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### On algebras admitting a complete set of near weights, evaluation codes and Goppa codes [article]

Cicero Carvalho, Ercilio Silva
<span title="2008-07-21">2008</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
In this paper we show that an algebra admits a set of m near weight functions having a compatibility property, namely, the set is a "complete set", if and only if it is the ring of regular functions of  ...  In 1998 Hoholdt, van Lint and Pellikaan introduced the concept of a "weight function" defined on a F_q-algebra and used it to construct linear codes, obtaining among them the algebraic-geometric (AG) codes  ...  Codes from near weights In this section we show how to construct codes from algebras that admit a complete set of n-weights, and give a lower bound for their minimum distance.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/0807.3198v1">arXiv:0807.3198v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/xiy3a36x3rc5xpcpwntatmqhr4">fatcat:xiy3a36x3rc5xpcpwntatmqhr4</a> </span>
<a target="_blank" rel="noopener" href="https://archive.org/download/arxiv-0807.3198/0807.3198.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> File Archive [PDF] </button> </a> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/0807.3198v1" title="arxiv.org access"> <button class="ui compact blue labeled icon button serp-button"> <i class="file alternate outline icon"></i> arxiv.org </button> </a>

### Code Based Cryptography and Steganography [chapter]

Pascal Véron
<span title="">2013</span> <i title="Springer Berlin Heidelberg"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/2w3awgokqne6te4nvlofavy5a4" style="color: black;">Lecture Notes in Computer Science</a> </i> &nbsp;
The aim of this paper is to show how algebraic coding theory offers ways to define secure cryptographic primitives and efficient steganographic schemes. Cryptography  ...  Nowadays, coding theory plays an important role in the area of cryptography and steganography.  ...  A good starting point on Steganography and matrix embedding is [51] .  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/978-3-642-40663-8_5">doi:10.1007/978-3-642-40663-8_5</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/t4a63pdm2bbtddmfm7uimzlmdu">fatcat:t4a63pdm2bbtddmfm7uimzlmdu</a> </span>
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### A New Rank Metric Codes Based Encryption Scheme [chapter]

Pierre Loidreau
<span title="">2017</span> <i title="Springer International Publishing"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/2w3awgokqne6te4nvlofavy5a4" style="color: black;">Lecture Notes in Computer Science</a> </i> &nbsp;
For a given security our public-keys are more compact than for the same security in the Hamming metric based settings.  ...  We design a new McEliece-like rank metric based encryption scheme from Gabidulin codes. We explain why it is not affected by the invariant subspace attacks also known as Overbeck's attacks.  ...  Though GRS codes are unsuitable for use in cryptosystems, Goppa codes are widely admitted as being suitable and even the best choice for the design of secure code-based primitives and even PQ primitives  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/978-3-319-59879-6_1">doi:10.1007/978-3-319-59879-6_1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/kqdhhgfszbfinpdfbdu24af6wy">fatcat:kqdhhgfszbfinpdfbdu24af6wy</a> </span>
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### δ-sequences and evaluation codes defined by plane valuations at infinity

Carlos Galindo, Francisco Monserrat
<span title="2008-11-06">2008</span> <i title="Oxford University Press (OUP)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/fnog2v763jbmxj5syfg42vo57u" style="color: black;">Proceedings of the London Mathematical Society</a> </i> &nbsp;
This leads us to a deeper knowledge of the involved evaluation codes what allows us to get explicitly a large set of codes (and associated parameters) from a simple input which can be easily determined  ...  We show how to construct (and compute parameters) for the dual code of any evaluation code associated with a weight function defined by ∆ from the polynomial ring in two indeterminates to a semigroup S  ...  Dual evaluation codes given by classical weight functions admit another lower bound of its minimal distance, called Goppa distance.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1112/plms/pdn042">doi:10.1112/plms/pdn042</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/ixmx5oq2lzak7mjd3cunj2ayei">fatcat:ixmx5oq2lzak7mjd3cunj2ayei</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20170808195921/http://repositori.uji.es/xmlui/bitstream/handle/10234/18553/31356.pdf?sequence=1" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/9d/c1/9dc15abe0fd255dec6fefc22fc6db897beb7dcd1.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1112/plms/pdn042"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> Publisher / doi.org </button> </a>

