to the guessed support, the algorithm succeeds as long as there is sufficient overlap between the guessed support and the actual error support. As a result, the proposed method improves the success probability and reduces computational complexity compared to generic decoding algorithms. Our contributions offer more accurate complexity estimates than previous works, which are essential for understanding the computational challenges involved in decoding sum-rank-metric codes. This improved complexity analysis, along with optimized support-guessing distributions, provides valuable insights for the design and evaluation of code-based cryptosystems using the sum-rank metric. This is particularly important in the context of quantum-resistant cryptography.

cs.IT↗

arXiv2024

An Error-Code Perspective on Metzner--Kapturowski-like Decoders

In this paper we consider a Metzner-Kapturowski-like decoding algorithm for high-order interleaved sum-rank-metric codes, offering a novel perspective on the decoding process through the concept of an error code. The error code, defined as the linear code spanned by the vectors forming the error matrix, provides a more intuitive understanding of the decoder's functionality and new insights. The proposed algorithm can correct errors of sum-rank weight up to $d-2$, where $d$ is the minimum distance of the constituent code, given a sufficiently large interleaving order. The decoder's versatility is highlighted by its applicability to any linear constituent code, including unstructured or random codes. The computational complexity is $O(\max\{n^3, n^2 s\})$ operations over $\mathbb{F}_{q^m}$, where $n$ is the code length and $s$ is the interleaving order. We further explore the success probability of the decoder for random errors, providing an efficient algorithm to compute an upper bound on this probability. Additionally, we derive bounds and approximations for the success probability when the error weight exceeds the unique decoding radius, showing that the decoder maintains a high success probability in this regime. Our findings suggest that this decoder could be a valuable tool for the design and security analysis of code-based cryptosystems using interleaved sum-rank-metric codes. The new insights into the decoding process and the high success probability of the algorithm even beyond the unique decoding radius underscore its potential to contribute to various coding-related applications.

cs.IT↗

arXiv2024

Error-Correction Performance of Regular Ring-Linear LDPC Codes over Lee Channels

Most low-density parity-check (LDPC) code constructions are considered over finite fields. In this work, we focus on regular LDPC codes over integer residue rings and analyze their performance with respect to the Lee metric. Their error-correction performance is studied over two channel models, in the Lee metric. The first channel model is a discrete memoryless channel, whereas in the second channel model an error vector is drawn uniformly at random from all vectors of a fixed Lee weight. It is known that the two channel laws coincide in the asymptotic regime, meaning that their marginal distributions match. For both channel models, we derive upper bounds on the block error probability in terms of a random coding union bound as well as sphere packing bounds that make use of the marginal distribution of the considered channels. We estimate the decoding error probability of regular LDPC code ensembles over the channels using the marginal distribution and determining the expected Lee weight distribution of a random LDPC code over a finite integer ring. By means of density evolution and finite-length simulations, we estimate the error-correction performance of selected LDPC code ensembles under belief propagation decoding and a low-complexity symbol message passing decoding algorithm and compare the performances. The analysis developed in this paper may serve to design regular LDPC codes over integer residue rings for storage and cryptographic application.

cs.IT↗

arXiv2023

Fast Gao-like Decoding of Horizontally Interleaved Linearized Reed-Solomon Codes

Both horizontal interleaving as well as the sum-rank metric are currently attractive topics in the field of code-based cryptography, as they could mitigate the problem of large key sizes. In contrast to vertical interleaving, where codewords are stacked vertically, each codeword of a horizontally $s$-interleaved code is the horizontal concatenation of $s$ codewords of $s$ component codes. In the case of horizontally interleaved linearized Reed-Solomon (HILRS) codes, these component codes are chosen to be linearized Reed-Solomon (LRS) codes. We provide a Gao-like decoder for HILRS codes that is inspired by the respective works for non-interleaved Reed-Solomon and Gabidulin codes. By applying techniques from the theory of minimal approximant bases, we achieve a complexity of $\tilde{\mathcal{O}}(s^{2.373} n^{1.635})$ operations in $\mathbb{F}_{q^m}$, where $\tilde{\mathcal{O}}(\cdot)$ neglects logarithmic factors, $s$ is the interleaving order and $n$ denotes the length of the component codes. For reasonably small interleaving order $s \ll n$, this is subquadratic in the component-code length $n$ and improves over the only known syndrome-based decoder for HILRS codes with quadratic complexity. Moreover, it closes the performance gap to vertically interleaved LRS codes for which a decoder of complexity $\tilde{\mathcal{O}}(s^{2.373} n^{1.635})$ is already known. We can decode beyond the unique-decoding radius and handle errors of sum-rank weight up to $\frac{s}{s + 1} (n - k)$ for component-code dimension $k$. We also give an upper bound on the failure probability in the zero-derivation setting and validate its tightness via Monte Carlo simulations.

