arXiv subjects

Alexander Zeh

Publications and source records attributed to Alexander Zeh.

At least 19 recordsLinked to original sources

arXiv2021

Decoding of (Interleaved) Generalized Goppa Codes

Generalized Goppa codes are defined by a code locator set $\mathcal{L}$ of polynomials and a Goppa polynomial $G(x)$. When the degree of all code locator polynomials in $\mathcal{L}$ is one, generalized Goppa codes are classical Goppa codes. In this work, binary generalized Goppa codes are investigated. First, a parity-check matrix for these codes with code locators of any degree is derived. A careful selection of the code locators leads to a lower bound on the minimum Hamming distance of generalized Goppa codes which improves upon previously known bounds. A quadratic-time decoding algorithm is presented which can decode errors up to half of the minimum distance. Interleaved generalized Goppa codes are introduced and a joint decoding algorithm is presented which can decode errors beyond half the minimum distance with high probability. Finally, some code parameters and how they apply to the Classic McEliece post-quantum cryptosystem are shown.

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arXiv2019

Parity-Based Concurrent Error Detection Schemes for the ChaCha Stream Cipher

We propose two parity-based concurrent error detection schemes for the Quarterround of the ChaCha stream cipher to protect from transient and permanent faults. They offer a trade-off between implementation overhead and error coverage. The second approach can detect any odd-weight error on the in-/output and intermediate signals of a Quarterround, while the first one requires less logic.

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arXiv2018

A Public-Key Cryptosystem from Interleaved Goppa Codes

In this paper, a code-based public-key cryptosystem based on interleaved Goppa codes is presented. The scheme is based on encrypting several ciphertexts with the same Goppa code and adding a burst error to them. Possible attacks are outlined and the key size of several choices of parameters is compared to those of known schemes for the same security level. For example, for security level 128 bits, we obtain a key size of 696 Kbits whereas the classical McEliece scheme based on Goppa codes using list decoding requires a key size of 1935 Kbits.

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arXiv2016

Bounds and Constructions of Codes with Multiple Localities

This paper studies bounds and constructions of locally repairable codes (LRCs) with multiple localities so-called multiple-locality LRCs (ML-LRCs). In the simplest case of two localities some code symbols of an ML-LRC have a certain locality while the remaining code symbols have another one. We extend two bounds, the Singleton and the alphabet-dependent upper bound on the dimension of Cadambe--Mazumdar for LRCs, to the case of ML-LRCs with more than two localities. Furthermore, we construct Singleton-optimal ML-LRCs as well as codes that achieve the extended alphabet-dependent bound. We give a family of binary ML-LRCs based on generalized code concatenation that is optimal with respect to the alphabet-dependent bound.

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arXiv2015

Spectral Analysis of Quasi-Cyclic Product Codes

This paper considers a linear quasi-cyclic product code of two given quasi-cyclic codes of relatively prime lengths over finite fields. We give the spectral analysis of a quasi-cyclic product code in terms of the spectral analysis of the row- and the column-code. Moreover, we provide a new lower bound on the minimum Hamming distance of a given quasi-cyclic code and present a new algebraic decoding algorithm.More specifically, we prove an explicit (unreduced) basis of an l\_a l\_b-quasi-cyclic product code in terms of the generator matrix in reduced Gr{ö}bner basis with respect to the position-over-term order (RGB/POT) form of the l\_a-quasi-cyclic row- and the l\_b-quasi-cyclic column-code, respectively. This generalizes the work of Burton and Weldon for the generator polynomial of a cyclic product code (where l\_a =l\_b=1). Furthermore, we derive the generator matrix in Pre-RGB/POT form of an l\_a l\_b-quasi-cyclic product code for two special cases: (i) for l\_a=2 and l\_b=1, and (ii) if the row-code is a 1-level l\_a-quasi-cyclic code (for arbitrary l\_a) and l\_b=1.For arbitrary l\_a and l\_b, the Pre-RGB/POT form of the generator matrix of an l\_a l\_b-quasi-cyclic product code is conjectured.The spectral analysis is applied to the generator matrix of the product of an l-quasi-cyclic and a cyclic code, and we propose a new lower bound on the minimum Hamming distance of a given l-quasi-cyclic code. In addition, we develop an efficient syndrome-based decoding algorithm for l-phased burst errors with guaranteed decoding radius.

