Decoding Reed-Solomon codes up to the Sudan radius with the Euclidean algorithm

International audienceWe modify the Euclidean algorithm of Feng and Tzeng to decode Reed-Solomon (RS) codes up to the Sudan radius. The basic steps are the virtual extension to an Interleaved RS code and the reformulation of the multi-sequence shift-register problem of varying length to a multi-sequence problem of equal length. We prove the reformulation and analyze the complexity of our new decoding approach. Furthermore, the extended key equation, that describes the multi-sequence problem, is derived in an alternative polynomial way

Fast Multi-Sequence Shift-Register Synthesis with the Euclidean Algorithm

International audienceFeng and Tzeng's generalization of the Extended Euclidean Algorithm synthesizes the shortest--length linear feedback shift--register for \$s \geq 1\$ sequences, where each sequence has the the same length \$n\$. In this contribution, it is shown that Feng and Tzeng's algorithm which solves this multi--sequence shift--register problem has time complexity \$\ONsn^2\$. An acceleration based on the Divide and Conquer strategy is proposed and it is proven that subquadratic time complexity is achieved

Iterative List-Decoding of Gabidulin Codes via Gr\"obner Based Interpolation

We show how Gabidulin codes can be list decoded by using an iterative parametrization approach. For a given received word, our decoding algorithm processes its entries one by one, constructing four polynomials at each step. This then yields a parametrization of interpolating solutions for the data so far. From the final result a list of all codewords that are closest to the received word with respect to the rank metric is obtained.Comment: Submitted to IEEE Information Theory Workshop 2014 in Hobart, Australi

Topics on Register Synthesis Problems

Pseudo-random sequences are ubiquitous in modern electronics and information technology. High speed generators of such sequences play essential roles in various engineering applications, such as stream ciphers, radar systems, multiple access systems, and quasi-Monte-Carlo simulation. Given a short prefix of a sequence, it is undesirable to have an efficient algorithm that can synthesize a generator which can predict the whole sequence. Otherwise, a cryptanalytic attack can be launched against the system based on that given sequence. Linear feedback shift registers (LFSRs) are the most widely studied pseudorandom sequence generators. The LFSR synthesis problem can be solved by the Berlekamp-Massey algorithm, by constructing a system of linear equations, by the extended Euclidean algorithm, or by the continued fraction algorithm. It is shown that the linear complexity is an important security measure for pseudorandom sequences design. So we investigate lower bounds of the linear complexity of different kinds of pseudorandom sequences. Feedback with carry shift registers (FCSRs) were first described by Goresky and Klapper. They have many good algebraic properties similar to those of LFSRs. FCSRs are good candidates as building blocks of stream ciphers. The FCSR synthesis problem has been studied in many literatures but there are no FCSR synthesis algorithms for multi-sequences. Thus one of the main contributions of this dissertation is to adapt an interleaving technique to develop two algorithms to solve the FCSR synthesis problem for multi-sequences. Algebraic feedback shift registers (AFSRs) are generalizations of LFSRs and FCSRs. Based on a choice of an integral domain R and π ∈ R, an AFSR can produce sequences whose elements can be thought of elements of the quotient ring R/(π). A modification of the Berlekamp-Massey algorithm, Xu\u27s algorithm solves the synthesis problem for AFSRs over a pair (R, π) with certain algebraic properties. We propose two register synthesis algorithms for AFSR synthesis problem. One is an extension of lattice approximation approach but based on lattice basis reduction and the other one is based on the extended Euclidean algorithm

Row Reduction Applied to Decoding of Rank Metric and Subspace Codes

We show that decoding of $\ell$-Interleaved Gabidulin codes, as well as list-$\ell$ decoding of Mahdavifar--Vardy codes can be performed by row reducing skew polynomial matrices. Inspired by row reduction of \F[x] matrices, we develop a general and flexible approach of transforming matrices over skew polynomial rings into a certain reduced form. We apply this to solve generalised shift register problems over skew polynomial rings which occur in decoding $\ell$-Interleaved Gabidulin codes. We obtain an algorithm with complexity $O(\ell \mu^2)$ where $\mu$ measures the size of the input problem and is proportional to the code length $n$ in the case of decoding. Further, we show how to perform the interpolation step of list-$\ell$-decoding Mahdavifar--Vardy codes in complexity $O(\ell n^2)$, where $n$ is the number of interpolation constraints.Comment: Accepted for Designs, Codes and Cryptograph

Systematische Analyse der Sequenz-Struktur-Funktions Zusammenhänge bei Thiamindiphosphat-abhängigen Enzymen

