1,402 research outputs found
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
Orbital Decay of the PSR J0045-7319/B Star Binary System: Age of Radio Pulsar and Initial Spin of Neutron Star
Recent timing observations of PSR J0045-7319 reveal that the neutron star/B
star binary orbit is decaying on a time scale of |\Porb/\dot\Porb|=0.5 Myr,
shorter than the characteristic age ( Myr) of the pulsar (Kaspi et
al.~1996a). We study mechanisms for the orbital decay. The standard weak
friction theory based on static tide requires far too short a viscous time to
explain the observed \dot\Porb. We show that dynamical tidal excitation of
g-modes in the B star can be responsible for the orbital decay. However, to
explain the observed short decay timescale, the B star must have some
significant retrograde rotation with respect to the orbit --- The retrograde
rotation brings lower-order g-modes, which couple much more strongly to the
tidal potential, into closer ``resonances'' with the orbital motion, thus
significantly enhancing the dynamical tide. A much less likely possibility is
that the g-mode damping time is much shorter than the ordinary radiative
damping time. The observed orbital decay timescale combined with a generic
orbital evolution model based on dynamical tide can be used as a ``timer'',
giving an upper limit of Myr for the age of the binary system since the
neutron star formation. Thus the characteristic age of the pulsar is not a good
age indicator. Assuming standard magnetic dipole braking for the pulsar and no
significant magnetic field decay on a timescale \lo 1 Myr, the upper limit
for the age implies that the initial spin of the neutron star at birth was
close to its current value.Comment: AASTeX, 9 pages, 3 ps figures. ApJ Letters, in pres
- …