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arXiv2015

Decoding of Repeated-Root Cyclic Codes up to New Bounds on Their Minimum Distance

The well-known approach of Bose, Ray-Chaudhuri and Hocquenghem and its generalization by Hartmann and Tzeng are lower bounds on the minimum distance of simple-root cyclic codes. We generalize these two bounds to the case of repeated-root cyclic codes and present a syndrome-based burst error decoding algorithm with guaranteed decoding radius based on an associated folded cyclic code. Furthermore, we present a third technique for bounding the minimum Hamming distance based on the embedding of a given repeated-root cyclic code into a repeated-root cyclic product code. A second quadratic-time probabilistic burst error decoding procedure based on the third bound is outlined. Index Terms Bound on the minimum distance, burst error, efficient decoding, folded code, repeated-root cyclic code, repeated-root cyclic product code

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arXiv2015

Optimal Linear and Cyclic Locally Repairable Codes over Small Fields

We consider locally repairable codes over small fields and propose constructions of optimal cyclic and linear codes in terms of the dimension for a given distance and length. Four new constructions of optimal linear codes over small fields with locality properties are developed. The first two approaches give binary cyclic codes with locality two. While the first construction has availability one, the second binary code is characterized by multiple available repair sets based on a binary Simplex code. The third approach extends the first one to q-ary cyclic codes including (binary) extension fields, where the locality property is determined by the properties of a shortened first-order Reed-Muller code. Non-cyclic optimal binary linear codes with locality greater than two are obtained by the fourth construction.

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arXiv2015

Optimal Binary Locally Repairable Codes via Anticodes

This paper presents a construction for several families of optimal binary locally repairable codes (LRCs) with small locality (2 and 3). This construction is based on various anticodes. It provides binary LRCs which attain the Cadambe-Mazumdar bound. Moreover, most of these codes are optimal with respect to the Griesmer bound.

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arXiv2015

Construction of Quasi-Cyclic Product Codes

Linear quasi-cyclic product codes over finite fields are investigated. Given the generating set in the form of a reduced Gr{ö}bner basis of a quasi-cyclic component code and the generator polynomial of a second cyclic component code, an explicit expression of the basis of the generating set of the quasi-cyclic product code is given. Furthermore, the reduced Gr{ö}bner basis of a one-level quasi-cyclic product code is derived.

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arXiv2014

List and Unique Error-Erasure Decoding of Interleaved Gabidulin Codes with Interpolation Techniques

A new interpolation-based decoding principle for interleaved Gabidulin codes is presented. The approach consists of two steps: First, a multi-variate linearized polynomial is constructed which interpolates the coefficients of the received word and second, the roots of this polynomial have to be found. Due to the specific structure of the interpolation polynomial, both steps (interpolation and root-finding) can be accomplished by solving a linear system of equations. This decoding principle can be applied as a list decoding algorithm (where the list size is not necessarily bounded polynomially) as well as an efficient probabilistic unique decoding algorithm. For the unique decoder, we show a connection to known unique decoding approaches and give an upper bound on the failure probability. Finally, we generalize our approach to incorporate not only errors, but also row and column erasures.

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arXiv2014

Multi-Trial Guruswami-Sudan Decoding for Generalised Reed--Solomon Codes

An iterated refinement procedure for the Guruswami-Sudan list decoding algorithm for Generalised Reed-Solomon codes based on Alekhnovich's module minimisation is proposed. The method is parametrisable and allows variants of the usual list decoding approach. In particular, finding the list of closest codewords within an intermediate radius can be performed with improved average-case complexity while retaining the worst-case complexity. We provide a detailed description of the module minimisation, reanalysing the Mulders-Storjohann algorithm and drawing new connections to both Alekhnovich's algorithm and Lee-O'Sullivan's. Furthermore, we show how to incorporate the re-encoding technique of Kötter and Vardy into our iterative algorithm.

