arXiv subjects

Sven Puchinger

Publications and source records attributed to Sven Puchinger.

At least 37 records · Page 2Linked to original sources

arXiv2021

Bounds and Code Constructions for Partially Defect Memory Cells

This paper considers coding for so-called partially stuck memory cells. Such memory cells can only store partial information as some of their levels cannot be used due to, e.g., wear out. First, we present a new code construction for masking such partially stuck cells while additionally correcting errors. This construction (for cells with $q >2$ levels) is achieved by generalizing an existing masking-only construction in [1] (based on binary codes) to correct errors as well. Compared to previous constructions in [2], our new construction achieves larger rates for many sets of parameters. Second, we derive a sphere-packing (any number of $u$ partially stuck cells) and a Gilbert-Varshamov bound ($u<q$ partially stuck cells) for codes that can mask a certain number of partially stuck cells and correct errors additionally. A numerical comparison between the new bounds and our previous construction of PSMCs for the case $u<q$ in [2] shows that our construction lies above the Gilbert-Varshamov-like bound for several code parameters.

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arXiv2021

On the Gap between Scalar and Vector Solutions of Generalized Combination Networks

We study scalar-linear and vector-linear solutions of the generalized combination network. We derive new upper and lower bounds on the maximum number of nodes in the middle layer, depending on the network parameters and the alphabet size. These bounds improve and extend the parameter range of known bounds. Using these new bounds we present a lower bound and an upper bound on the gap in the alphabet size between optimal scalar-linear and optimal vector-linear network coding solutions. For a fixed network structure, while varying the number of middle-layer nodes $r$, the asymptotic behavior of the upper and lower bounds shows that the gap is in $Θ(\log(r))$.

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arXiv2021

Fast Decoding of Codes in the Rank, Subspace, and Sum-Rank Metric

We speed up existing decoding algorithms for three code classes in different metrics: interleaved Gabidulin codes in the rank metric, lifted interleaved Gabidulin codes in the subspace metric, and linearized Reed-Solomon codes in the sum-rank metric. The speed-ups are achieved by new algorithms that reduce the cores of the underlying computational problems of the decoders to one common tool: computing left and right approximant bases of matrices over skew polynomial rings. To accomplish this, we describe a skew-analogue of the existing PM-Basis algorithm for matrices over ordinary polynomials. This captures the bulk of the work in multiplication of skew polynomials, and the complexity benefit comes from existing algorithms performing this faster than in classical quadratic complexity. The new algorithms for the various decoding-related computational problems are interesting in their own and have further applications, in particular parts of decoders of several other codes and foundational problems related to the remainder-evaluation of skew polynomials.

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arXiv2021

Correctable Erasure Patterns in Product Topologies

Locality enables storage systems to recover failed nodes from small subsets of surviving nodes. The setting where nodes are partitioned into subsets, each allowing for local recovery, is well understood. In this work we consider a generalization introduced by Gopalan et al., where, viewing the codewords as arrays, constraints are imposed on the columns and rows in addition to some global constraints. Specifically, we present a generic method of adding such global parity-checks and derive new results on the set of correctable erasure patterns. Finally, we relate the set of correctable erasure patterns in the considered topology to those correctable in tensor-product codes.

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arXiv2021

Bounds on List Decoding of Linearized Reed-Solomon Codes

Linearized Reed-Solomon (LRS) codes are sum-rank metric codes that fulfill the Singleton bound with equality. In the two extreme cases of the sum-rank metric, they coincide with Reed-Solomon codes (Hamming metric) and Gabidulin codes (rank metric). List decoding in these extreme cases is well-studied, and the two code classes behave very differently in terms of list size, but nothing is known for the general case. In this paper, we derive a lower bound on the list size for LRS codes, which is, for a large class of LRS codes, exponential directly above the Johnson radius. Furthermore, we show that some families of linearized Reed-Solomon codes with constant numbers of blocks cannot be list decoded beyond the unique decoding radius.

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arXiv2021

Efficient Decoding of Gabidulin Codes over Galois Rings

This paper presents the first decoding algorithm for Gabidulin codes over Galois rings with provable quadratic complexity. The new method consists of two steps: (1) solving a syndrome-based key equation to obtain the annihilator polynomial of the error and therefore the column space of the error, (2) solving a key equation based on the received word in order to reconstruct the error vector. This two-step approach became necessary since standard solutions as the Euclidean algorithm do not properly work over rings.

