arXiv subjects

Martin Bossert

Publications and source records attributed to Martin Bossert.

At least 37 records · Page 2Linked to original sources

arXiv2016

An Alternative Decoding Method for Gabidulin Codes in Characteristic Zero

Gabidulin codes, originally defined over finite fields, are an important class of rank metric codes with various applications. Recently, their definition was generalized to certain fields of characteristic zero and a Welch--Berlekamp like algorithm with complexity $O(n^3)$ was given. We propose a new application of Gabidulin codes over infinite fields: low-rank matrix recovery. Also, an alternative decoding approach is presented based on a Gao type key equation, reducing the complexity to at least $O(n^2)$. This method immediately connects the decoding problem to well-studied problems, which have been investigated in terms of coefficient growth and numerical stability.

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arXiv2015

Coherence Optimization and Best Complex Antipodal Spherical Codes

Vector sets with optimal coherence according to the Welch bound cannot exist for all pairs of dimension and cardinality. If such an optimal vector set exists, it is an equiangular tight frame and represents the solution to a Grassmannian line packing problem. Best Complex Antipodal Spherical Codes (BCASCs) are the best vector sets with respect to the coherence. By extending methods used to find best spherical codes in the real-valued Euclidean space, the proposed approach aims to find BCASCs, and thereby, a complex-valued vector set with minimal coherence. There are many applications demanding vector sets with low coherence. Examples are not limited to several techniques in wireless communication or to the field of compressed sensing. Within this contribution, existing analytical and numerical approaches for coherence optimization of complex-valued vector spaces are summarized and compared to the proposed approach. The numerically obtained coherence values improve previously reported results. The drawback of increased computational effort is addressed and a faster approximation is proposed which may be an alternative for time critical cases.

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arXiv2015

Code-Based Cryptosystems Using Generalized Concatenated Codes

The security of public-key cryptosystems is mostly based on number theoretic problems like factorization and the discrete logarithm. There exists an algorithm which solves these problems in polynomial time using a quantum computer. Hence, these cryptosystems will be broken as soon as quantum computers emerge. Code-based cryptography is an alternative which resists quantum computers since its security is based on an NP-complete problem, namely decoding of random linear codes. The McEliece cryptosystem is the most prominent scheme to realize code-based cryptography. Many codeclasses were proposed for the McEliece cryptosystem, but most of them are broken by now. Sendrier suggested to use ordinary concatenated codes, however, he also presented an attack on such codes. This work investigates generalized concatenated codes to be used in the McEliece cryptosystem. We examine the application of Sendrier's attack on generalized concatenated codes and present alternative methods for both partly finding the code structure and recovering the plaintext from a cryptogram. Further, we discuss modifications of the cryptosystem making it resistant against these attacks.

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arXiv2015

Error Correction for Differential Linear Network Coding in Slowly-Varying Networks

Differential linear network coding (DLNC) is a precoding scheme for information transmission over random linear networks. By using differential encoding and decoding, the conventional approach of lifting, required for inherent channel sounding, can be omitted and in turn higher transmission rates are supported. However, the scheme is sensitive to variations in the network topology. In this paper, we derive an extended DLNC channel model which includes slow network changes. Based on this, we propose and analyze a suitable channel coding scheme matched to the situation at hand using rank-metric convolutional codes.

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arXiv2015

On Error Correction for Physical Unclonable Functions

Physical Unclonable Functions evaluate manufacturing variations to generate secure cryptographic keys for embedded systems without secure key storage. It is explained how methods from coding theory are applied in order to ensure reliable key reproduction. We show how better results can be obtained using code classes and decoding principles not used for this scenario before. These methods are exemplified by specific code constructions which improve existing codes with respect to error probability, decoding complexity and codeword length.

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arXiv2014

Error Correction for Physical Unclonable Functions Using Generalized Concatenated Codes

Physical Unclonable Functions can be used for secure key generation in cryptographic applications. It is explained how methods from coding theory must be applied in order to ensure reliable key regeneration. Based on previous work, we show ways how to obtain better results with respect to error probability and codeword length. Also, an example based on Generalized Concatenated codes is given, which improves upon used coding schemes for PUFs.

