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Algebraic Constructions and Decoding of Rank Metric Codes with Applications to Network Coding and Code based Cryptography

English title Algebraic Constructions and Decodingof Rank Metric Codes with Applications to Network Coding and Code based Cryptography
Applicant Rosenthal Joachim
Number 169510
Funding scheme Project funding (Div. I-III)
Research institution Institut für Mathematik Universität Zürich
Institution of higher education University of Zurich - ZH
Main discipline Mathematics
Start/End 01.10.2016 - 30.09.2019
Approved amount 241'812.00
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All Disciplines (3)

Discipline
Mathematics
Information Technology
Electrical Engineering

Keywords (5)

Subspace Codes ; Code Based Cryptography; Rank Metric Codes; Coding Theory; McEliece cryptographic systems

Lay Summary (German)

Lead
Heute werden oft Informationen über ganze Netzwerke übermitteltund abgespeichert. Ziel des Projektes ist die Entwicklung bessererCodierungsmethoden welche Fehler entdeckt und auch korrigieren kann.Fortschritte in dieser Forschung wird Auswirkungen auf die nextenGenerationen von kommunikationssystemen haben. Diese Codes haben auch Anwendungen für öffentliche Verschlüsselungsverfahren.
Lay summary
Codierungstheorie befasst sich mit der Beschreibung von Information in
einer Weise welche das Entdecken und das Korrigieren von Fehlern
gewährt so lange die Anzahl der Fehler nicht zu gross ist.
Traditionelle Kommunikationssysteme betrachteten den Informationsfluss
zwischen zwei Parteien und traditionelle Codierungstheorie
beschäftigte sich mit Codierungsmethoden für solche Systeme.

Moderne Kommunikationssysteme und Speichersysteme machen oft Gebrauch
von einem ganzen Netzwerk von Übertragungsstationen. Es kommen oft
auch Szenarien vor wo eine Partei Information über eine Vielzahl von
Übertragungsstationen an eine Vielzahl von Empfängern schickt. Zum
Beispiel ist die TV Übertragung über das Internet und das Handy-Netz
von dieser Natur. Mit der Technik von Netzwerk Codierung versuchen
Forscher Methoden zu entwickeln welche garantieren dass Fehler
entdeckt und korrigiert werden. Die angewandte Algebragruppe der
Universität Zürich entwickelt neue Codierungsmethoden für die Netzwerk
Übertragung.

Die Codes die für Netzwerke entwickelt werden haben auch einen
potentiellen Nutzen in der Kryptographie. Das Decodieren eines allgemeinen
Codes hat eine riesige Komplexität und basierend darauf entwickelten
Forscher sogenannt Code basierende kryptographische Systeme. Es besteht
die Hoffnung dass solche Systeme auch noch sicher sind wenn einmal
ein praktisch brauchbarer Quantum Computer gebaut werden kann. In diesem
Projekt wird sich die Forschungsgruppe auch  mit der Entwicklung neuer
Code basierender kryptographischen Systeme beschäftigen.
Direct link to Lay Summary Last update: 10.10.2016

