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Sven Puchinger

Publications and source records attributed to Sven Puchinger.

At least 55 records · Page 3Linked to original sources
arXiv2019

On Error Decoding of Locally Repairable and Partial MDS Codes

We consider error decoding of locally repairable codes (LRC) and partial MDS (PMDS) codes through interleaved decoding. For a specific class of LRCs we investigate the success probability of interleaved decoding. For PMDS codes we show that there is a wide range of parameters for which interleaved decoding can increase their decoding radius beyond the minimum distance with the probability of successful decoding approaching $1$, when the code length goes to infinity.

cs.IT↗
arXiv2019

Reed-Solomon Codes over Fields of Characteristic Zero

We study Reed--Solomon codes over arbitrary fields, inspired by several recent papers dealing with Gabidulin codes over fields of characteristic zero. Over the field of rational numbers, we derive bounds on the coefficient growth during encoding and the bit complexity of decoding, which is polynomial in the code length and in the bit width of error and codeword values. The results can be generalized to arbitrary number fields.

cs.IT↗
arXiv2019

Invariants and Inequivalence of Linear Rank-Metric Codes

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. These invariants give rise to an easily computable criterion to check if two codes are inequivalent. With this criterion we then derive bounds on the number of equivalence classes of classical and twisted Gabidulin codes.

cs.IT↗
arXiv2019

Decoding High-Order Interleaved Rank-Metric Codes

This paper presents an algorithm for decoding homogeneous interleaved codes of high interleaving order in the rank metric. The new decoder is an adaption of the Hamming-metric decoder by Metzner and Kapturowski (1990) and guarantees to correct all rank errors of weight up to $d-2$ whose rank over the large base field of the code equals the number of errors, where $d$ is the minimum rank distance of the underlying code. In contrast to previously-known decoding algorithms, the new decoder works for any rank-metric code, not only Gabidulin codes. It is purely based on linear-algebraic computations, and has an explicit and easy-to-handle success condition. Furthermore, a lower bound on the decoding success probability for random errors of a given weight is derived. The relation of the new algorithm to existing interleaved decoders in the special case of Gabidulin codes is given.

cs.IT↗
arXiv2018

Constructing an LDPC Code Containing a Given Vector

The coding problem considered in this work is to construct a linear code $\mathcal{C}$ of given length $n$ and dimension $k<n$ such that a given binary vector $\mathbf{r} \in \mathbb{F}^{n}$ is contained in the code. We study a recent solution of this problem by Müelich and Bossert, which is based on LDPC codes. We address two open questions of this construction. First, we show that under certain assumptions, this code construction is possible with high probability if $\mathbf{r}$ is chosen uniformly at random. Second, we calculate the uncertainty of $\mathbf{r}$ given the constructed code $\mathcal{C}$. We present an application of this problem in the field of Physical Unclonable Functions (PUFs).

cs.IT↗
arXiv2018

Twisted Gabidulin Codes in the GPT Cryptosystem

In this paper, we investigate twisted Gabidulin codes in the GPT code-based public-key cryptosystem. We show that Overbeck's attack is not feasible for a subfamily of twisted Gabidulin codes. The resulting key sizes are significantly lower than in the original McEliece system and also slightly smaller than in Loidreau's unbroken GPT variant.

cs.IT↗
arXiv2018

Structural Properties of Twisted Reed-Solomon Codes with Applications to Cryptography

We present a generalisation of Twisted Reed-Solomon codes containing a new large class of MDS codes. We prove that the code class contains a large subfamily that is closed under duality. Furthermore, we study the Schur squares of the new codes and show that their dimension is often large. Using these structural properties, we single out a subfamily of the new codes which could be considered for code-based cryptography: These codes resist some existing structural attacks for Reed-Solomon-like codes, i.e. methods for retrieving the code parameters from an obfuscated generator matrix.

cs.IT↗
arXiv2018

Repairing the Faure-Loidreau Public-Key Cryptosystem

A repair of the Faure-Loidreau (FL) public-key code-based cryptosystem is proposed. The FL cryptosystem is based on the hardness of list decoding Gabidulin codes which are special rank-metric codes. We prove that the recent structural attack on the system by Gaborit et al. is equivalent to decoding an interleaved Gabidulin code. Since all known polynomial-time decoders for these codes fail for a large constructive class of error patterns, we are able to construct public keys that resist the attack. It is also shown that all other known attacks fail for our repair and parameter choices. Compared to other code-based cryptosystems, we obtain significantly smaller key sizes for the same security level.

cs.CR↗
arXiv2018

Improved Power Decoding of Interleaved One-Point Hermitian Codes

We propose a new partial decoding algorithm for $h$-interleaved one-point Hermitian codes that can decode-under certain assumptions-an error of relative weight up to $1-(\tfrac{k+g}{n})^{\frac{h}{h+1}}$, where $k$ is the dimension, $n$ the length, and $g$ the genus of the code. Simulation results for various parameters indicate that the new decoder achieves this maximal decoding radius with high probability. The algorithm is based on a recent generalization of Rosenkilde's improved power decoder to interleaved Reed-Solomon codes, does not require an expensive root-finding step, and improves upon the previous best decoding radius by Kampf at all rates. In the special case $h=1$, we obtain an adaption of the improved power decoding algorithm to one-point Hermitian codes, which for all simulated parameters achieves a similar observed failure probability as the Guruswami-Sudan decoder above the latter's guaranteed decoding radius.