Thiamine diphosphate (ThDP)-dependent enzymes form a vast and diverse protein family, both in the sequence space and in their functional potential. Of particular interest are the enantioselective C-C bond forming and cleavage reactions catalyzed by those enzymes. In these reaction, different ThDP-dependent enzymes provide distinct enantio- and chemoselectivities with often narrow substrate and product ranges. This specificity, which is beneficial for the enantiopure synthesis of fine chemicals like 2-hydroxy ketones, limits the scope of accessible products. Investigations of crystal structures of different ThDP-dependent decarboxylases revealed steric properties in the active sites of those enzymes to control the enantio- and chemoselectivity (S-pocket and donor-acceptor concept). Subsequent application of those concepts by modulation of the steric properties of enzymes’ active sites enabled rational engineering of biocatalysts with desired, but often only moderate, non-physiological enantioselectivities. The major objective of this thesis was to systematically analyze the sequences and structures of this enzyme family and to elucidate the relationships between sequence, structure and function. Detailed understanding of those relationships is pivotal for rational engineering and therefore necessary for the design of biocatalysts with desired selectivities. As compared to the enormous size of this enzyme family only a small number of representatives were experimentally characterized. Even less ThDP-dependent enzymes were modified by mutations in order to analyze effects of distinct amino acid residues and still less were structurally determined. Since the systematic analysis of the sequence-structure-function relationships requires information on the structure and function of a major fraction of family members, methods were developed and applied to increase the amount of available structure and function information. By making use of homology modeling, putative atom coordinates for enzymes lacking experimentally determined structure information were predicted. In addition, by development of a new database system that combines sequence, structure and function information, the acquisition of accurate and comparable biochemical data unambiguously linked to the biocatalysts’ amino acid sequences was enabled. Comparability of biochemical data and deduction of functional roles of certain residues requires comparable biochemical data on the one hand and methods to compare residues from different enzymes on the other hand. Introduction of standard numbering schemes for ThDP-dependent enzymes facilitated fast and accurate comparison of structurally equivalent positions without the need for structure information. The findings derived from those analyses accelerated the engineering of enzymes with desired enantio- and chemoselectivities and inter alia enabled the enzymatic, direct asymmetric synthesis of (S)-benzoins with excellent ees.Die Familie der Thiamindiphosphat (ThDP)-abhängigen Enzyme ist gleichermaßen sequenziell als auch funktionell vielfältig. Besonderes Interesse wird dieser Familie aufgrund ihrer Fähigkeit zuteil, C-C Bindungs- und Spaltungsreaktionen zu katalysieren. Für einen Einsatz in der Biokatalyse und der Synthese von Feinchemikalien (wie beispielsweise alpha-Hydroxyketone) zeichnen sie sich zudem durch ihre definierten Substratspektren als auch ihre Enantioselektivität in zahlreichen Reaktionen aus. Allerdings schränken diese Spezifitäten das Spektrum an enzymatisch zugänglichen Produkten ein. Vergleichende Untersuchungen vorhandener Proteinstrukturen verschiedener ThDP-abhängiger Enzyme zeigten Unterschiede in der Form der Substrat-Bindetaschen der unterschiedlichen Vertreter. Die daraus abgeleiteten ’S-pocket’- und ’Donor/Akzeptor’-Konzepte führen diese sterischen Unterschiede und die resultierenden verschiedenen räumlichen Anordnungen der beiden Substrate in Ligationsreaktionen als die Ursache verschiedener Enantio- und Substratpräferenzen an. Auf dieser Grundlage konnten, durch Anpassung der Form der aktiven Taschen, Decarboxylasen mit geänderten Selektivitäten erzeugt werden. Oft allerdings einhergehend mit nur moderaten Stereoselektivitäten in der Katalyse nicht-natürlicher Reaktionen. Für den Erfolg von Rationalem Design von Biokatalysatoren mit gewünschten Eigenschaften sind detaillierte Kenntnisse über die Sequenz-Struktur-Funktions Zusammenhänge der jeweiligen Proteinfamilie von Bedeutung. Diese Doktorarbeit hatte die systematische Analyse dieser Zusammenhänge in ThDP-abhängigen Enzymen zum Ziel. Eine systematische Analyse von Sequenz-Struktur-Funktions Zusammenhängen erfordert implizit Sequenz-, Struktur- und Funktionsinformation für einen Großteil der zur Familie gehörenden Enzyme. In bisherigen Arbeiten wurden - relativ zu den enormen Ausmaßen dieser Proteinfamilie - nur wenige Vertreter experimentell charakterisiert. Für weiterführende Untersuchungen bezüglich des Einflusses bestimmter Aminosäure-Positionen auf die katalytische Aktivität oder Selektivität wurden nochmals nur wenige dieser Enzyme herangezogen. Eine experimentelle Bestimmung der Proteinstruktur, welche für Rationales Design von Biokatalysatoren von besonderer Bedeutung ist, wurde nur für einen noch geringeren Bruchteil der ThDP-abhängigen Enzyme durchgeführt. Um dem bestehenden Mangel an Informationen über die Struktur und Funktion von Enzymen zu begegnen, wurden im Rahmen dieser Arbeit Proteinstrukturen per Homologie-Modellierung vorhergesagt und Methoden zur Erfassung und Auswertung von Funktionsdaten entwickelt. Mit Hilfe eines neuartigen Datenbank-Systems zur Erfassung verlässlicher und vergleichbarer Daten über die Funktion und Sequenz von Enzymen, wurde die Basis für eine systematische Analyse der genannten Zusammenhänge geschaffen. Neben der Verfügbarkeit von Funktionsinformation, eindeutig mit der Sequenz des entsprechenden Enzyms verknüpft, erfordert die systematische Analyse möglicher funktioneller Bedeutungen einzelner Aminosäure-Positionen eine Methode zum Vergleich von Aminosäuren aus verschiedenen Enzymen. Eine solche Methode wurde mit dem hier präsentierten ’standard numbering scheme’ (Standard-Nummerierungs System) zur Verfügung gestellt. Die Anwendung dieser Methode erlaubt die schnelle und akkurate Identifikation strukturell äquivalenter Positionen in verschiedenen Enzymen ohne Abhängigkeit von Strukturinformation zu den jeweils analysierten Proteinen. Die aus diesen Analysen gezogenen Erkenntnisse wurden eingesetzt, um Biokatalysatoren mit gewünschten Enantio- und Chemoselektivitäten zu erzeugen und erstmals die enzymatische, direkte asymmetrische Synthese von (S)-Benzoinen zu ermöglichen