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On simple ringoids [article]

Jens Zumbrägel
2009 arXiv   pre-print
A ringoid is a set with two binary operations that are linked by the distributive laws. We study special classes of ringoids that are congruence-simple or ideal-simple. In particular, we examine generalised parasemifields and non-associative semirings.
arXiv:0910.4760v1 fatcat:g5u6vhkbwna55gnd7cyq77zasy

On the Pseudocodeword Redundancy [article]

Jens Zumbragel, Mark F. Flanagan, Vitaly Skachek
2010 arXiv   pre-print
We define the AWGNC, BSC, and max-fractional pseudocodeword redundancy of a code as the smallest number of rows in a parity-check matrix such that the corresponding minimum pseudoweight is equal to the minimum Hamming distance. We show that most codes do not have a finite pseudocodeword redundancy. We also provide bounds on the pseudocodeword redundancy for some families of codes, including codes based on designs.
arXiv:1001.1705v1 fatcat:cedhiqfwyrar3ddjhxkv5x7re4

Indiskrete Logarithmen?

Jens Zumbrägel
2018 Mitteilungen der DMV  
Zusammenfassung Die moderne Public-Key-Kryptographie, begründet durch die bahnbrechende Arbeit von Diffie und Hellman, ist seit jeher mit der Schwierigkeit des diskreten Logarithmusproblems verbunden. Allerdings ist in Körpern kleiner Charakteristik die Hartnäckigkeit dieses Problems nicht so hoch, wie lange Zeit angenommen. Daraus resultieren beträchtliche Rekordberechnungen, sowie Konsequenzen für die Sicherheit einiger Kryptosysteme.
doi:10.1515/dmvm-2018-0027 fatcat:jss3yfoy6na5xjiz4zx2etm3vm

Classification of finite congruence-simple semirings with zero [article]

Jens Zumbrägel
2007 arXiv   pre-print
Our main result states that a finite semiring of order >2 with zero which is not a ring is congruence-simple if and only if it is isomorphic to a 'dense' subsemiring of the endomorphism semiring of a finite idempotent commutative monoid. We also investigate those subsemirings further, addressing e.g. the question of isomorphy.
arXiv:math/0702416v1 fatcat:62bxsrqjmvddnhur43i7n6o3je

New Results on the Pseudoredundancy [article]

Zihui Liu, Jens Zumbrägel, Marcus Greferath, Xin-Wen Wu
2014 arXiv   pre-print
The concepts of pseudocodeword and pseudoweight play a fundamental role in the finite-length analysis of LDPC codes. The pseudoredundancy of a binary linear code is defined as the minimum number of rows in a parity-check matrix such that the corresponding minimum pseudoweight equals its minimum Hamming distance. By using the value assignment of Chen and Kløve we present new results on the pseudocodeword redundancy of binary linear codes. In particular, we give several upper bounds on the
more » ... edundancies of certain codes with repeated and added coordinates and of certain shortened subcodes. We also investigate several kinds of k-dimensional binary codes and compute their exact pseudocodeword redundancy.
arXiv:1410.1627v1 fatcat:zxezwai5c5eghh6v2kplksbjni

On Simpleness of Semirings and Complete Semirings [article]