cs.IT↗
arXiv2018

Using Convolutional Codes for Key Extraction in SRAM Physical Unclonable Functions

Physical Unclonable Functions (PUFs) exploit variations in the manufacturing process to derive bit sequences from integrated circuits, which can be used as secure cryptographic keys. Instead of storing the keys in an insecure, non-volatile memory, they can be reproduced when needed. Since the reproduced sequences are not stable due to physical reasons, error correction must be applied. Recently, convolutional codes were shown to be suitable for key reproduction in PUFs based on SRAM. This work shows how to further decrease the reconstruction failure probability and PUF implementation size using codes with larger memory length and decoding concepts such as soft-information and list decoding.

cs.IT↗
arXiv2017

Further Generalisations of Twisted Gabidulin Codes

We present a new family of maximum rank distance (MRD) codes. The new class contains codes that are neither equivalent to a generalised Gabidulin nor to a twisted Gabidulin code, the only two known general constructions of linear MRD codes.

cs.IT↗
arXiv2017

Fast Operations on Linearized Polynomials and their Applications in Coding Theory

This paper considers fast algorithms for operations on linearized polynomials. We propose a new multiplication algorithm for skew polynomials (a generalization of linearized polynomials) which has sub-quadratic complexity in the polynomial degree $s$, independent of the underlying field extension degree~$m$. We show that our multiplication algorithm is faster than all known ones when $s \leq m$. Using a result by Caruso and Le Borgne (2017), this immediately implies a sub-quadratic division algorithm for linearized polynomials for arbitrary polynomial degree $s$. Also, we propose algorithms with sub-quadratic complexity for the $q$-transform, multi-point evaluation, computing minimal subspace polynomials, and interpolation, whose implementations were at least quadratic before. Using the new fast algorithm for the $q$-transform, we show how matrix multiplication over a finite field can be implemented by multiplying linearized polynomials of degrees at most $s=m$ if an elliptic normal basis of extension degree $m$ exists, providing a lower bound on the cost of the latter problem. Finally, it is shown how the new fast operations on linearized polynomials lead to the first error and erasure decoding algorithm for Gabidulin codes with sub-quadratic complexity.

cs.SC↗
arXiv2017

Space-Time Codes Based on Rank-Metric Codes and Their Decoding

We propose a new class of space-time block codes based on finite-field rank-metric codes in combination with a rank-metric-preserving mapping to the set of Eisenstein integers. It is shown that these codes achieve maximum diversity order and improve upon certain existing constructions. Moreover, we present a new decoding algorithm for these codes which utilizes the algebraic structure of the underlying finite-field rank-metric codes and employs lattice-reduction-aided equalization. This decoder does not achieve the same performance as the classical maximum-likelihood decoding methods, but has polynomial complexity in the matrix dimension, making it usable for large field sizes and numbers of antennas.

cs.IT↗
arXiv2017

Decoding of Interleaved Reed-Solomon Codes Using Improved Power Decoding

We propose a new partial decoding algorithm for $m$-interleaved Reed--Solomon (IRS) codes that can decode, with high probability, a random error of relative weight $1-R^{\frac{m}{m+1}}$ at all code rates $R$, in time polynomial in the code length $n$. For $m>2$, this is an asymptotic improvement over the previous state-of-the-art for all rates, and the first improvement for $R>1/3$ in the last $20$ years. The method combines collaborative decoding of IRS codes with power decoding up to the Johnson radius.

cs.IT↗
arXiv2017

Twisted Reed-Solomon Codes

We present a new general construction of MDS codes over a finite field $\mathbb{F}_q$. We describe two explicit subclasses which contain new MDS codes of length at least $q/2$ for all values of $q \ge 11$. Moreover, we show that most of the new codes are not equivalent to a Reed-Solomon code.

cs.IT↗
arXiv2017

Improved Power Decoding of One-Point Hermitian Codes

We propose a new partial decoding algorithm for one-point Hermitian codes that can decode up to the same number of errors as the Guruswami--Sudan decoder. Simulations suggest that it has a similar failure probability as the latter one. The algorithm is based on a recent generalization of the power decoding algorithm for Reed--Solomon codes and does not require an expensive root-finding step. In addition, it promises improvements for decoding interleaved Hermitian codes.

cs.IT↗
arXiv2017

Multi-Block Interleaved Codes for Local and Global Read Access

We define multi-block interleaved codes as codes that allow reading information from either a small sub-block or from a larger full block. The former offers faster access, while the latter provides better reliability. We specify the correction capability of the sub-block code through its gap $t$ from optimal minimum distance, and look to have full-block minimum distance that grows with the parameter $t$. We construct two families of such codes when the number of sub-blocks is $3$. The codes match the distance properties of known integrated-interleaving codes, but with the added feature of mapping the same number of information symbols to each sub-block. As such, they are the first codes that provide read access in multiple size granularities and correction capabilities.

cs.IT↗