cs.IT↗

arXiv2023

Fast Decoding of Lifted Interleaved Linearized Reed-Solomon Codes for Multishot Network Coding

Mart{\'ı}nez-Pe{ñ}as and Kschischang (IEEE Trans.\ Inf.\ Theory, 2019) proposed lifted linearized Reed--Solomon codes as suitable codes for error control in multishot network coding. We show how to construct and decode \ac{LILRS} codes. Compared to the construction by Mart{\'ı}nez-Pe{ñ}as--Kschischang, interleaving allows to increase the decoding region significantly and decreases the overhead due to the lifting (i.e., increases the code rate), at the cost of an increased packet size. We propose two decoding schemes for \ac{LILRS} that are both capable of correcting insertions and deletions beyond half the minimum distance of the code by either allowing a list or a small decoding failure probability. We propose a probabilistic unique {\LOlike} decoder for \ac{LILRS} codes and an efficient interpolation-based decoding scheme that can be either used as a list decoder (with exponential worst-case list size) or as a probabilistic unique decoder. We derive upper bounds on the decoding failure probability of the probabilistic-unique decoders which show that the decoding failure probability is very small for most channel realizations up to the maximal decoding radius. The tightness of the bounds is verified by Monte Carlo simulations.

cs.IT↗

arXiv2023

Randomized Decoding of Linearized Reed-Solomon Codes Beyond the Unique Decoding Radius

In this paper we address the problem of decoding linearized Reed-Solomon (LRS) codes beyond their unique decoding radius. We analyze the complexity in order to evaluate if the considered problem is of cryptographic relevance, i.e., can be used to design cryptosystems that are computationally hard to break. We show that our proposed algorithm improves over other generic algorithms that do not take into account the underlying code structure.

cs.IT↗

arXiv2023

Distinguishing and Recovering Generalized Linearized Reed-Solomon Codes

We study the distinguishability of linearized Reed-Solomon (LRS) codes by defining and analyzing analogs of the square-code and the Overbeck distinguisher for classical Reed-Solomon and Gabidulin codes, respectively. Our main results show that the square-code distinguisher works for generalized linearized Reed-Solomon (GLRS) codes defined with the trivial automorphism, whereas the Overbeck-type distinguisher can handle LRS codes in the general setting. We further show how to recover defining code parameters from any generator matrix of such codes in the zero-derivation case. For other choices of automorphisms and derivations simulations indicate that these distinguishers and recovery algorithms do not work. The corresponding LRS and GLRS codes might hence be of interest for code-based cryptography.

cs.IT↗

arXiv2023

On Decoding High-Order Interleaved Sum-Rank-Metric Codes

We consider decoding of vertically homogeneous interleaved sum-rank-metric codes with high interleaving order $s$, that are constructed by stacking $s$ codewords of a single constituent code. We propose a Metzner--Kapturowski-like decoding algorithm that can correct errors of sum-rank weight $t <= d-2$, where $d$ is the minimum distance of the code, if the interleaving order $s > t$ and the error matrix fulfills a certain rank condition. The proposed decoding algorithm generalizes the Metzner--Kapturowski(-like) decoders in the Hamming metric and the rank metric and has a computational complexity of $\tilde{O}(\max(n^3, n^2 s))$ operations in $\mathbb{F}_{q^m}$, where $n$ is the length of the code. The scheme performs linear-algebraic operations only and thus works for any interleaved linear sum-rank-metric code. We show how the decoder can be used to decode high-order interleaved codes in the skew metric. Apart from error control, the proposed decoder allows to determine the security level of code-based cryptosystems based on interleaved sum-rank metric codes.

cs.IT↗

arXiv2023

Interpolation-Based Decoding of Folded Variants of Linearized and Skew Reed-Solomon Codes

The sum-rank metric is a hybrid between the Hamming metric and the rank metric and suitable for error correction in multishot network coding and distributed storage as well as for the design of quantum-resistant cryptosystems. In this work, we consider the construction and decoding of folded linearized Reed-Solomon (FLRS) codes, which are shown to be maximum sum-rank distance (MSRD) for appropriate parameter choices. We derive an efficient interpolation-based decoding algorithm for FLRS codes that can be used as a list decoder or as a probabilistic unique decoder. The proposed decoding scheme can correct sum-rank errors beyond the unique decoding radius with a computational complexity that is quadratic in the length of the unfolded code. We show how the error-correction capability can be optimized for high-rate codes by an alternative choice of interpolation points. We derive a heuristic upper bound on the decoding failure probability of the probabilistic unique decoder and verify its tightness by Monte Carlo simulations. Further, we study the construction and decoding of folded skew Reed-Solomon codes in the skew metric. Up to our knowledge, FLRS codes are the first MSRD codes with different block sizes that come along with an efficient decoding algorithm.

cs.IT↗

arXiv2022

Analysis of Low-Density Parity-Check Codes over Finite Integer Rings for the Lee Channel