### δ-sequences and Evaluation Codes defined by Plane Valuations at Infinity [article]

C. Galindo, F.Monserrat
<span title="2008-07-14">2008</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We show how to construct (and compute parameters) for the dual code of any evaluation code associated with a weight function defined by Δ from the polynomial ring in two indeterminates to a semigroup S  ...  We also give algorithms to construct an unlimited number of δ-sequences of the different existing types, and so this paper provides the tools to know and use a new large set of codes.  ...  Dual evaluation codes given by classical weight functions admit another lower bound of its minimal distance, called Goppa distance.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/0704.3536v2">arXiv:0704.3536v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/dlzyexkl3bgnnbdpwbj5ktrrei">fatcat:dlzyexkl3bgnnbdpwbj5ktrrei</a> </span>
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### Improved evaluation codes defined by plane valuations

M. Borges-Quintana, M.A. Borges-Trenard, C. Galindo
<span title="">2010</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/vppj2ymhunarjnozole4nystdi" style="color: black;">Finite Fields and Their Applications</a> </i> &nbsp;
We show minimal sets of generators of the semigroups of these structures and provide parameters for the corresponding improved evaluation codes.  ...  We study improved evaluation codes associated with finitely generated order structures given by plane valuations.  ...  Well-known codes as the one-point geometric Goppa codes or weighted Reed-Muller codes can be regarded as codes given by order structures.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.ffa.2010.03.002">doi:10.1016/j.ffa.2010.03.002</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/jdkc7ras2zchlm6gli2ms33nja">fatcat:jdkc7ras2zchlm6gli2ms33nja</a> </span>
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### Evaluation codes defined by finite families of plane valuations at infinity

C. Galindo, F. Monserrat
<span title="2012-08-30">2012</span> <i title="Springer Nature"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/c45m6ttnaje4xbjsq7m2c6df2a" style="color: black;">Designs, Codes and Cryptography</a> </i> &nbsp;
We construct evaluation codes given by weight functions defined over polynomial rings in m ≥ 2 indeterminates.  ...  These weight functions are determined by sets of m − 1 weight functions over polynomial rings in two indeterminates defined by plane valuations at infinity.  ...  Recall that order functions only reproduce one-point geometric Goppa codes; to study other geometric Goppa codes (on curves), with a similar machinery, a weak version of order functions, the so-called  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/s10623-012-9738-7">doi:10.1007/s10623-012-9738-7</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/6uoz52plyfauhmttkbgzkrvnua">fatcat:6uoz52plyfauhmttkbgzkrvnua</a> </span>
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### How to Construct Curves over Finite Fields With Many Points [article]

Gerard van der Geer, Marcel van der Vlugt
<span title="1996-05-23">1996</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
In this paper we give several methods to construct curves over finite fields with many points and illustrate this with examples of the results.  ...  by evaluating (m−w)/2 i=1 Tr(a i x)Tr(b i x) on F * q is a codeword in C * h .  ...  It implies the existence of an isogeny Jac(C D ) ∼ f ∈P(L D ) Jac(C f ), (5) where the summation is over a complete set of representatives for the canonical action of F * p on L D − {0}.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/alg-geom/9511005v2">arXiv:alg-geom/9511005v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/blvtyycf7nhirocoxiz7t6rt4q">fatcat:blvtyycf7nhirocoxiz7t6rt4q</a> </span>
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### LDPC codes in the McEliece cryptosystem: attacks and countermeasures [article]

Marco Baldi
<span title="2007-09-30">2007</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
The original version, based on Goppa codes, is able to guarantee a high level of security, and is faster than competing solutions, like RSA.  ...  Quasi-Cyclic (QC) LDPC codes are a particular class of LDPC codes, able to join low complexity encoding of QC codes with high-performing and low-complexity decoding of LDPC codes.  ...  His scheme uses a large set of subcodes of a given BCH code in place of the Goppa code of the original system; so, though the idea of using subcodes is very interesting and maybe reusable also in different  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/0710.0142v1">arXiv:0710.0142v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/7frqnapqyrdyfowupqkk4yknim">fatcat:7frqnapqyrdyfowupqkk4yknim</a> </span>
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### On the security of subspace subcodes of Reed-Solomon codes for public key encryption [article]

Alain Couvreur, Matthieu Lequesne
<span title="2021-10-08">2021</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
These codes appear to be a natural generalisation of Goppa and alternant codes and provide a broader flexibility in designing code based encryption schemes.  ...  For the security analysis, we introduce a new operation on codes called the twisted product which yields a polynomial time distinguisher on such subspace subcodes as soon as the chosen 𝔽_q-subspaces have  ...  The use of subspace subcodes of Reed-Solomon codes is of particular interest in code based cryptography since it includes McEliece's original proposal based on Goppa codes on the one hand and encryption  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2009.05826v2">arXiv:2009.05826v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/nb5hv6ytz5fuxfuu2wg67n3d7m">fatcat:nb5hv6ytz5fuxfuu2wg67n3d7m</a> </span>
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### A formula on the weight distribution of linear codes with applications to AMDS codes [article]

