Equivalence Of Codes Research Articles

Equivalence of constacyclic codes with shift constants of different orders

Equivalence of constacyclic codes with shift constants of different orders

Representations of group actions and their applications in cryptography

Cryptographic group actions provide a flexible framework that allows the instantiation of several primitives, ranging from key exchange protocols to PRFs and digital signatures. The security of such constructions is based on the intractability of some computational problems. For example, given the group action (G,X,⋆), the weak unpredictability assumption (Alamati et al. (2020) [1]) requires that, given random xi's in X, no probabilistic polynomial time algorithm can compute, on input {(xi,g⋆xi)}i=1,…,Q and y, the set element g⋆y.In this work, we study such assumptions, aided by the definition of group action representations and a new metric, the q-linear dimension, that estimates the “linearity” of a group action, or in other words, how much it is far from being linear. We show that under some hypotheses on the group action representation, and if the q-linear dimension is polynomial in the security parameter, then the weak unpredictability and other related assumptions cannot hold. This technique is applied to some actions from cryptography, like the ones arising from the equivalence of linear codes, as a result, we obtain the impossibility of using such actions for the instantiation of certain primitives.As an additional result, some bounds on the q-linear dimension are given for classical groups, such as Sn, GL(Fn) and the cyclic group Zn acting on itself.

Monomial isomorphism for tensors and applications to code equivalence problems

Starting from the problem of d-tensor isomorphism (d-TI\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extsf {TI}$$\\end{document}), we study the relation between various code equivalence problems in different metrics. In particular, we show a reduction from the sum-rank metric (CEsr\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extsf {CE}_{\ extsf {sr}}$$\\end{document}) to the rank metric (CErk\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extsf {CE}_{\ extsf {rk}}$$\\end{document}). To obtain this result, we investigate reductions between tensor problems. We define the monomial isomorphism problem for d-tensors (d-TI∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extsf {TI}^*$$\\end{document}), where, given two d-tensors, we ask if there are d-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d-1$$\\end{document} invertible matrices and a monomial matrix sending one tensor into the other. We link this problem to the well-studied d-TI\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extsf {TI}$$\\end{document} and the TI\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extsf {TI}$$\\end{document}-completeness of d-TI∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extsf {TI}^*$$\\end{document} is shown. Due to this result, we obtain a reduction from CEsr\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extsf {CE}_{\ extsf {sr}}$$\\end{document} to CErk\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extsf {CE}_{\ extsf {rk}}$$\\end{document}. In the literature, a similar result was known, but it needs an additional assumption on the automorphisms of matrix codes. Since many constructions based on the hardness of Code Equivalence problems are emerging in cryptography, we analyze how such reductions can be taken into account in the design of cryptosystems based on CEsr\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extsf {CE}_{\ extsf {sr}}$$\\end{document}.

Local disentanglers for the equivalence of two-dimensional topological quantum codes

Local disentanglers for the equivalence of two-dimensional topological quantum codes

How to Find the Equivalence Classes in a Set of Linear Codes in Practice?

An algorithm for equivalence of linear codes over finite fields is presented. Its main advantage is that it can extract exactly one representative from each equivalence class among a large number of linear codes. It can also be used as a test for isomorphism of binary matrices. The algorithm is implemented in the program LCequivalence, which is designed to obtain the inequivalent codes in a set of linear codes over a finite field with q<64 elements. This program is a module of the free software package QextNewEdition for constructing, classifying and studying linear codes.

Hardness estimates of the code equivalence problem in the rank metric

In this paper, we analyze the hardness of the Matrix Code Equivalence (MCE) problem for matrix codes endowed with the rank metric, and provide the first algorithms for solving it. We do this by making a connection to another well-known equivalence problem from multivariate cryptography—the Isomorphism of Polynomials (IP). Under mild assumptions, we give tight reductions from MCE to the homogenous version of the Quadratic Maps Linear Equivalence (QMLE) problem, and vice versa. Furthermore, we present reductions to and from similar problems in the sum-rank metric, showing that MCE is at the core of code equivalence problems. On the practical side, using birthday techniques known for IP, we present two algorithms: a probabilistic algorithm for MCE running in time q23(n+m)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$q^{\\frac{2}{3}(n+m)}$$\\end{document} up to a polynomial factor, and a deterministic algorithm for MCE with roots, running in time qmin{m,n,k}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$q^{\\min \\{m,n,k\\}}$$\\end{document} up to a polynomial factor. Lastly, to confirm these findings, we solve randomly-generated instances of MCE using these two algorithms.

