arXiv subjects

Joachim Rosenthal

Publications and source records attributed to Joachim Rosenthal.

At least 19 recordsLinked to original sources

arXiv2024

Error-Correction Performance of Regular Ring-Linear LDPC Codes over Lee Channels

Most low-density parity-check (LDPC) code constructions are considered over finite fields. In this work, we focus on regular LDPC codes over integer residue rings and analyze their performance with respect to the Lee metric. Their error-correction performance is studied over two channel models, in the Lee metric. The first channel model is a discrete memoryless channel, whereas in the second channel model an error vector is drawn uniformly at random from all vectors of a fixed Lee weight. It is known that the two channel laws coincide in the asymptotic regime, meaning that their marginal distributions match. For both channel models, we derive upper bounds on the block error probability in terms of a random coding union bound as well as sphere packing bounds that make use of the marginal distribution of the considered channels. We estimate the decoding error probability of regular LDPC code ensembles over the channels using the marginal distribution and determining the expected Lee weight distribution of a random LDPC code over a finite integer ring. By means of density evolution and finite-length simulations, we estimate the error-correction performance of selected LDPC code ensembles under belief propagation decoding and a low-complexity symbol message passing decoding algorithm and compare the performances. The analysis developed in this paper may serve to design regular LDPC codes over integer residue rings for storage and cryptographic application.

cs.IT↗

arXiv2024

A Survey on Code-Based Cryptography

The improvements on quantum technology are threatening our daily cybersecurity, as a capable quantum computer can break all currently employed asymmetric cryptosystems. In preparation for the quantum-era the National Institute of Standards and Technology (NIST) has initiated in 2016 a standardization process for public-key encryption (PKE) schemes, key-encapsulation mechanisms (KEM) and digital signature schemes. In 2023, NIST made an additional call for post-quantum signatures. With this chapter we aim at providing a survey on code-based cryptography, focusing on PKEs and signature schemes. We cover the main frameworks introduced in code-based cryptography and analyze their security assumptions. We provide the mathematical background in a lecture notes style, with the intention of reaching a wider audience.

cs.CR↗

arXiv2024

Schubert Subspace Codes

In this paper, we initiate the study of constant dimension subspace codes restricted to Schubert varieties, which we call Schubert subspace codes. These codes have a very natural geometric description, as objects that we call intersecting sets with respect to a fixed subspace. We provide a geometric construction of maximum size constant dimension subspace codes in some Schubert varieties with the largest possible value for the minimum subspace distance. Finally, we generalize the problem to different values of the minimum distance.

cs.IT↗

arXiv2024

Algorithms for Computing the Free Distance of Convolutional Codes

The free distance of a convolutional code is a reliable indicator of its performance. However its computation is not an easy task. In this paper, we present some algorithms to compute the free distance with good efficiency that work for convolutional codes of all rates and over any field. Furthermore we discuss why an algorithm which is claimed to be very efficient is incorrect.

cs.IT↗

arXiv2023

Cryptanalysis of some Nonabelian Group-Based Key Exchange Protocols

In the recently emerging field of nonabelian group-based cryptography, a prominently used one-way function is the Conjugacy Search Problem (CSP), and two important classes of platform groups are polycyclic and matrix groups. In this paper, we discuss the complexity of the conjugacy search problem (CSP) in these two classes of platform groups using the three protocols in [10], [26], and [29] as our starting point. We produce a polynomial time solution for the CSP in a finite polycyclic group with two generators, and show that a restricted CSP is reducible to a DLP. In matrix groups over finite fields, we usedthe Jordan decomposition of a matrix to produce a polynomial time reduction of an A-restricted CSP, where A is a cyclic subgroup of the general linear group, to a set of DLPs over an extension of Fq. We use these general methods and results to describe concrete cryptanalysis algorithms for these three systems. In particular, we show that in the group of invertible matrices over finite fields and in polycyclic groups with two generators, a CSP where conjugators are restricted to a cyclic subgroup is reducible to a set of O(n2) discrete logarithm problems. Using our general results, we demonstrate concrete cryptanalysis algorithms for each of these three schemes. We believe that our methods and findings are likely to allow for several other heuristic attacks in the general case.

cs.CR↗

arXiv2023

Criteria for the construction of MDS convolutional codes with good column distances

Maximum-distance separable (MDS) convolutional codes are characterized by the property that their free distance reaches the generalized Singleton bound. In this paper, new criteria to construct MDS convolutional codes are presented. Additionally, the obtained convolutional codes have optimal first (reverse) column distances and the criteria allow to relate the construction of MDS convolutional codes to the construction of reverse superregular Toeplitz matrices. Moreover, we present some construction examples for small code parameters over small finite fields.

