392 research outputs found
Fast Computation of Minimal Interpolation Bases in Popov Form for Arbitrary Shifts
We compute minimal bases of solutions for a general interpolation problem,
which encompasses Hermite-Pad\'e approximation and constrained multivariate
interpolation, and has applications in coding theory and security.
This problem asks to find univariate polynomial relations between vectors
of size ; these relations should have small degree with respect to an
input degree shift. For an arbitrary shift, we propose an algorithm for the
computation of an interpolation basis in shifted Popov normal form with a cost
of field operations, where
is the exponent of matrix multiplication and the notation
indicates that logarithmic terms are omitted.
Earlier works, in the case of Hermite-Pad\'e approximation and in the general
interpolation case, compute non-normalized bases. Since for arbitrary shifts
such bases may have size , the cost bound
was feasible only with restrictive
assumptions on the shift that ensure small output sizes. The question of
handling arbitrary shifts with the same complexity bound was left open.
To obtain the target cost for any shift, we strengthen the properties of the
output bases, and of those obtained during the course of the algorithm: all the
bases are computed in shifted Popov form, whose size is always . Then, we design a divide-and-conquer scheme. We recursively reduce
the initial interpolation problem to sub-problems with more convenient shifts
by first computing information on the degrees of the intermediate bases.Comment: 8 pages, sig-alternate class, 4 figures (problems and algorithms
Fast Computation of Shifted Popov Forms of Polynomial Matrices via Systems of Modular Polynomial Equations
We give a Las Vegas algorithm which computes the shifted Popov form of an nonsingular polynomial matrix of degree in expected
field operations, where is the
exponent of matrix multiplication and
indicates that logarithmic factors are omitted. This is the first algorithm in
for shifted row reduction with arbitrary
shifts.
Using partial linearization, we reduce the problem to the case where is the generic determinant bound, with bounded from above by both the average row degree and the average column
degree of the matrix. The cost above becomes , improving upon the cost of the fastest previously
known algorithm for row reduction, which is deterministic.
Our algorithm first builds a system of modular equations whose solution set
is the row space of the input matrix, and then finds the basis in shifted Popov
form of this set. We give a deterministic algorithm for this second step
supporting arbitrary moduli in
field operations, where is the number of unknowns and is the sum
of the degrees of the moduli. This extends previous results with the same cost
bound in the specific cases of order basis computation and M-Pad\'e
approximation, in which the moduli are products of known linear factors.Comment: 8 pages, sig-alternate class, 5 figures (problems and algorithms
Computing minimal interpolation bases
International audienceWe consider the problem of computing univariate polynomial matrices over afield that represent minimal solution bases for a general interpolationproblem, some forms of which are the vector M-Pad\'e approximation problem in[Van Barel and Bultheel, Numerical Algorithms 3, 1992] and the rationalinterpolation problem in [Beckermann and Labahn, SIAM J. Matrix Anal. Appl. 22,2000]. Particular instances of this problem include the bivariate interpolationsteps of Guruswami-Sudan hard-decision and K\"otter-Vardy soft-decisiondecodings of Reed-Solomon codes, the multivariate interpolation step oflist-decoding of folded Reed-Solomon codes, and Hermite-Pad\'e approximation. In the mentioned references, the problem is solved using iterative algorithmsbased on recurrence relations. Here, we discuss a fast, divide-and-conquerversion of this recurrence, taking advantage of fast matrix computations overthe scalars and over the polynomials. This new algorithm is deterministic, andfor computing shifted minimal bases of relations between vectors of size it uses field operations, where is the exponent of matrix multiplication, and is the sum of theentries of the input shift , with . This complexity boundimproves in particular on earlier algorithms in the case of bivariateinterpolation for soft decoding, while matching fastest existing algorithms forsimultaneous Hermite-Pad\'e approximation
Algorithms for Simultaneous Pad\'e Approximations
We describe how to solve simultaneous Pad\'e approximations over a power
series ring for a field using operations in
, where is the sought precision and is the number of power series to
approximate. We develop two algorithms using different approaches. Both
algorithms return a reduced sub-bases that generates the complete set of
solutions to the input approximations problem that satisfy the given degree
constraints. Our results are made possible by recent breakthroughs in fast
computations of minimal approximant bases and Hermite Pad\'e approximations.Comment: ISSAC 201
Fast Decoding of Codes in the Rank, Subspace, and Sum-Rank Metric
We speed up existing decoding algorithms for three code classes in different
metrics: interleaved Gabidulin codes in the rank metric, lifted interleaved
Gabidulin codes in the subspace metric, and linearized Reed-Solomon codes in
the sum-rank metric. The speed-ups are achieved by reducing the core of the
underlying computational problems of the decoders to one common tool: computing
left and right approximant bases of matrices over skew polynomial rings. To
accomplish this, we describe a skew-analogue of the existing PM-Basis algorithm
for matrices over usual polynomials. This captures the bulk of the work in
multiplication of skew polynomials, and the complexity benefit comes from
existing algorithms performing this faster than in classical quadratic
complexity. The new faster algorithms for the various decoding-related
computational problems are interesting in their own and have further
applications, in particular parts of decoders of several other codes and
foundational problems related to the remainder-evaluation of skew polynomials
Fast, deterministic computation of the Hermite normal form and determinant of a polynomial matrix
Given a nonsingular matrix of univariate polynomials over a
field , we give fast and deterministic algorithms to compute its
determinant and its Hermite normal form. Our algorithms use
operations in ,
where is bounded from above by both the average of the degrees of the rows
and that of the columns of the matrix and is the exponent of matrix
multiplication. The soft- notation indicates that logarithmic factors in the
big- are omitted while the ceiling function indicates that the cost is
when . Our algorithms are based
on a fast and deterministic triangularization method for computing the diagonal
entries of the Hermite form of a nonsingular matrix.Comment: 34 pages, 3 algorithm
Row Reduction Applied to Decoding of Rank Metric and Subspace Codes
We show that decoding of -Interleaved Gabidulin codes, as well as
list- decoding of Mahdavifar--Vardy codes can be performed by row
reducing skew polynomial matrices. Inspired by row reduction of \F[x]
matrices, we develop a general and flexible approach of transforming matrices
over skew polynomial rings into a certain reduced form. We apply this to solve
generalised shift register problems over skew polynomial rings which occur in
decoding -Interleaved Gabidulin codes. We obtain an algorithm with
complexity where measures the size of the input problem
and is proportional to the code length in the case of decoding. Further, we
show how to perform the interpolation step of list--decoding
Mahdavifar--Vardy codes in complexity , where is the number of
interpolation constraints.Comment: Accepted for Designs, Codes and Cryptograph
Computing syzygies in finite dimension using fast linear algebra
We consider the computation of syzygies of multivariate polynomials in afinite-dimensional setting: for a -module of finite dimension as a -vector space, andgiven elements in , the problem is to computesyzygies between the 's, that is, polynomials in such that in. Assuming that the multiplication matrices of the variables with respect to some basis of are known, we give analgorithm which computes the reduced Gr\"obner basis of the module of thesesyzygies, for any monomial order, using operations in the base field , where is theexponent of matrix multiplication. Furthermore, assuming that is itself given as ,under some assumptions on we show that these multiplicationmatrices can be computed from a Gr\"obner basis of within thesame complexity bound. In particular, taking , and in, this yields a change of monomial order algorithm along thelines of the FGLM algorithm with a complexity bound which is sub-cubic in
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