Responsible applicant and co-applicants

Employees

Publications

Publication
A complete classification of partial MDS (maximally recoverable) codes with one global parity
Horlemann-Trautmann Anna-Lena, Neri Alessandro (2020), A complete classification of partial MDS (maximally recoverable) codes with one global parity, in Advances in Mathematics of Communications, 14(1), 69-88.
Equivalence and characterizations of linear rank-metric codes based on invariants
Neri Alessandro, Puchinger Sven, Horlemann-Trautmann Anna-Lena (2020), Equivalence and characterizations of linear rank-metric codes based on invariants, in Linear Algebra and its Applications, 603, 418-469.
New Lower Bounds for Permutation Codes Using Linear Block Codes
Micheli Giacomo, Neri Alessandro (2020), New Lower Bounds for Permutation Codes Using Linear Block Codes, in IEEE Transactions on Information Theory, 66(7), 4019-4025.
Random construction of partial MDS codes
Neri A., Horlemann-Trautmann A.-L. (2020), Random construction of partial MDS codes, in Designs, Codes, and Cryptography, 88(4), 711-725.
Systematic encoders for generalized Gabidulin codes and the q-analogue of Cauchy matrices
Neri A. (2020), Systematic encoders for generalized Gabidulin codes and the q-analogue of Cauchy matrices, in Linear Algebra and Its Applications, 593, 116-149.
Security of generalised Reed–Solomon code-based cryptosystems
Baldi Marco, Chiaraluce Franco, Rosenthal Joachim, Santini Paolo, Schipani Davide (2019), Security of generalised Reed–Solomon code-based cryptosystems, in IET Information Security, 13(4), 404-410.
How many weights can a cyclic code have?
Shi Minjia, Li Xiaoxiao, Neri Alessandro, Solé Patrick (2019), How many weights can a cyclic code have?, in IEEE Transactions on Information Theory, 66(2), 1449-1459.
Invariants and inequivalence of linear rank-metric codes
Neri Alessandro, Puchinger Sven, Horlemann-Trautmann Anna-Lena (2019), Invariants and inequivalence of linear rank-metric codes, in 2019 IEEE International Symposium on Information Theory (ISIT), IEEE, Paris, France.
Maximum weight spectrum codes
Alderson Tim, Neri Alessandro (2019), Maximum weight spectrum codes, in Advances in Mathematics of Communications, 13(1), 101-119.
Tensor representation of rank-metric codes
Byrne Eimear, Neri Alessandro, Ravagnani Alberto, Sheekey John (2019), Tensor representation of rank-metric codes, in SIAM Journal of Applied Algebra and Geometry, 3(4), 614-643.
On q -Steiner systems from rank metric codes
Arias Francisco, de la Cruz Javier, Rosenthal Joachim, Willems Wolfgang (2018), On q -Steiner systems from rank metric codes, in Discrete Mathematics, 341(10), 2729-2734.
Construction of Constant Dimension Codes
Horlemann-Trautmann Anna-Lena, Rosenthal Joachim (2018), Construction of Constant Dimension Codes, Springer International Publishing, Cham.
Fuzzy authentication using rank distance
Neri Alessandro, Rosenthal Joachim, Schipani Davide (2018), Fuzzy authentication using rank distance, Springer, Cham, Switzerland 447, 97-108.
Hartmann–Tzeng bound and skew cyclic codes of designed Hamming distance
Gómez-Torrecillas Josè, Lobillo Fco Javier, Navarro Gabriel, Neri Alessandro (2018), Hartmann–Tzeng bound and skew cyclic codes of designed Hamming distance, in Finite Fields and their Applications, 50, 84-112.
On the genericity of maximum rank distance and Gabidulin codes
Neri Alessandro, Horlemann-Trautmann Anna-Lena, Randrianarisoa Tovohery, Rosenthal Joachim (2018), On the genericity of maximum rank distance and Gabidulin codes, in Designs, Codes, and Cryptography, 86(2), 341-363.
Variations of the McEliece Cryptosystem
Bolkema Jessalyn, Gluesing-Luerssen Heide, Kelley Christine A., Lauter Kristin E., Malmskog Beth, Rosenthal Joachim (2017), Variations of the McEliece Cryptosystem, Springer International Publishing, Cham.
MRD rank metric convolutional codes
Napp Diego, Pinto Raquel, Rosenthal Joachim, Vettori Paolo (2017), MRD rank metric convolutional codes, in 2017 IEEE International Symposium on Information Theory (ISIT), Aachen, GermanyIEEE, Aachen, Germany.
A decoding algorithm for twisted Gabidulin codes
Randrianarisoa Tovohery, Rosenthal Jaochim (2017), A decoding algorithm for twisted Gabidulin codes, in 2017 IEEE International Symposium on Information Theory (ISIT), Aachen, GermanyIEEE, Aachen, Germany.
An application of group theory in confidential network communications
López-Ramos Juan Antonio, Rosenthal Joachim, Schipani Davide, Schnyder Reto (2016), An application of group theory in confidential network communications, in Mathematical Methods in the Applied Sciences, 41(6), 2294-2298.