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arXiv2014

Decoding of Quasi-Cyclic Codes up to A New Lower Bound on the Minimum Distance

A new lower bound on the minimum Hamming distance of linear quasi-cyclic codes over finite fields is proposed. It is based on spectral analysis and generalizes the Semenov- Trifonov bound in a similar way as the Hartmann-Tzeng bound extends the BCH approach for cyclic codes. Furthermore, a syndrome-based algebraic decoding algorithm is given.

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arXiv2013

Generalizing Bounds on the Minimum Distance of Cyclic Codes Using Cyclic Product Codes

Two generalizations of the Hartmann--Tzeng (HT) bound on the minimum distance of q-ary cyclic codes are proposed. The first one is proven by embedding the given cyclic code into a cyclic product code. Furthermore, we show that unique decoding up to this bound is always possible and outline a quadratic-time syndrome-based error decoding algorithm. The second bound is stronger and the proof is more involved. Our technique of embedding the code into a cyclic product code can be applied to other bounds, too and therefore generalizes them.

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arXiv2013

Multi-Trial Guruswami--Sudan Decoding for Generalised Reed--Solomon Codes

An iterated refinement procedure for the Guruswami--Sudan list decoding algorithm for Generalised Reed--Solomon codes based on Alekhnovich's module minimisation is proposed. The method is parametrisable and allows variants of the usual list decoding approach. In particular, finding the list of \emph{closest} codewords within an intermediate radius can be performed with improved average-case complexity while retaining the worst-case complexity.

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arXiv2012

A New Bound on the Minimum Distance of Cyclic Codes Using Small-Minimum-Distance Cyclic Codes

A new bound on the minimum distance of q-ary cyclic codes is proposed. It is based on the description by another cyclic code with small minimum distance. The connection to the BCH bound and the Hartmann--Tzeng (HT) bound is formulated explicitly. We show that for many cases our approach improves the HT bound. Furthermore, we refine our bound for several families of cyclic codes. We define syndromes and formulate a Key Equation that allows an efficient decoding up to our bound with the Extended Euclidean Algorithm. It turns out that lowest-code-rate cyclic codes with small minimum distances are useful for our approach. Therefore, we give a sufficient condition for binary cyclic codes of arbitrary length to have minimum distance two or three and lowest code-rate

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arXiv2012

Describing A Cyclic Code by Another Cyclic Code

A new approach to bound the minimum distance of $q$-ary cyclic codes is presented. The connection to the BCH and the Hartmann--Tzeng bound is formulated and it is shown that for several cases an improvement is achieved. We associate a second cyclic code to the original one and bound its minimum distance in terms of parameters of the associated code.

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arXiv2012

Decoding Cyclic Codes up to a New Bound on the Minimum Distance

A new lower bound on the minimum distance of q-ary cyclic codes is proposed. This bound improves upon the Bose-Chaudhuri-Hocquenghem (BCH) bound and, for some codes, upon the Hartmann-Tzeng (HT) bound. Several Boston bounds are special cases of our bound. For some classes of codes the bound on the minimum distance is refined. Furthermore, a quadratic-time decoding algorithm up to this new bound is developed. The determination of the error locations is based on the Euclidean Algorithm and a modified Chien search. The error evaluation is done by solving a generalization of Forney's formula.

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arXiv2011

An Interpolation Procedure for List Decoding Reed--Solomon codes Based on Generalized Key Equations

The key step of syndrome-based decoding of Reed-Solomon codes up to half the minimum distance is to solve the so-called Key Equation. List decoding algorithms, capable of decoding beyond half the minimum distance, are based on interpolation and factorization of multivariate polynomials. This article provides a link between syndrome-based decoding approaches based on Key Equations and the interpolation-based list decoding algorithms of Guruswami and Sudan for Reed-Solomon codes. The original interpolation conditions of Guruswami and Sudan for Reed-Solomon codes are reformulated in terms of a set of Key Equations. These equations provide a structured homogeneous linear system of equations of Block-Hankel form, that can be solved by an adaption of the Fundamental Iterative Algorithm. For an $(n,k)$ Reed-Solomon code, a multiplicity $s$ and a list size $\listl$, our algorithm has time complexity \ON{\listl s^4n^2}.

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