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arXiv2014

Reduced List-Decoding of Reed--Solomon Codes Using Reliability Information

We decode Reed-Solomon codes using soft information provided at the receiver. The Extended Euclidean Algorithm (EEA) is considered as an initial step to obtain an intermediate result. The final decoding result is obtained by interpolating the output of the EEA at the least reliable positions of the received word. We refer to this decoding method as reduced list-decoding, since not all received positions are used in the interpolation as in other list-decoding methods, such as the Guruswami-Sudan and Wu algorithms. Consequently the complexity of the interpolation step is reduced considerably. The probability of failure can be minimised by adjusting certain parameters, making it comparable with the Kötter-Vardy algorithm but having a much lower complexity.

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arXiv2013

Dictionary Adaptation in Sparse Recovery Based on Different Types of Coherence

In sparse recovery, the unique sparsest solution to an under-determined system of linear equations is of main interest. This scheme is commonly proposed to be applied to signal acquisition. In most cases, the signals are not sparse themselves, and therefore, they need to be sparsely represented with the help of a so-called dictionary being specific to the corresponding signal family. The dictionaries cannot be used for optimization of the resulting under-determined system because they are fixed by the given signal family. However, the measurement matrix is available for optimization and can be adapted to the dictionary. Multiple properties of the resulting linear system have been proposed which can be used as objective functions for optimization. This paper discusses two of them which are both related to the coherence of vectors. One property aims for having incoherent measurements, while the other aims for insuring the successful reconstruction. In the following, the influences of both criteria are compared with different reconstruction approaches.

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arXiv2013

Harmonic Analysis of Boolean Networks: Determinative Power and Perturbations

Consider a large Boolean network with a feed forward structure. Given a probability distribution on the inputs, can one find, possibly small, collections of input nodes that determine the states of most other nodes in the network? To answer this question, a notion that quantifies the determinative power of an input over the states of the nodes in the network is needed. We argue that the mutual information (MI) between a given subset of the inputs X = {X_1, ..., X_n} of some node i and its associated function f_i(X) quantifies the determinative power of this set of inputs over node i. We compare the determinative power of a set of inputs to the sensitivity to perturbations to these inputs, and find that, maybe surprisingly, an input that has large sensitivity to perturbations does not necessarily have large determinative power. However, for unate functions, which play an important role in genetic regulatory networks, we find a direct relation between MI and sensitivity to perturbations. As an application of our results, we analyze the large-scale regulatory network of Escherichia coli. We identify the most determinative nodes and show that a small subset of those reduces the overall uncertainty of the network state significantly. Furthermore, the network is found to be tolerant to perturbations of its inputs.

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arXiv2013

Computing preimages of Boolean Networks

In this paper we present an algorithm to address the predecessor problem of feed-forward Boolean networks. We propose an probabilistic algorithm, which solves this problem in linear time with respect to the number of nodes in the network. Finally, we evaluate our algorithm for random Boolean networks and the regulatory network of Escherichia coli.

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arXiv2012

Canalizing Boolean Functions Maximize the Mutual Information

The ability of information processing in biologically motivated Boolean networks is of interest in recent information theoretic research. One measure to quantify this ability is the well known mutual information. Using Fourier analysis we show that canalizing functions maximize the mutual information between an input variable and the outcome of the function. We proof our result for Boolean functions with uniform distributed as well as product distributed input variables.

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arXiv2012

Efficient Decoding of Partial Unit Memory Codes of Arbitrary Rate

Partial Unit Memory (PUM) codes are a special class of convolutional codes, which are often constructed by means of block codes. Decoding of PUM codes may take advantage of existing decoders for the block code. The Dettmar--Sorger algorithm is an efficient decoding algorithm for PUM codes, but allows only low code rates. The same restriction holds for several known PUM code constructions. In this paper, an arbitrary-rate construction, the analysis of its distance parameters and a generalized decoding algorithm for PUM codes of arbitrary rate are provided. The correctness of the algorithm is proven and it is shown that its complexity is cubic in the length.