Collaboration

Group / person Country
Types of collaboration
University of Kentucky United States of America (North America)
- in-depth/constructive exchanges on approaches, methods or results
Shannon Institute, University College Dublin Ireland (Europe)
- in-depth/constructive exchanges on approaches, methods or results
University of Neuchatel Switzerland (Europe)
- in-depth/constructive exchanges on approaches, methods or results
- Publication
University of Almeria Spain (Europe)
- in-depth/constructive exchanges on approaches, methods or results
- Publication
University of Ancona Italy (Europe)
- in-depth/constructive exchanges on approaches, methods or results

Scientific events

Active participation

Title Type of contribution Title of article or contribution Date Place Persons involved
Invited talk at Technische Universität München Individual talk Rank-Metric Codes and 3-Tensors 18.07.2019 Technische Universität München (Munich), Germany Neri Alessandro;
Munich Workshop on Coding and Cryptography (MWCC 2019) Talk given at a conference Code Based Crypto Involving Expanded Reed-Solomon Codes 15.07.2019 Munich Technical University, Germany Rosenthal Joachim;
Invited talk at SIAM Conference on Applied Algebraic Geometry - Network Coding and Subspace Designs Talk given at a conference Invariants of rank-metric codes via Galois group action 12.07.2019 University of Bern, Switzerland Neri Alessandro;
SIAM Conference on Applied Algebraic Geometry Talk given at a conference Rank Metric Codes and Subspace Codes in a Convolutional Setting 09.07.2019 Bern, Switzerland Rosenthal Joachim;
PhD Defense Individual talk Algebraic Theory of Rank-Metric Codes: Representations, Invariants and Density Results 04.07.2019 University of Zurich, Switzerland Neri Alessandro;
Invited talk at Universidad de Alicante Individual talk Representations of Rank-Metric and Gabidulin Codes 04.06.2019 Universidad de Alicante, Spain Neri Alessandro;
Talk at Zurich Graduate Colloquium Individual talk What is... the MDS conjecture? 21.05.2019 University of Zurich, Switzerland Neri Alessandro;
Oberwolfach Workshop 1912: Contemporary Coding Theory Talk given at a conference Identifying rank-metric codes using Galois group action 17.03.2019 Mathematisches Forschungsinstitut Oberwolfach, Germany Rosenthal Joachim; Neri Alessandro;
Second Colombian Workshop on Coding Theory (CWC 19) Talk given at a conference On the Cauchy-like structure of generator matrices for Gabidulin codes 15.01.2019 Universidad del Norte, Barranquilla, Colombia Rosenthal Joachim; Neri Alessandro;
Dagstuhl Seminar 18511: Algebraic Coding Theory for Networks, Storage, and Security Talk given at a conference Parametrizing Systematic Gabidulin Codes and Applications 16.12.2018 Schloss Dagstuhl, Germany Neri Alessandro; Rosenthal Joachim;
Invited Talk at Technische Universität München Individual talk Encoding schemes for Gabidulin codes with applications 12.11.2018 Technische Universität München (Munich), Germany Neri Alessandro;
Invited talk at the Algebra and Number Theory (ANT), University College Dublin Individual talk Properties of rank metric codes and analogies with the Hamming metric 18.10.2018 University College Dublin (Dublin), Ireland Neri Alessandro;
CIMPA Research School and Workshop: Quasi-Cyclic and Related Algebraic Codes Individual talk One of main speaker of the Research School 27.08.2018 Ankara, Turkey, Turkey Rosenthal Joachim;
CIMPA Research School and Workshop: Quasi-Cyclic and Related Algebraic Codes Talk given at a conference Rank Metric vs. Hamming Metric: Evaluation Codes and their Encoders 27.08.2018 METU (Ankara), Turkey Neri Alessandro;
Latin American Week on Coding and Information 2018 Poster Generalized Gabidulin Codes and their Generator Matrices 22.07.2018 Unicamp, Campinas, Brazil Neri Alessandro;
MTNS 2018 Talk given at a conference Weight Two Masking of the Reed-Solomon Structure in Conjunction with List Decoding 16.07.2018 Hong Kong., Hongkong Rosenthal Joachim;
Invited talk at Seminari Università di Trento Individual talk The Systematic Encoder for Generalized Gabidulin Codes 11.06.2018 Università di Trento , Italy Neri Alessandro;
Combinatorics 2018 Talk given at a conference Maximum Weight Spectrum Codes 03.06.2018 Arco (Trento), Italy Neri Alessandro;
Invited talk at Séminaire Mathématiques Discrètes, Codes et Cryptographie, Université Paris 13 Individual talk Analogies between Rank-Metric and Hamming Codes: Old and New Results 19.04.2018 Université Paris 13, France Neri Alessandro;
Munich Workshop on Coding and Cryptography 2018 Poster The Generator Matrix in Standard Form of Gabidulin Codes 10.04.2018 Technische Universität München (Munich), Germany Neri Alessandro; Rosenthal Joachim;
Talk at Arbeitsgemeinschaft in Codierungstheorie und Kryptographie Individual talk A Decoding Algorithm for Rank Metric Codes 06.12.2017 Zurich, Switzerland Randrianarisoa Tovohery Hajatiana;
Talk at Arbeitsgemeinschaft in Codierungstheorie und Kryptographie Individual talk MRD codes: further properties and standard form of Gabidulin codes 22.11.2017 University of Neuchatel, Switzerland Neri Alessandro;
Munich Workshop on Coding and Applications 2017 Talk given at a conference Hiding Distinguishers in Code Based Cryptography' 03.07.2017 Munich Germany, Germany Rosenthal Joachim;
2017 IEEE International Symposium on Information Theory (ISIT) Talk given at a conference A decoding algorithm for twisted Gabidulin codes 25.06.2017 Aachen, Germany, Germany Rosenthal Joachim; Randrianarisoa Tovohery Hajatiana;
The 13th International Conference on Finite Fields and their Applications Talk given at a conference On linear Generalized Twisted Gabidulin codes and the existence of new MRD codes 04.06.2017 Gaeta (Rome), Italy Rosenthal Joachim; Neri Alessandro;
2nd Workshop on Communication Security Talk given at a conference Fuzzy Authentication using Rank Distance 30.04.2017 Paris, France Neri Alessandro;
Invited talk at the Algebra, Number Theory, and Applications (ANTA) Seminar, AALTO University Individual talk On the Genericity of Maximum Rank Distance and Gabidulin Codes 16.11.2016 AALTO University (Helsinki), Finland Neri Alessandro;


