7,799 research outputs found
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
A space-time pseudospectral discretization method for solving diffusion optimal control problems with two-sided fractional derivatives
We propose a direct numerical method for the solution of an optimal control
problem governed by a two-side space-fractional diffusion equation. The
presented method contains two main steps. In the first step, the space variable
is discretized by using the Jacobi-Gauss pseudospectral discretization and, in
this way, the original problem is transformed into a classical integer-order
optimal control problem. The main challenge, which we faced in this step, is to
derive the left and right fractional differentiation matrices. In this respect,
novel techniques for derivation of these matrices are presented. In the second
step, the Legendre-Gauss-Radau pseudospectral method is employed. With these
two steps, the original problem is converted into a convex quadratic
optimization problem, which can be solved efficiently by available methods. Our
approach can be easily implemented and extended to cover fractional optimal
control problems with state constraints. Five test examples are provided to
demonstrate the efficiency and validity of the presented method. The results
show that our method reaches the solutions with good accuracy and a low CPU
time.Comment: This is a preprint of a paper whose final and definite form is with
'Journal of Vibration and Control', available from
[http://journals.sagepub.com/home/jvc]. Submitted 02-June-2018; Revised
03-Sept-2018; Accepted 12-Oct-201
An elementary approach to the quasipolynomiality of the Kronecker coefficients
The Kronecker coefficients are the structure constants for the restriction of
irreducible representations of the general linear group GL(nm) into
irreducibles for the subgroup GL(n)xGL(m). In this work we study the piecewise
quasipolynomial nature of the Kronecker function using tools from polyhedral
geometry. By bounding the lengths of the partitions, we write the Kronecker
function in terms of coefficients of vector partition functions. We illustrate
the power of this approach in the first nontrivial case: we give exact formulas
and an upper bound for the Kronecker coefficients, and derive other properties.
An additional advantage of this approach is that asymptotic estimates for
dilations are computable using techniques of analytic combinatorics in several
variables. We also show how our approach can be applied to other families of
structure constants.Comment: 44 pages; some results on chambers added, revised introductio
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