5,320 research outputs found
Note on Integer Factoring Methods IV
This note continues the theoretical development of deterministic integer
factorization algorithms based on systems of polynomials equations. The main
result establishes a new deterministic time complexity bench mark in integer
factorization.Comment: 20 Pages, New Versio
On tori triangulations associated with two-dimensional continued fractions of cubic irrationalities
We show several properties related to the structure of the family of classes
of two-dimensional periodic continued fractions. This approach to the study of
the family of classes of nonequivalent two dimexsional periodic continued
fractions leads to the visualization of special subfamilies of continued
fractions with torus triangulations (i.e. combinatorics of their fundamental
domains) that possess explicit regularities.Several cases of such subfamilies
are studied in detail; the method to construct other similar subfamilies is
given
Computing sparse multiples of polynomials
We consider the problem of finding a sparse multiple of a polynomial. Given f
in F[x] of degree d over a field F, and a desired sparsity t, our goal is to
determine if there exists a multiple h in F[x] of f such that h has at most t
non-zero terms, and if so, to find such an h. When F=Q and t is constant, we
give a polynomial-time algorithm in d and the size of coefficients in h. When F
is a finite field, we show that the problem is at least as hard as determining
the multiplicative order of elements in an extension field of F (a problem
thought to have complexity similar to that of factoring integers), and this
lower bound is tight when t=2.Comment: Extended abstract appears in Proc. ISAAC 2010, pp. 266-278, LNCS 650
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
The Minimal Resultant Locus
Let K be a complete, algebraically closed nonarchimedean valued field, and
let f(z) in K(z) be a rational function of degree d at least 2. We give an
algorithm to determine whether f(z) has potential good reduction over K, based
on a geometric reformulation of the problem using the Berkovich Projective
Line. We show the minimal resultant is is either achieved at a single point in
the Berkovich line, or on a segment, and that minimal resultant locus is
contained in the tree in spanned by the fixed points and the poles of f(z).
When f(z) is defined over the rationals, the algorithm runs in probabilistic
polynomial time. If f(z) has potential good reduction, and is defined over a
subfield H of K, we show there is an extension L/H in K with degree at most (d
+ 1)^2 such that f(z) achieves good reduction over L.Comment: 37 page
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