Awards

Title Year
Fellow of IEEE 2019
Honorary Professor, Universidad del Norte, Colombia. 2019

Associated projects

Number Title Start Funding scheme
188430 New Constructions of Convolutional Codes 01.01.2020 Project funding (Div. I-III)
181446 Code based cryptosystems and linear complexity of sequences 01.09.2018 Early Postdoc.Mobility
187711 Algebraic Methods for Rank-Metric Codes in Post-Quantum Cryptography and Communications 01.09.2019 Early Postdoc.Mobility
149716 Algebraic Constructions and Decoding of Subspace Codes 01.10.2013 Project funding (Div. I-III)
149716 Algebraic Constructions and Decoding of Subspace Codes 01.10.2013 Project funding (Div. I-III)

Abstract

Coding theory has emerged out of the need for bettercommunication and has rapidly developed as a mathematical theoryin strong relationship with algebra, combinatorics and algebraicgeometry. Nowadays error-correcting-codes are used in everydaypractical applications such as digital-storage media, wire-lineand wireless networks, and satellite and deep-space communicationsystems. Example of simple block codes are the internationalstandard book numbers (ISBN), the ASCII code and various encodingschemes used to identify bank accounts.Network coding theory is concerned with the encoding andtransmission of information where there may be many informationsources and possibly many receivers. R.~K\"otter andF.~Kschischang identified a fundamental mathematicalquestion which lies at the heart of network coding. Thisformulation seeks the construction of good subsets of thefinite Grassmann variety and it is the intended plan of theproposed research to use algebraic techniques to come up with newnetwork codes which have better performance.An important class of codes used in network coding theory are the so called rank metric codes. These codes have also been used to construct public key cryptographic systems. It is an open questionif such constructed systems are also secure in case a quantum computer is available. The project aims to come up with promising new classes of subspace codes and rank metric codes with potentialuse in network coding, distributed storage and cryptography.
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