Yefim Katsov, Tran Giang Nam, Jens Zumbrägel
2011 arXiv   pre-print
In this paper, among other results, there are described (complete) simple - simultaneously ideal- and congruence-simple - endomorphism semirings of (complete) idempotent commutative monoids; it is shown that the concepts of simpleness, congruence-simpleness and ideal-simpleness for (complete) endomorphism semirings of projective semilattices (projective complete lattices) in the category of semilattices coincide iff those semilattices are finite distributive lattices; there are described
more » ... nce-simple complete hemirings and left artinian congruence-simple complete hemirings. Considering the relationship between the concepts of "Morita equivalence" and "simpleness" in the semiring setting, we have obtained the following results: The ideal-simpleness, congruence-simpleness and simpleness of semirings are Morita invariant properties; A complete description of simple semirings containing the infinite element; The representation theorem - "Double Centralizer Property" - for simple semirings; A complete description of simple semirings containing a projective minimal one-sided ideal; A characterization of ideal-simple semirings having either infinite elements or a projective minimal one-sided ideal; A confirmation of Conjecture of [Kat04a] and solving Problem 3.9 of [Kat04b] in the classes of simple semirings containing either infinite elements or projective minimal left (right) ideals, showing, respectively, that semirings of those classes are not perfect and the concepts of "mono-flatness" and "flatness" for semimodules over semirings of those classes are the same. Finally, we give a complete description of ideal-simple, artinian additively idempotent chain semirings, as well as of congruence-simple, lattice-ordered semirings.
arXiv:1105.5591v1 fatcat:t7dmlixsu5a65oykheqfotyamy

Finite simple additively idempotent semirings [article]

Andreas Kendziorra, Jens Zumbrägel
2012 arXiv   pre-print
Since for the classification of finite (congruence-)simple semirings it remains to classify the additively idempotent semirings, we progress on the characterization of finite simple additively idempotent semirings as semirings of join-morphisms of a semilattice. We succeed in doing this for many cases, amongst others for every semiring of this kind with an additively neutral element. As a consequence we complete the classification of finite simple semirings with an additively neutral element.
more » ... complete the classification of all finite simple semirings it remains to classify some very specific semirings, which will be discussed here. Our results employ the theory of idempotent irreducible semimodules, which we develop further.
arXiv:1201.0272v2 fatcat:ropwddoeajf75dbhnbvnjlitni

List Decoding of Quaternary Codes in the Lee Metric [article]

Marcus Greferath, Jens Zumbrägel
2022 arXiv   pre-print
We present a list decoding algorithm for quaternary negacyclic codes over the Lee metric. To achieve this result, we use a Sudan-Guruswami type list decoding algorithm for Reed-Solomon codes over certain ring alphabets. Our decoding strategy for negacyclic codes over the ring ℤ_4 combines the list decoding algorithm by Wu with the Gröbner basis approach for solving a key equation due to Byrne and Fitzpatrick.
arXiv:2202.03977v1 fatcat:vigflvxyxnbtxagryedrpawabm

Indiscreet logarithms in finite fields of small characteristic [article]

Robert Granger, Thorsten Kleinjung, Jens Zumbrägel
2016 arXiv   pre-print
was later developed into an alternative algorithm by Granger, Kleinjung and Zumbrägel, which also has quasi-polynomial complexity, but is rigorous for a large family of fields [GKZ14b, GKZ15] .  ...  Indeed, nearly thirty years after Coppersmith's algorithm the L(1/3) barrier was broken in a series of remarkable results, starting with the work of Göloglu, Granger, McGuire and Zumbrägel [GGMZ13] ,  ... 
arXiv:1604.03837v1 fatcat:tafikapvajb57kwzxixbkfdc2u

Exploration of AWGNC and BSC Pseudocodeword Redundancy [article]

Jens Zumbragel, Mark F. Flanagan, Vitaly Skachek
2010 arXiv   pre-print
Zumbrägel and M.F. Flanagan are with the Claude Shannon Institute, University College Dublin, Belfield, Dublin 4, Ireland. Emails: jens.zumbragel@ucd.ie, mark.flanagan@ieee.org. V.  ... 
arXiv:1005.3486v1 fatcat:5syloiijh5hzpahdsmlaquqosq

Designs and codes in affine geometry [article]