Przekład intersemiotyczny. Przypadek znaku drogowego, ekfrazy i adaptacji filmowej


 
 
 The article is an attempt to clarify and refine the ways of understanding two basic types of translation: from one language into another ethnic language (translation proper) – i.e. following the principles of translating homo- geneous systems of signs – and from one system of signs to another, e.g. verbal to iconic, or vice versa, which is sometimes called transmutation. Therefore, the starting point for the discussion is Roman Jakobson’s well-known typology of translation, which also took into account cases of intralingual translation within the same language, called rewording. However, the article mainly focuses on various cases of intersemiotic translation and thus transmutation, meaning situations where a text expressed in one system of signs is translated into a system of a different nature. The primary research question concerns what is actually being translated in such a situation and how this type of translation differs from a typical language-to-language translation. The article is chiefly theoretical, and the examples concern the cases of translating verbal signs into iconic ones, such as the rules governing traffic into a road sign, the translation of an image (a painting, photography) into a literary text (ekphrasis and its varieties), and a film adaptation. The equivalence of codes secures the adequacy of a linguistic translation. In the case of ekphrasis, the image is usually replaced by a verbal narrative, and in a film adaptation, the type and sequence of iconic signs must simply be invented because these sign systems are not directly equivalent, as Benveniste and Lotman believe. Examples that illustrate the main theses of the article are Jacek Kaczmarski’s art-inspired songs, several film adaptations of Shakespeare’s Macbeth (Polański, Kurosawa), and Agnieszka Holland’s film adaptation (Pokot [Spoor]) of one of Olga Tokarczuk’s novels.
 
 

Generalized column weights and equivalences of convolutional codes

We revisit the notion of [Formula: see text]th generalized column weights for the [Formula: see text]-truncation of a convolutional code. Taking the limit as [Formula: see text] tends to infinity, we define [Formula: see text]th generalized column weights of a convolutional code. We introduce suitable notions of [Formula: see text]-equivalence and equivalence, with respect to which generalized column weights are code invariants. We establish some properties of these invariants and compare them with other definitions of generalized distance and weight which appear in the literature.

Constructions and equivalence of Sidon spaces

Sidon spaces have been introduced by Bachoc et al. (in: Mathematical Proceedings of the Cambridge Philosophical Society, Cambridge University Press, 2017) as the q-analogue of Sidon sets. The interest on Sidon spaces has increased quickly, especially after the work of Roth et al. (IEEE Trans Inform Theory 64(6):4412–4422, 2017), in which they highlighted the correspondence between Sidon spaces and cyclic subspace codes. Up to now, the known constructions of Sidon Spaces may be divided in three families: the ones contained in the sum of two multiplicative cosets of a fixed subfield of mathbb {F}_{q^n}, the ones contained in the sum of more than two multiplicative cosets of a fixed subfield of mathbb {F}_{q^n} and finally the ones obtained as the kernel of subspace polynomials. In this paper, we will mainly focus on the first class of examples, for which we provide characterization results and we will show some new examples, arising also from some well-known combinatorial objects. Moreover, we will give a quite natural definition of equivalence among Sidon spaces, which relies on the notion of equivalence of cyclic subspace codes and we will discuss about the equivalence of the known examples.

Equivalence and duality of polycyclic codes associated with trinomials over finite fields

In this paper, all of the conjectures in Aydin et al. (2022) [6] are settled: some proven correct, some proven correct with a modification, and some disproven, involving the equivalence and duality of polycyclic codes associated with trinomials. We also give methods to construct isodual and self-dual polycyclic codes, and study the self-orthogonal and dual-containing polycyclic codes over F2.

On the equivalence of linear cyclic and constacyclic codes

We introduce new sufficient conditions for permutation and monomial equivalence of linear cyclic codes over various finite fields. We recall that monomial equivalence and isometric equivalence are the same relation for linear codes over finite fields. A necessary and sufficient condition for the monomial equivalence of linear cyclic codes through a shift map on their defining set is also given. Finally, we prove that if gcd⁡(3n,ϕ(3n))=1, where ϕ is the Euler's totient function, then all permutation equivalent constacyclic codes of length n over F4 are given by the action of multipliers. The results of this work allow us to prune the search algorithm for new linear codes and discover record-breaking linear and quantum codes.

On the computational hardness of the code equivalence problem in cryptography

<p style='text-indent:20px;'>Code equivalence is a well-known concept in coding theory. Recently, literature saw an increased interest in this notion, due to the introduction of protocols based on the hardness of finding the equivalence between two linear codes. In this paper, we analyze the security of code equivalence, with a special focus on the hardest instances, in the interest of cryptographic usage. Our work stems from a thorough review of existing literature, identifies the various types of solvers for the problem, and provides a precise complexity analysis, where previously absent. Furthermore, we are able to improve on the state of the art, providing more efficient algorithm variations, for which we include numerical simulation data. In the end, the goal of this paper is to provide a complete, single point of access, which can be used as a tool for designing schemes that rely on the code equivalence problem.</p>

An Attack on a Non-Interactive Key Exchange from Code Equivalence

Abstract A recent paper by Zhang and Zhang claims to construct the first code-based non-interactive key exchange protocol, using a modified version of the Code Equivalence Problem. In this paper we explain why this approach is flawed. Namely, we describe an attack which involves only linear algebra and completely breaks the protocol with overwhelming probability. A simple Magma script confirms our results.