cs.IT↗

arXiv2023

Binary convolutional codes with optimal column distances

There exists a large literature of construction of convolutional codes with maximal or near maximal free distance. Much less is known about constructions of convolutional codes having optimal or near optimal column distances. In this paper, a new construction of convolutional codes over the binary field with optimal column distances is presented.

cs.IT↗

arXiv2022

Analysis of Low-Density Parity-Check Codes over Finite Integer Rings for the Lee Channel

We study the performance of nonbinary low-density parity-check (LDPC) codes over finite integer rings over two channels that arise from the Lee metric. The first channel is a discrete memory-less channel (DMC) matched to the Lee metric. The second channel adds to each codeword an error vector of constant Lee weight, where the error vector is picked uniformly at random from the set of vectors of constant Lee weight. It is shown that the marginal conditional distributions of the two channels coincide, in the limit of large block length. Random coding union bounds on the block error probability are derived for both channels. Moreover, the performance of selected LDPC code ensembles is analyzed by means of density evolution and finite-length simulations, with belief propagation decoding and with a low-complexity symbol message passing algorithm and it is compared to the derived bounds.

cs.IT↗

arXiv2022

Efficient Description of some Classes of Codes using Group Algebras

Circulant matrices are an important tool widely used in coding theory and cryptography. A circulant matrix is a square matrix whose rows are the cyclic shifts of the first row. Such a matrix can be efficiently stored in memory because it is fully specified by its first row. The ring of $n \times n$ circulant matrices can be identified with the quotient ring $\mathbb{F}[x]/(x^n-1)$. In consequence, the strong algebraic structure of the ring $\mathbb{F}[x]/(x^n-1)$ can be used to study properties of the collection of all $n\times n$ circulant matrices. The ring $\mathbb{F}[x]/(x^n-1)$ is a special case of a group algebra and elements of any finite dimensional group algebra can be represented with square matrices which are specified by a single column. In this paper we study this representation and prove that it is an injective Hamming weight preserving homomorphism of $\mathbb{F}$-algebras and classify it in the case where the underlying group is abelian. Our work is motivated by the desire to generalize the BIKE cryptosystem (a contender in the NIST competition to get a new post-quantum standard for asymmetric cryptography). Group algebras can be used to design similar cryptosystems or, more generally, to construct low density or moderate density parity-check matrices for linear codes.

cs.IT↗

arXiv2022

Self-Dual Convolutional Codes

This paper investigates the concept of self-dual convolutional code. We derive the basic properties of this interesting class of codes and we show how some of the techniques to construct self-dual linear block codes generalize to self-dual convolutional codes. As for self-dual linear block codes we are able to give a complete classification for some small parameters.

cs.IT↗

arXiv2022

Convolutional codes over finite chain rings, MDP codes and their characterization

In this paper, we develop the theory of convolutional codes over finite commutative chain rings. In particular, we focus on maximum distance profile (MDP) convolutional codes and we provide a characterization of these codes, generalizing the one known for fields. Moreover, we relate (reverse) MDP convolutional codes over a finite chain ring with (reverse) MDP convolutional codes over its residue field. Finally, we provide a construction of (reverse) MDP convolutional codes over finite chain rings generalizing the notion of (reverse) superregular matrices.

cs.IT↗

arXiv2022

A Deterministic Algorithm for the Discrete Logarithm Problem in a Semigroup

The discrete logarithm problem in a finite group is the basis for many protocols in cryptography. The best general algorithms which solve this problem have time complexity of $\mathcal{O}(\sqrt{N}\log N)$, and a space complexity of $\mathcal{O}(\sqrt{N})$ where $N$ is the order of the group. (If $N$ is unknown, a simple modification would achieve a time complexity of $\mathcal{O}(\sqrt{N}(\log N)^2)$.) These algorithms require the inversion of some group elements or rely on finding collisions and the existence of inverses, and thus do not adapt to work in the general semigroup setting. For semigroups, probabilistic algorithms with similar time complexity have been proposed. The main result of this paper is a deterministic algorithm for solving the discrete logarithm problem in a semigroup. Specifically, let $x$ be an element in a semigroup having finite order $N_x$. The paper provides an algorithm, which, given any element $y\in \langle x \rangle $, provides all natural numbers $m$ with $x^m=y$, and has time complexity $O(\sqrt{N_x}(\log N_x)^2)$ steps. The paper also gives an analysis of the success rates of the existing probabilistic algorithms, which were so far only conjectured or stated loosely.