We study the performance of nonbinary low-density parity-check (LDPC) codes over finite integer rings over two channels that arise from the Lee metric. The first channel is a discrete memory-less channel (DMC) matched to the Lee metric. The second channel adds to each codeword an error vector of constant Lee weight, where the error vector is picked uniformly at random from the set of vectors of constant Lee weight. It is shown that the marginal conditional distributions of the two channels coincide, in the limit of large block length. Random coding union bounds on the block error probability are derived for both channels. Moreover, the performance of selected LDPC code ensembles is analyzed by means of density evolution and finite-length simulations, with belief propagation decoding and with a low-complexity symbol message passing algorithm and it is compared to the derived bounds.

cs.IT↗

arXiv2022

Efficient Decoding of Folded Linearized Reed-Solomon Codes in the Sum-Rank Metric

Recently, codes in the sum-rank metric attracted attention due to several applications in e.g. multishot network coding, distributed storage and quantum-resistant cryptography. The sum-rank analogs of Reed-Solomon and Gabidulin codes are linearized Reed-Solomon codes. We show how to construct $h$-folded linearized Reed-Solomon (FLRS) codes and derive an interpolation-based decoding scheme that is capable of correcting sum-rank errors beyond the unique decoding radius. The presented decoder can be used for either list or probabilistic unique decoding and requires at most $\mathcal{O}(sn^2)$ operations in $\mathbb{F}_{q^m}$, where $s \leq h$ is an interpolation parameter and $n$ denotes the length of the unfolded code. We derive a heuristic upper bound on the failure probability of the probabilistic unique decoder and verify the results via Monte Carlo simulations.

cs.IT↗

arXiv2022

Error-Erasure Decoding of Linearized Reed-Solomon Codes in the Sum-Rank Metric

Codes in the sum-rank metric have various applications in error control for multishot network coding, distributed storage and code-based cryptography. Linearized Reed-Solomon (LRS) codes contain Reed-Solomon and Gabidulin codes as subclasses and fulfill the Singleton-like bound in the sum-rank metric with equality. We propose the first known error-erasure decoder for LRS codes to unleash their full potential for multishot network coding. The presented syndrome-based Berlekamp-Massey-like error-erasure decoder can correct $t_F$ full errors, $t_R$ row erasures and $t_C$ column erasures up to $2t_F + t_R + t_C \leq n-k$ in the sum-rank metric requiring at most $\mathcal{O}(n^2)$ operations in $\mathbb{F}_{q^m}$, where $n$ is the code's length and $k$ its dimension. We show how the proposed decoder can be used to correct errors in the sum-subspace metric that occur in (noncoherent) multishot network coding.

cs.IT↗

arXiv2022

Fast Kötter-Nielsen-Høholdt Interpolation over Skew Polynomial Rings and its Application in Coding Theory

Skew polynomials are a class of non-commutative polynomials that have several applications in computer science, coding theory and cryptography. In particular, skew polynomials can be used to construct and decode evaluation codes in several metrics, like e.g. the Hamming, rank, sum-rank and skew metric. We propose a fast divide-and-conquer variant of Kötter-Nielsen-Høholdt (KNH) interpolation algorithm: it inputs a list of linear functionals on skew polynomial vectors, and outputs a reduced Gröbner basis of their kernel intersection. We show, that the proposed KNH interpolation can be used to solve the interpolation step of interpolation-based decoding of interleaved Gabidulin codes in the rank-metric, linearized Reed-Solomon codes in the sum-rank metric and skew Reed-Solomon codes in the skew metric requiring at most $\tilde{O}(s^ω M(n))$ operations in $\mathbb{F}_{q^m}$ , where $n$ is the length of the code, $s$ the interleaving order, $M(n)$ the complexity for multiplying two skew polynomials of degree at most $n$, $ω$ the matrix multiplication exponent and $\tilde{O}(\cdot)$ the soft-O notation which neglects log factors. This matches the previous best speeds for these tasks, which were obtained by top-down minimal approximant bases techniques, and complements the theory of efficient interpolation over free skew polynomial modules by the bottom-up KNH approach. In contrast to the top-down approach the bottom-up KNH algorithm has no requirements on the interpolation points and thus does not require any pre-processing.

cs.IT↗

arXiv2022

Rank-Metric Codes and Their Applications

The rank metric measures the distance between two matrices by the rank of their difference. Codes designed for the rank metric have attracted considerable attention in recent years, reinforced by network coding and further motivated by a variety of applications. In code-based cryptography, the hardness of the corresponding generic decoding problem can lead to systems with reduced public-key size. In distributed data storage, codes in the rank metric have been used repeatedly to construct codes with locality, and in coded caching, they have been employed for the placement of coded symbols. This survey gives a general introduction to rank-metric codes, explains their most important applications, and highlights their relevance to these areas of research.

cs.IT↗