36 research outputs found
Feng-Rao decoding of primary codes
We show that the Feng-Rao bound for dual codes and a similar bound by
Andersen and Geil [H.E. Andersen and O. Geil, Evaluation codes from order
domain theory, Finite Fields Appl., 14 (2008), pp. 92-123] for primary codes
are consequences of each other. This implies that the Feng-Rao decoding
algorithm can be applied to decode primary codes up to half their designed
minimum distance. The technique applies to any linear code for which
information on well-behaving pairs is available. Consequently we are able to
decode efficiently a large class of codes for which no non-trivial decoding
algorithm was previously known. Among those are important families of
multivariate polynomial codes. Matsumoto and Miura in [R. Matsumoto and S.
Miura, On the Feng-Rao bound for the L-construction of algebraic geometry
codes, IEICE Trans. Fundamentals, E83-A (2000), pp. 926-930] (See also [P.
Beelen and T. H{\o}holdt, The decoding of algebraic geometry codes, in Advances
in algebraic geometry codes, pp. 49-98]) derived from the Feng-Rao bound a
bound for primary one-point algebraic geometric codes and showed how to decode
up to what is guaranteed by their bound. The exposition by Matsumoto and Miura
requires the use of differentials which was not needed in [Andersen and Geil
2008]. Nevertheless we demonstrate a very strong connection between Matsumoto
and Miura's bound and Andersen and Geil's bound when applied to primary
one-point algebraic geometric codes.Comment: elsarticle.cls, 23 pages, no figure. Version 3 added citations to the
works by I.M. Duursma and R. Pellikaa
On the complexity of computing Gr\"obner bases for weighted homogeneous systems
Solving polynomial systems arising from applications is frequently made
easier by the structure of the systems. Weighted homogeneity (or
quasi-homogeneity) is one example of such a structure: given a system of
weights , -homogeneous polynomials are polynomials
which are homogeneous w.r.t the weighted degree
. Gr\"obner bases for weighted homogeneous systems can be
computed by adapting existing algorithms for homogeneous systems to the
weighted homogeneous case. We show that in this case, the complexity estimate
for Algorithm~\F5 \left(\binom{n+\dmax-1}{\dmax}^{\omega}\right) can be
divided by a factor . For zero-dimensional
systems, the complexity of Algorithm~\FGLM (where is the
number of solutions of the system) can be divided by the same factor
. Under genericity assumptions, for
zero-dimensional weighted homogeneous systems of -degree
, these complexity estimates are polynomial in the
weighted B\'ezout bound .
Furthermore, the maximum degree reached in a run of Algorithm \F5 is bounded by
the weighted Macaulay bound , and this bound is
sharp if we can order the weights so that . For overdetermined
semi-regular systems, estimates from the homogeneous case can be adapted to the
weighted case. We provide some experimental results based on systems arising
from a cryptography problem and from polynomial inversion problems. They show
that taking advantage of the weighted homogeneous structure yields substantial
speed-ups, and allows us to solve systems which were otherwise out of reach
Polynomial time attack on high rate random alternant codes
A long standing open question is whether the distinguisher of high rate
alternant codes or Goppa codes \cite{FGOPT11} can be turned into an algorithm
recovering the algebraic structure of such codes from the mere knowledge of an
arbitrary generator matrix of it. This would allow to break the McEliece scheme
as soon as the code rate is large enough and would break all instances of the
CFS signature scheme. We give for the first time a positive answer for this
problem when the code is {\em a generic alternant code} and when the code field
size is small : and for {\em all} regime of other
parameters for which the aforementioned distinguisher works. This breakthrough
has been obtained by two different ingredients : (i) a way of using code
shortening and the component-wise product of codes to derive from the original
alternant code a sequence of alternant codes of decreasing degree up to getting
an alternant code of degree (with a multiplier and support related to those
of the original alternant code);
(ii) an original Gr\"obner basis approach which takes into account the non
standard constraints on the multiplier and support of an alternant code which
recovers in polynomial time the relevant algebraic structure of an alternant
code of degree from the mere knowledge of a basis for it
On the matrix code of quadratic relationships for a Goppa code
In this article, we continue the analysis started in \cite{CMT23} for the
matrix code of quadratic relationships associated with a Goppa code. We provide
new sparse and low-rank elements in the matrix code and categorize them
according to their shape. Thanks to this description, we prove that the set of
rank 2 matrices in the matrix codes associated with square-free binary Goppa
codes, i.e. those used in Classic McEiece, is much larger than what is
expected, at least in the case where the Goppa polynomial degree is 2. We build
upon the algebraic determinantal modeling introduced in \cite{CMT23} to derive
a structural attack on these instances. Our method can break in just a few
seconds some recent challenges about key-recovery attacks on the McEliece
cryptosystem, consistently reducing their estimated security level. We also
provide a general method, valid for any Goppa polynomial degree, to transform a
generic pair of support and multiplier into a pair of support and Goppa
polynomial
A Combinatorial Commutative Algebra Approach to Complete Decoding
Esta tesis pretende explorar el nexo de unión que existe entre la estructura algebraica de un código lineal y el proceso de descodificación completa. Sabemos que el proceso de descodificación completa para códigos lineales arbitrarios es NP-completo, incluso si se admite preprocesamiento de los datos. Nuestro objetivo es realizar un análisis algebraico del proceso de la descodificación, para ello asociamos diferentes estructuras matemáticas a ciertas familias de códigos. Desde el punto de vista computacional, nuestra descripción no proporciona un algoritmo eficiente pues nos enfrentamos a un problema de naturaleza NP. Sin embargo, proponemos algoritmos alternativos y nuevas técnicas que permiten relajar las condiciones del problema reduciendo los recursos de espacio y tiempo necesarios para manejar dicha estructura algebraica.Departamento de Algebra, Geometría y Topologí