36,046 research outputs found
On Solving Systems of Diagonal Polynomial Equations Over Finite Fields
We present an algorithm to solve a system of diagonal polynomial equations
over finite fields when the number of variables is greater than some fixed
polynomial of the number of equations whose degree depends only on the degree
of the polynomial equations. Our algorithm works in time polynomial in the
number of equations and the logarithm of the size of the field, whenever the
degree of the polynomial equations is constant. As a consequence we design
polynomial time quantum algorithms for two algebraic hidden structure problems:
for the hidden subgroup problem in certain semidirect product p-groups of
constant nilpotency class, and for the multi-dimensional univariate hidden
polynomial graph problem when the degree of the polynomials is constant.Comment: A preliminary extended abstract of this paper has appeared in
Proceedings of FAW 2015, Springer LNCS vol. 9130, pp. 125-137 (2015
Computational linear algebra over finite fields
We present here algorithms for efficient computation of linear algebra
problems over finite fields
Polynomial-Time Algorithms for Quadratic Isomorphism of Polynomials: The Regular Case
Let and be
two sets of nonlinear polynomials over
( being a field). We consider the computational problem of finding
-- if any -- an invertible transformation on the variables mapping
to . The corresponding equivalence problem is known as {\tt
Isomorphism of Polynomials with one Secret} ({\tt IP1S}) and is a fundamental
problem in multivariate cryptography. The main result is a randomized
polynomial-time algorithm for solving {\tt IP1S} for quadratic instances, a
particular case of importance in cryptography and somewhat justifying {\it a
posteriori} the fact that {\it Graph Isomorphism} reduces to only cubic
instances of {\tt IP1S} (Agrawal and Saxena). To this end, we show that {\tt
IP1S} for quadratic polynomials can be reduced to a variant of the classical
module isomorphism problem in representation theory, which involves to test the
orthogonal simultaneous conjugacy of symmetric matrices. We show that we can
essentially {\it linearize} the problem by reducing quadratic-{\tt IP1S} to
test the orthogonal simultaneous similarity of symmetric matrices; this latter
problem was shown by Chistov, Ivanyos and Karpinski to be equivalent to finding
an invertible matrix in the linear space of matrices over and to compute the square root in a matrix
algebra. While computing square roots of matrices can be done efficiently using
numerical methods, it seems difficult to control the bit complexity of such
methods. However, we present exact and polynomial-time algorithms for computing
the square root in for various fields (including
finite fields). We then consider \\#{\tt IP1S}, the counting version of {\tt
IP1S} for quadratic instances. In particular, we provide a (complete)
characterization of the automorphism group of homogeneous quadratic
polynomials. Finally, we also consider the more general {\it Isomorphism of
Polynomials} ({\tt IP}) problem where we allow an invertible linear
transformation on the variables \emph{and} on the set of polynomials. A
randomized polynomial-time algorithm for solving {\tt IP} when
is presented. From an algorithmic point
of view, the problem boils down to factoring the determinant of a linear matrix
(\emph{i.e.}\ a matrix whose components are linear polynomials). This extends
to {\tt IP} a result of Kayal obtained for {\tt PolyProj}.Comment: Published in Journal of Complexity, Elsevier, 2015, pp.3
Factorizing the Stochastic Galerkin System
Recent work has explored solver strategies for the linear system of equations
arising from a spectral Galerkin approximation of the solution of PDEs with
parameterized (or stochastic) inputs. We consider the related problem of a
matrix equation whose matrix and right hand side depend on a set of parameters
(e.g. a PDE with stochastic inputs semidiscretized in space) and examine the
linear system arising from a similar Galerkin approximation of the solution. We
derive a useful factorization of this system of equations, which yields bounds
on the eigenvalues, clues to preconditioning, and a flexible implementation
method for a wide array of problems. We complement this analysis with (i) a
numerical study of preconditioners on a standard elliptic PDE test problem and
(ii) a fluids application using existing CFD codes; the MATLAB codes used in
the numerical studies are available online.Comment: 13 pages, 4 figures, 2 table
Syndrome decoding of Reed-Muller codes and tensor decomposition over finite fields
Reed-Muller codes are some of the oldest and most widely studied
error-correcting codes, of interest for both their algebraic structure as well
as their many algorithmic properties. A recent beautiful result of Saptharishi,
Shpilka and Volk showed that for binary Reed-Muller codes of length and
distance , one can correct random errors
in time (which is well beyond the worst-case error
tolerance of ).
In this paper, we consider the problem of `syndrome decoding' Reed-Muller
codes from random errors. More specifically, given the
-bit long syndrome vector of a codeword corrupted in
random coordinates, we would like to compute the
locations of the codeword corruptions. This problem turns out to be equivalent
to a basic question about computing tensor decomposition of random low-rank
tensors over finite fields.
Our main result is that syndrome decoding of Reed-Muller codes (and the
equivalent tensor decomposition problem) can be solved efficiently, i.e., in
time. We give two algorithms for this problem:
1. The first algorithm is a finite field variant of a classical algorithm for
tensor decomposition over real numbers due to Jennrich. This also gives an
alternate proof for the main result of Saptharishi et al.
2. The second algorithm is obtained by implementing the steps of the
Berlekamp-Welch-style decoding algorithm of Saptharishi et al. in
sublinear-time. The main new ingredient is an algorithm for solving certain
kinds of systems of polynomial equations.Comment: 24 page
Solving the "Isomorphism of Polynomials with Two Secrets" Problem for all Pairs of Quadratic Forms
We study the Isomorphism of Polynomial (IP2S) problem with m=2 homogeneous
quadratic polynomials of n variables over a finite field of odd characteristic:
given two quadratic polynomials (a, b) on n variables, we find two bijective
linear maps (s,t) such that b=t . a . s. We give an algorithm computing s and t
in time complexity O~(n^4) for all instances, and O~(n^3) in a dominant set of
instances.
The IP2S problem was introduced in cryptography by Patarin back in 1996. The
special case of this problem when t is the identity is called the isomorphism
with one secret (IP1S) problem. Generic algebraic equation solvers (for example
using Gr\"obner bases) solve quite well random instances of the IP1S problem.
For the particular cyclic instances of IP1S, a cubic-time algorithm was later
given and explained in terms of pencils of quadratic forms over all finite
fields; in particular, the cyclic IP1S problem in odd characteristic reduces to
the computation of the square root of a matrix.
We give here an algorithm solving all cases of the IP1S problem in odd
characteristic using two new tools, the Kronecker form for a singular quadratic
pencil, and the reduction of bilinear forms over a non-commutative algebra.
Finally, we show that the second secret in the IP2S problem may be recovered in
cubic time
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