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arXiv2020

Low-Rank Parity-Check Codes over Galois Rings

Low-rank parity-check (LRPC) are rank-metric codes over finite fields, which have been proposed by Gaborit et al. (2013) for cryptographic applications. Inspired by a recent adaption of Gabidulin codes to certain finite rings by Kamche et al. (2019), we define and study LRPC codes over Galois rings - a wide class of finite commutative rings. We give a decoding algorithm similar to Gaborit et al.'s decoder, based on simple linear-algebraic operations. We derive an upper bound on the failure probability of the decoder, which is significantly more involved than in the case of finite fields. The bound depends only on the rank of an error, i.e., is independent of its free rank. Further, we analyze the complexity of the decoder. We obtain that there is a class of LRPC codes over a Galois ring that can decode roughly the same number of errors as a Gabidulin code with the same code parameters, but faster than the currently best decoder for Gabidulin codes. However, the price that one needs to pay is a small failure probability, which we can bound from above.

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arXiv2020

Equivalence and Characterizations of Linear Rank-Metric Codes Based on Invariants

We show that the sequence of dimensions of the linear spaces, generated by a given rank-metric code together with itself under several applications of a field automorphism, is an invariant for the whole equivalence class of the code. The same property is proven for the sequence of dimensions of the intersections of itself under several applications of a field automorphism. These invariants give rise to easily computable criteria to check if two codes are inequivalent. We derive some concrete values and bounds for these dimension sequences for some known families of rank-metric codes, namely Gabidulin and (generalized) twisted Gabidulin codes. We then derive conditions on the length of the codes with respect to the field extension degree, such that codes from different families cannot be equivalent. Furthermore, we derive upper and lower bounds on the number of equivalence classes of Gabidulin codes and twisted Gabidulin codes, improving a result of Schmidt and Zhou for a wider range of parameters. In the end we use the aforementioned sequences to determine a characterization result for Gabidulin codes.

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arXiv2020

Error Decoding of Locally Repairable and Partial MDS Codes

In this work it is shown that locally repairable codes (LRCs) can be list-decoded efficiently beyond the Johnson radius for a large range of parameters by utilizing the local error-correction capabilities. The corresponding decoding radius is derived and the asymptotic behavior is analyzed. A general list-decoding algorithm for LRCs that achieves this radius is proposed along with an explicit realization for LRCs that are subcodes of Reed--Solomon codes (such as, e.g., Tamo--Barg LRCs). Further, a probabilistic algorithm of low complexity for unique decoding of LRCs is given and its success probability is analyzed. The second part of this work considers error decoding of LRCs and partial maximum distance separable (PMDS) codes through interleaved decoding. For a specific class of LRCs the success probability of interleaved decoding is investigated. For PMDS codes, it is shown that there is a wide range of parameters for which interleaved decoding can increase their decoding radius beyond the minimum distance such that the probability of successful decoding approaches $1$ when the code length goes to infinity.

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arXiv2020

LIGA: A Cryptosystem Based on the Hardness of Rank-Metric List and Interleaved Decoding

We propose the new rank-metric code-based cryptosystem LIGA which is based on the hardness of list decoding and interleaved decoding of Gabidulin codes. LIGA is an improved variant of the Faure-Loidreau (FL) system, which was broken in a structural attack by Gaborit, Otmani, and Talé Kalachi (GOT, 2018). We keep the FL encryption and decryption algorithms, but modify the insecure key generation algorithm. Our crucial observation is that the GOT attack is equivalent to decoding an interleaved Gabidulin code. The new key generation algorithm constructs public keys for which all polynomial-time interleaved decoders fail---hence LIGA resists the GOT attack. We also prove that the public-key encryption version of LIGA is IND-CPA secure in the standard model and the KEM version is IND-CCA2 secure in the random oracle model, both under hardness assumptions of formally defined problems related to list decoding and interleaved decoding of Gabidulin codes. We propose and analyze various exponential-time attacks on these problems, calculate their work factors, and compare the resulting parameters to NIST proposals. The strengths of LIGA are short ciphertext sizes and (relatively) small key sizes. Further, LIGA guarantees correct decryption and has no decryption failure rate. It is not based on hiding the structure of a code. Since there are efficient and constant-time algorithms for encoding and decoding Gabidulin codes, timing attacks on the encryption and decryption algorithms can be easily prevented.