Jens Zumbrägel
2016 arXiv   pre-print
Classical designs and their (projective) q-analogs can both be viewed as designs in matroids, using the matroid of all subsets of a set and the matroid of linearly independent subsets of a vector space, respectively. Another natural matroid is given by the point sets in general position of an affine space, leading to the concept of an affine design. Accordingly, a t-(n, k, λ) affine design of order q is a collection B of (k-1)-dimensional spaces in the affine geometry A = AG(n-1, q) such that
more » ... ch (t-1)-dimensional space in A is contained in exactly λ spaces of B. In the case λ = 1, as usual, one also refers to an affine Steiner system S(t, k, n). In this work we examine the relationship between the affine and the projective q-analogs of designs. The existence of affine Steiner systems with various parameters is shown, including the affine q-analog S(2, 3, 7) of the Fano plane. Moreover, we consider various distances in matroids and geometries, and we discuss the application of codes in affine geometry for error-control in a random network coding scenario.
arXiv:1605.03789v2 fatcat:s4xog2np3vgczpauzhh3da5pci

Characteristics of Invariant Weights Related to Code Equivalence over Rings [article]

Marcus Greferath, Cathy Mc Fadden, Jens Zumbrägel
2011 arXiv   pre-print
The Equivalence Theorem states that, for a given weight on the alphabet, every linear isometry between linear codes extends to a monomial transformation of the entire space. This theorem has been proved for several weights and alphabets, including the original MacWilliams' Equivalence Theorem for the Hamming weight on codes over finite fields. The question remains: What conditions must a weight satisfy so that the Extension Theorem will hold? In this paper we provide an algebraic framework for
more » ... etermining such conditions, generalising the approach taken in [Greferath, Honold '06].
arXiv:1110.1538v1 fatcat:iweebpwj7rf6vnw7shirpiski4

Efficient recovering of operation tables of black box groups and rings [article]

Jens Zumbragel, Gerard Maze, Joachim Rosenthal
2008 arXiv   pre-print
People have been studying the following problem: Given a finite set S with a hidden (black box) binary operation * on S which might come from a group law, and suppose you have access to an oracle that you can ask for the operation x*y of single pairs (x,y) you choose. What is the minimal number of queries to the oracle until the whole binary operation is recovered, i.e. you know x*y for all x,y in S? This problem can trivially be solved by using |S|^2 queries to the oracle, so the question
more » ... s under which circumstances you can succeed with a significantly smaller number of queries. In this presentation we give a lower bound on the number of queries needed for general binary operations. On the other hand, we present algorithms solving this problem by using |S| queries, provided that * is an abelian group operation. We also investigate black box rings and give lower and upper bounds for the number of queries needed to solve product recovering in this case.
arXiv:0805.0514v1 fatcat:kh356egys5aujo6dsxhfbmqhwu

Computation of a 30750-Bit Binary Field Discrete Logarithm [article]

Robert Granger, Thorsten Kleinjung, Arjen K. Lenstra, Benjamin Wesolowski, Jens Zumbrägel
2020 arXiv   pre-print
The present computation made essential use of the elimination step of the quasi-polynomial algorithm due to Granger, Kleinjung and Zumbrägel, and is the first large-scale experiment to truly test and successfully  ...  independent and distinct quasi-polynomial algorithms for the DLP in fixed characteristic: the first due to Barbulescu, Gaudry, Joux and Thomé (BGJT) [1] ; and the second due to Granger, Kleinjung and Zumbrägel  ... 
arXiv:2008.02717v1 fatcat:yk6xgbsua5gqlfb7hna55nr4bq

Efficient Decoding of Gabidulin Codes over Galois Rings [article]

Sven Puchinger and Julian Renner and Antonia Wachter-Zeh and Jens Zumbrägel
2021 arXiv   pre-print
This paper presents the first decoding algorithm for Gabidulin codes over Galois rings with provable quadratic complexity. The new method consists of two steps: (1) solving a syndrome-based key equation to obtain the annihilator polynomial of the error and therefore the column space of the error, (2) solving a key equation based on the received word in order to reconstruct the error vector. This two-step approach became necessary since standard solutions as the Euclidean algorithm do not properly work over rings.
arXiv:2102.02157v1 fatcat:ek7z746aojadndtjpkmtgqf5aa
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