12,374 research outputs found
Functional Decomposition using Principal Subfields
Let be a univariate rational function. It is well known that any
non-trivial decomposition , with , corresponds to a
non-trivial subfield and vice-versa. In
this paper we use the idea of principal subfields and fast
subfield-intersection techniques to compute the subfield lattice of
. This yields a Las Vegas type algorithm with improved complexity
and better run times for finding all non-equivalent complete decompositions of
.Comment: 8 pages, accepted for ISSAC'1
Computing discrete logarithms in subfields of residue class rings
Recent breakthrough methods \cite{gggz,joux,bgjt} on computing discrete
logarithms in small characteristic finite fields share an interesting feature
in common with the earlier medium prime function field sieve method \cite{jl}.
To solve discrete logarithms in a finite extension of a finite field \F, a
polynomial h(x) \in \F[x] of a special form is constructed with an
irreducible factor g(x) \in \F[x] of the desired degree. The special form of
is then exploited in generating multiplicative relations that hold in
the residue class ring \F[x]/h(x)\F[x] hence also in the target residue class
field \F[x]/g(x)\F[x]. An interesting question in this context and addressed
in this paper is: when and how does a set of relations on the residue class
ring determine the discrete logarithms in the finite fields contained in it? We
give necessary and sufficient conditions for a set of relations on the residue
class ring to determine discrete logarithms in the finite fields contained in
it. We also present efficient algorithms to derive discrete logarithms from the
relations when the conditions are met. The derived necessary conditions allow
us to clearly identify structural obstructions intrinsic to the special
polynomial in each of the aforementioned methods, and propose
modifications to the selection of so as to avoid obstructions.Comment: arXiv admin note: substantial text overlap with arXiv:1312.167
Computation of Galois groups of rational polynomials
Computational Galois theory, in particular the problem of computing the
Galois group of a given polynomial is a very old problem. Currently, the best
algorithmic solution is Stauduhar's method. Computationally, one of the key
challenges in the application of Stauduhar's method is to find, for a given
pair of groups H<G a G-relative H-invariant, that is a multivariate polynomial
F that is H-invariant, but not G-invariant. While generic, theoretical methods
are known to find such F, in general they yield impractical answers. We give a
general method for computing invariants of large degree which improves on
previous known methods, as well as various special invariants that are derived
from the structure of the groups. We then apply our new invariants to the task
of computing the Galois groups of polynomials over the rational numbers,
resulting in the first practical degree independent algorithm.Comment: Improved version and new titl
A précis of philosophy of computing and information technology
The authors recently finished a comprehensive chapter on “Philosophy of Computing and Information Technology” for the forthcoming (fall 2009) Philosophy of Technology and Engineering Sciences (Ed.: A. Meijers), Volume IX in the Elsevier series Handbook of the Philosophy of Science (Eds.: D. Gabbay, P. Thagard and J. Woods). The purpose of the chapter is to review and discuss the main developments, concepts, topics, and contributors in the intersection between philosophy and computing, as well as provide some suggestions on how to structure the many subcategories within what is loosely referred to as philosophy of computing. In this short synopsis, we will give an outline of the kinds of issues raised in this chapter
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