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arXiv2020

Low-Rank Parity-Check Codes over the Ring of Integers Modulo a Prime Power

We define and analyze low-rank parity-check (LRPC) codes over extension rings of the finite chain ring $\mathbb{Z}_{p^r}$, where $p$ is a prime and $r$ is a positive integer. LRPC codes have originally been proposed by Gaborit et al.(2013) over finite fields for cryptographic applications. The adaption to finite rings is inspired by a recent paper by Kamche et al. (2019), which constructed Gabidulin codes over finite principle ideal rings with applications to space-time codes and network coding. We give a decoding algorithm based on simple linear-algebraic operations. Further, we derive an upper bound on the failure probability of the decoder. The upper bound is valid for errors whose rank is equal to the free rank.

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arXiv2020

On the Gap between Scalar and Vector Solutions of Generalized Combination Networks

We study scalar-linear and vector-linear solutions to the generalized combination network. We derive new upper and lower bounds on the maximum number of nodes in the middle layer, depending on the network parameters. These bounds improve and extend the parameter range of known bounds. Using these new bounds we present a general lower bound on the gap in the alphabet size between scalar-linear and vector-linear solutions.

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arXiv2020

Partial MDS Codes with Local Regeneration

Partial MDS (PMDS) and sector-disk (SD) codes are classes of erasure codes that combine locality with strong erasure correction capabilities. We construct PMDS and SD codes where each local code is a bandwidth-optimal regenerating MDS code. The constructions require significantly smaller field size than the only other construction known in literature.

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arXiv2020

Achievable Rates of Concatenated Codes in DNA Storage under Substitution Errors

In this paper, we study achievable rates of concatenated coding schemes over a deoxyribonucleic acid (DNA) storage channel. Our channel model incorporates the main features of DNA-based data storage. First, information is stored on many, short DNA strands. Second, the strands are stored in an unordered fashion inside the storage medium and each strand is replicated many times. Third, the data is accessed in an uncontrollable manner, i.e., random strands are drawn from the medium and received, possibly with errors. As one of our results, we show that there is a significant gap between the channel capacity and the achievable rate of a standard concatenated code in which one strand corresponds to an inner block. This is in fact surprising as for other channels, such as $q$-ary symmetric channels, concatenated codes are known to achieve the capacity. We further propose a modified concatenated coding scheme by combining several strands into one inner block, which allows to narrow the gap and achieve rates that are close to the capacity.

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arXiv2020

Randomized Decoding of Gabidulin Codes Beyond the Unique Decoding Radius

We address the problem of decoding Gabidulin codes beyond their unique error-correction radius. The complexity of this problem is of importance to assess the security of some rank-metric code-based cryptosystems. We propose an approach that introduces row or column erasures to decrease the rank of the error in order to use any proper polynomial-time Gabidulin code error-erasure decoding algorithm. This approach improves on generic rank-metric decoders by an exponential factor.

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arXiv2019

Error Correction for Partially Stuck Memory Cells

We present code constructions for masking $u$ partially stuck memory cells with $q$ levels and correcting additional random errors. The results are achieved by combining the methods for masking and error correction for stuck cells in [1] with the masking-only results for partially stuck cells in [2]. We present two constructions for masking $u 1$ and $q>2$, all new constructions require less redundancy for masking partially stuck cells than previous work on stuck cells, which in turn can result in higher code rates at the same masking and error correction capability.

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arXiv2019

On Decoding and Applications of Interleaved Goppa Codes

Goppa Codes are a well-known class of codes with, among others, applications in code-based cryptography. In this paper, we present a collaborative decoding algorithm for interleaved Goppa codes (IGC). Collaborative decoding increases the decoding radius beyond half of the designed minimum distance. We consider wild Goppa codes and show that we can collaboratively correct more errors for binary Goppa codes than the Patterson decoder. We propose a modified version of the McEliece cryptosystem using wild IGC based on a recently proposed system by Elleuch et al., analyze attacks on the system and present some parameters with the corresponding key sizes.

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arXiv2019

Interleaving Loidreau's Rank-Metric Cryptosystem

We propose and analyze an interleaved variant of Loidreau's rank-metric cryptosystem based on rank multipliers. We analyze and adapt several attacks on the system, propose design rules, and study weak keys. Finding secure instances requires near-MRD rank-metric codes which are not investigated in the literature. Thus, we propose a random code construction that makes use of the fact that short random codes over large fields are MRD with high probability. We derive an upper bound on the decryption failure rate and give example parameters for potential key size reduction.

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