Alessio Meneghetti, Marco Pellegrini, Massimiliano Sala
<span title="2021-09-25">2021</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
There has been a lot of research on weight enumerators of special cases, such as self-dual codes and codes with small Singleton's defect.  ...  We propose a new set of linear relations that must be satisfied by the coefficients of the weight distribution.  ...  Preliminary results of this paper were presented on a talk given at Algebra for Cryptography 2019 in L'Aquila [Menss].  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2003.14063v2">arXiv:2003.14063v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/h4tk4i5otrcyrknekhx2xsjuva">fatcat:h4tk4i5otrcyrknekhx2xsjuva</a> </span>
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### Post-Quantum Cryptography: State of the Art [chapter]

Johannes A. Buchmann, Denis Butin, Florian Göpfert, Albrecht Petzoldt
<span title="">2016</span> <i title="Springer Berlin Heidelberg"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/2w3awgokqne6te4nvlofavy5a4" style="color: black;">Lecture Notes in Computer Science</a> </i> &nbsp;
This paper gives an overview of the alternative public-key schemes that have the capability to resist quantum computer attacks and compares them.  ...  However, as a result of Peter Shor shows, the public-key schemes that are being used today will become insecure once quantum computers reach maturity.  ...  Using such a Goppa polynomial, the decoding problem for Goppa codes can be solved in polynomial time for errors of weight at most t.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/978-3-662-49301-4_6">doi:10.1007/978-3-662-49301-4_6</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/ylrv42q33fegnidacywmkzmvme">fatcat:ylrv42q33fegnidacywmkzmvme</a> </span>
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### Tangent Codes [article]

Azniv Kasparian, Evgeniya Velikova
<span title="2015-05-08">2015</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We provide constructions of affine varieties with near MDS, cyclic or Hamming tangent codes.  ...  Tangent codes are shown to admit simultaneous decoding. The duals of the tangent codes to X are realized by gradients of polynomials from the ideal of X.  ...  imply that the averaged homogeneous weight enumerator of Goppa codes, associated with a complete set of representatives of the linear equivalence classes of divisors of fixed degree is related to the ζ-polynomial  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1409.4583v2">arXiv:1409.4583v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/l6b4yoc3pjbvxddqpfthrjuzca">fatcat:l6b4yoc3pjbvxddqpfthrjuzca</a> </span>
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### Toward good families of codes from towers of surfaces [article]

Alain Couvreur, Philippe Lebacque, Marc Perret
<span title="2020-06-08">2020</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We introduce in this article a new method to estimate the minimum distance of codes from algebraic surfaces.  ...  In the same direction, we establish a criterion for a surface with a fixed finite set of closed points P to have an infinite tower of ℓ–étale covers in which P splits totally.  ...  Acknowledgements The authors express a deep gratitude to Olivier Dudas, David Madore, Jade Nardi, Fabrice Orgogozo, Cédric Pépin, Michel Raynaud, Alexander Schmidt and Jakob Stix, for inspiring discussions  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2002.02220v2">arXiv:2002.02220v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/cayps5bgcffizhb6rl7slcwwue">fatcat:cayps5bgcffizhb6rl7slcwwue</a> </span>
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### Identity-Based Encryption from Codes with Rank Metric [chapter]

Philippe Gaborit, Adrien Hauteville, Duong Hieu Phan, Jean-Pierre Tillich
<span title="">2017</span> <i title="Springer International Publishing"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/2w3awgokqne6te4nvlofavy5a4" style="color: black;">Lecture Notes in Computer Science</a> </i> &nbsp;
Although this latter scheme has some advantages, like a short signature size, the small weight vector has a logarithmic weight in the length of the code, which implies a super polynomial complexity and  ...  considered as secure, and the indistinguishability of Goppa codes from random codes for standard encryption parameters remains unbroken.  ...  This work has been supported in part by the French ANR projects ALAMBIC (ANR-16-CE39-0006) and ID-FIX (ANR-16-CE39-0004).  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/978-3-319-63697-9_7">doi:10.1007/978-3-319-63697-9_7</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/wscwpb53mnbq3gvupqhivibf6e">fatcat:wscwpb53mnbq3gvupqhivibf6e</a> </span>
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