cs.CC↗

arXiv2022

Existence and Cardinality of $k$-Normal Elements in Finite Fields

Normal bases in finite fields constitute a vast topic of large theoretical and practical interest. Recently, $k$-normal elements were introduced as a natural extension of normal elements. The existence and the number of $k$-normal elements in a fixed extension of a finite field are both open problems in full generality, and comprise a promising research avenue. In this paper, we first formulate a general lower bound for the number of $k$-normal elements, assuming that they exist. We further derive a new existence condition for $k$-normal elements using the general factorization of the polynomial $x^m-1$ into cyclotomic polynomials. Finally, we provide an existence condition for normal elements in $\fqm$ with a non-maximal but high multiplicative order in the group of units of the finite field.

math.NT↗

arXiv2022

On the Properties of Error Patterns in the Constant Lee Weight Channel

The problem of scalar multiplication applied to vectors is considered in the Lee metric. Unlike in other metrics, the Lee weight of a vector may be increased or decreased by the product with a nonzero, nontrivial scalar. This problem is of particular interest for cryptographic applications, like for example Lee metric code-based cryptosystems, since an attacker may use scalar multiplication to reduce the Lee weight of the error vector and thus to reduce the complexity of the corresponding generic decoder. The scalar multiplication problem is analyzed in the asymptotic regime. Furthermore, the construction of a vector with constant Lee weight using integer partitions is analyzed and an efficient method for drawing vectors of constant Lee weight uniformly at random from the set of all such vectors is given.

cs.IT↗

arXiv2021

Moderate Density Parity-Check Codes from Projective Bundles

A new construction for moderate density parity-check (MDPC) codes using finite geometry is proposed. We design a parity-check matrix for this family of binary codes as the concatenation of two matrices: the incidence matrix between points and lines of the Desarguesian projective plane and the incidence matrix between points and ovals of a projective bundle. A projective bundle is a special collection of ovals which pairwise meet in a unique point. We determine minimum distance and dimension of these codes, showing that they have a natural quasi-cyclic structure. In addition, we analyze the error-correction performance within one round of a modification of Gallager's bit-flipping decoding algorithm. In this setting, our codes have the best possible error-correction performance for this range of parameters.

cs.IT↗

arXiv2021

On Bounds for Ring-Based Coding Theory

Coding Theory where the alphabet is identified with the elements of a ring or a module has become an important research topic over the last 30 years. Such codes over rings had important applications and many interesting mathematical problems are related to this line of research. It has been well established, that with the generalization of the algebraic structure to rings there is a need to also generalize the underlying metric beyond the usual Hamming weight used in traditional coding theory over finite fields. This paper introduces a new weight, called the overweight, which can be seen as a generalization of the Lee weight on the integers modulo $4$. For this new weight we provide a number of well-known bounds, like a Plotkin bound, a sphere-packing bound, and a Gilbert-Varshamov bound. A further highlight is the proof of a Johnson bound for the homogeneous weight on a general finite Frobenius ring.

cs.IT↗

arXiv2020

Construction of LDPC convolutional codes via difference triangle sets

In this paper, a construction of $(n,k,δ)$ LDPC convolutional codes over arbitrary finite fields, which generalizes the work of Robinson and Bernstein and the later work of Tong is provided. The sets of integers forming a $(k,w)$-(weak) difference triangle set are used as supports of some columns of the sliding parity-check matrix of an $(n,k,δ)$ convolutional code, where $n\in\mathbb{N}$, $n>k$. The parameters of the convolutional code are related to the parameters of the underlying difference triangle set. In particular, a relation between the free distance of the code and $w$ is established as well as a relation between the degree of the code and the scope of the difference triangle set. Moreover, we show that some conditions on the weak difference triangle set ensure that the Tanner graph associated to the sliding parity-check matrix of the convolutional code is free from $2\ell$-cycles not satisfying the full rank condition over any finite field. Finally, we relax these conditions and provide a lower bound on the field size, depending on the parity of $\ell$, that is sufficient to still avoid $2\ell$-cycles. This is important for improving the performance of a code and avoiding the presence of low-weight codewords and absorbing sets.

cs.IT↗

arXiv2020

Erasure decoding of convolutional codes using first order representations

In this paper, we employ the linear systems representation of a convolutional code to develop a decoding algorithm for convolutional codes over the erasure channel. We study the decoding problem using the state space description and this provides in a natural way additional information. With respect to previously known decoding algorithms, our new algorithm has the advantage that it is able to reduce the decoding delay as well as the computational effort in the erasure recovery process. We describe which properties a convolutional code should have in order to obtain a good decoding performance and illustrate it with an example.

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