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arXiv2012

Some Comments on the Strong Simplex Conjecture

In the disproof of the Strong Simplex Conjecture presented in [Steiner, 1994], a counterexample signal set was found that has higher average probability of correct optimal decoding than the corresponding regular simplex signal set, when compared at small values of the signal-to-noise ratio. The latter was defined as the quotient of average signal energy and average noise power. In this paper, it is shown that this interpretation of the signal-to-noise ratio is inappropriate for a comparison of signal sets, since it leads to a contradiction with the Channel Coding Theorem. A modified counterexample signal set is proposed and examined using the classical interpretation of the signal-to-noise ratio, i.e., as the quotient of average signal energy and average noise energy. This signal set outperforms the regular simplex signal set for small signal-to-noise ratios without contradicting the Channel Coding Theorem, hence the Strong Simplex Conjecture remains proven false.

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arXiv2011

Optimal Threshold-Based Multi-Trial Error/Erasure Decoding with the Guruswami-Sudan Algorithm

Traditionally, multi-trial error/erasure decoding of Reed-Solomon (RS) codes is based on Bounded Minimum Distance (BMD) decoders with an erasure option. Such decoders have error/erasure tradeoff factor L=2, which means that an error is twice as expensive as an erasure in terms of the code's minimum distance. The Guruswami-Sudan (GS) list decoder can be considered as state of the art in algebraic decoding of RS codes. Besides an erasure option, it allows to adjust L to values in the range 1 =1 times. We show that BMD decoders with z_BMD decoding trials can result in lower residual codeword error probability than GS decoders with z_GS trials, if z_BMD is only slightly larger than z_GS. This is of practical interest since BMD decoders generally have lower computational complexity than GS decoders.

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arXiv2011

Adaptive Single-Trial Error/Erasure Decoding of Reed-Solomon Codes

Algebraic decoding algorithms are commonly applied for the decoding of Reed-Solomon codes. Their main advantages are low computational complexity and predictable decoding capabilities. Many algorithms can be extended for correction of both errors and erasures. This enables the decoder to exploit binary quantized reliability information obtained from the transmission channel: Received symbols with high reliability are forwarded to the decoding algorithm while symbols with low reliability are erased. In this paper we investigate adaptive single-trial error/erasure decoding of Reed-Solomon codes, i.e. we derive an adaptive erasing strategy which minimizes the residual codeword error probability after decoding. Our result is applicable to any error/erasure decoding algorithm as long as its decoding capabilities can be expressed by a decoder capability function. Examples are Bounded Minimum Distance decoding with the Berlekamp-Massey- or the Sugiyama algorithms and the Guruswami-Sudan list decoder.

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arXiv2011

On (Partial) Unit Memory Codes Based on Gabidulin Codes

(Partial) Unit Memory ((P)UM) codes provide a powerful possibility to construct convolutional codes based on block codes in order to achieve a high decoding performance. In this contribution, a construction based on Gabidulin codes is considered. This construction requires a modified rank metric, the so-called sum rank metric. For the sum rank metric, the free rank distance, the extended row rank distance and its slope are defined analogous to the extended row distance in Hamming metric. Upper bounds for the free rank distance and the slope of (P)UM codes in the sum rank metric are derived and an explicit construction of (P)UM codes based on Gabidulin codes is given, achieving the upper bound for the free rank distance.

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arXiv2010

The Euclidean Algorithm for Generalized Minimum Distance Decoding of Reed-Solomon Codes

This paper presents a method to merge Generalized Minimum Distance decoding of Reed-Solomon codes with the extended Euclidean algorithm. By merge, we mean that the steps taken to perform the Generalized Minimum Distance decoding are similar to those performed by the extended Euclidean algorithm. The resulting algorithm has a complexity of O(n^2).

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arXiv2010

A Basis for all Solutions of the Key Equation for Gabidulin Codes

We present and prove the correctness of an efficient algorithm that provides a basis for all solutions of a key equation in order to decode Gabidulin (G-) codes up to a given radius tau. This algorithm is based on a symbolic equivalent of the Euclidean Algorithm (EA) and can be applied for decoding of G-codes beyond half the minimum rank distance. If the key equation has a unique solution, our algorithm reduces to Gabidulin's decoding algorithm up to half the minimum distance. If the solution is not unique, we provide a basis for all solutions of the key equation. Our algorithm has time complexity O(tau^2) and is a generalization of the modified EA by Bossert and Bezzateev for Reed-Solomon codes.

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