A generalization of cyclic code equivalence algorithm to constacyclic codes

Recently, a new algorithm to test equivalence of two cyclic codes has been introduced which is efficient and produced useful results. In this work, we generalize this algorithm to constacyclic codes. As an application of the algorithm we found many constacyclic codes with good parameters and properties. In particular, we found 22 new codes that improve the minimum distances of best known linear codes (BKLCs).

Some Binary Quasi-perfect Linear Codes Defined by APN Functions

In 2021, Tutdere proved that the covering radii R of a class of primitive binary cyclic codes with minimum distance strictly greater than an odd integer l satisfy r≤R≤l, where l, r are some integers depending on the given code. We here first discuss some equivalences of linear codes defined by Gold functions, which are quadratic APN (almost perfect nonlinear) functions. We then show that by applying the result of Tutdere one can find the covering radii of these quasi-perfect codes. In 2016, Li and Helleseth proved that the linear codes defined by the quadratic APN functions are quasi-perfect and they asked whether the linear codes defined by the non-quadratic APN functions are quasi-perfect or not. We here prove that the linear codes defined by some non-quadratic APN functions over the finite field〖 F〗_(2^m ) , for 1≤m≤8, are quasi-perfect, by computing the covering radii of these codes.

A polynomial-time reduction from the multi-graph isomorphism problem to additive code equivalence

We present a polynomial time reduction from the multi-graph isomorphism problem to the problem of code equivalence of additive codes over finite extensions of ${\mathbb F}_2$.

A new algorithm for equivalence of cyclic codes and its applications

Cyclic codes are among the most important families of codes in coding theory for both theoretical and practical reasons. Despite their prominence and intensive research on cyclic codes for over a half century, there are still open problems related to cyclic codes. In this work, we use recent results on the equivalence of cyclic codes to create a more efficient algorithm to partition cyclic codes by equivalence based on cyclotomic cosets. This algorithm is then implemented to carry out computer searches for both cyclic codes and quasi-cyclic (QC) codes with good parameters. We also generalize these results to repeated-root cases. We have found several new linear codes that are cyclic or QC as an application of the new approach, as well as more desirable constructions for linear codes with best known parameters. With the additional new codes obtained through standard constructions, we have found a total of 14 new linear codes.

Advanced Signature Functionalities from the Code Equivalence Problem

The LESS signature scheme, introduced in 2020, represented a fresh start for code-based signatures. In this paper we explore advanced functionalities for signature schemes, stemming from the work of LESS. First, we adapt a recent protocol of Beullens et al. to obtain a construction for (linkable) ring signatures. Then, we realize an identity-based signature scheme following the traditional approach by Joye and Neven. Our performance numbers confirm that signature schemes based on the code equivalence problem have considerable potential for practical applications.

ПОТЕНЦИАЛЬНЫЕ ВОЗМОЖНОСТИ ПЕРЕВОДА В АСПЕКТЕ ТЕОРИИ КОММУНИКАЦИИ

The presented article is written on a topical topic. In it, within the boundaries of the main acts of communication, emphasis is placed on some supporting factors. First of all, the two-part formula is subject to correction: translator-reader. Without in any way questioning the inviolability of this traditional formulation, the article attempts to saturate it with new content. In particular, based on the works of leading Russian and European scientists, it is shown that the translator, using different communicative capabilities, brings to the reader the meaning of the transmitted information, drawing ideas from the arsenal of several options. This is how the discussion about the variability of translation activity arises and develops, which in turn is closely related to the signification and denotation of the text. These terms are not only disclosed, but also subjected to a little critical analysis. The article is written exclusively in a theoretical way: a large place is occupied by the statements of scientists who are actively involved in the problems chosen in the article, and from various linguistic positions. Meanwhile, such an approach, in our opinion, fully justifies the goal set: to point out the broad potential possibilities of translation, using purely theoretical material for this. It includes: A) Definition of monolingual or bilingual communication as an integral part of translation; B) An indication of the equivalence, identity or difference of the language codes of the sender and the recipient of the information; C) Identification of the nature of language communication with its inherent factors; D) Definition of the role of significates and denotations. At the end of the article, the translation possibilities are subordinated to some cultural ideas.

On the equivalence of authentication codes and robust (2, 2)-threshold schemes

Abstract In this paper, we show a “direct” equivalence between certain authentication codes and robust threshold schemes. It was previously known that authentication codes and robust threshold schemes are closely related to similar types of designs, but direct equivalences had not been considered in the literature. Our new equivalences motivate the consideration of a certain “key-substitution attack.” We study this attack and analyze it in the setting of “dual authentication codes.” We also show how this viewpoint provides a nice way to prove properties and generalizations of some known constructions.