3,416 research outputs found
Sparse implicitization by interpolation: Characterizing non-exactness and an application to computing discriminants
We revisit implicitization by interpolation in order to examine its properties in the context of sparse elimination theory. Based on the computation of a superset of the implicit support, implicitization is reduced to computing the nullspace of a numeric matrix. The approach is applicable to polynomial and rational parameterizations of curves and (hyper)surfaces of any dimension, including the case of parameterizations with base points.
Our support prediction is based on sparse (or toric) resultant theory, in order to exploit the sparsity of the input and the output. Our method may yield a multiple of the implicit equation: we characterize and quantify this situation by relating the nullspace dimension to the predicted support and its geometry. In this case, we obtain more than one multiples of the implicit equation; the latter can be obtained via multivariate polynomial gcd (or factoring).
All of the above techniques extend to the case of approximate computation, thus yielding a method of sparse approximate implicitization, which is important in tackling larger problems. We discuss our publicly available Maple implementation through several examples, including the benchmark of bicubic surface.
For a novel application, we focus on computing the discriminant of a multivariate polynomial, which characterizes the existence of multiple roots and generalizes the resultant of a polynomial system.
This yields an efficient, output-sensitive algorithm for
computing the discriminant polynomial
Implicit factorization of unbalanced RSA moduli
International audienceLet N1 = p1q1 and N2 = p2q2 be two RSA moduli, not necessarily of the same bit-size. In 2009, May and Ritzenhofen proposed a method to factor N1 and N2 given the implicit information that p1 and p2 share an amount of least significant bits. In this paper, we propose a generalization of their attack as follows: suppose that some unknown multiples a1p1 and a2p2 of the prime factors p1 and p2 share an amount of their Most Significant Bits (MSBs) or an amount of their Least Significant Bits (LSBs). Using a method based on the continued fraction algorithm, we propose a method that leads to the factorization of N1 and N2. Using simultaneous diophantine approximations and lattice reduction , we extend the method to factor k ≥ 3 RSA moduli Ni = piqi, i = 1,. .. , k given the implicit information that there exist unknown multiples a1p1,. .. , ak pk sharing an amount of their MSBs or their LSBs. Also, this paper extends many previous works where similar results were obtained when the pi's share their MSBs or their LSBs
Generalized Implicit Factorization Problem
The Implicit Factorization Problem was first introduced by May and
Ritzenhofen at PKC'09. This problem aims to factorize two RSA moduli
and when their prime factors share a certain number
of least significant bits (LSBs). They proposed a lattice-based algorithm to
tackle this problem and extended it to cover RSA moduli. Since then,
several variations of the Implicit Factorization Problem have been studied,
including the cases where and share some most significant bits
(MSBs), middle bits, or both MSBs and LSBs at the same position.
In this paper, we explore a more general case of the Implicit Factorization
Problem, where the shared bits are located at different and unknown positions
for different primes. We propose a lattice-based algorithm and analyze its
efficiency under certain conditions. We also present experimental results to
support our analysis
Formal Desingularization of Surfaces - The Jung Method Revisited -
In this paper we propose the concept of formal desingularizations as a
substitute for the resolution of algebraic varieties. Though a usual resolution
of algebraic varieties provides more information on the structure of
singularities there is evidence that the weaker concept is enough for many
computational purposes. We give a detailed study of the Jung method and show
how it facilitates an efficient computation of formal desingularizations for
projective surfaces over a field of characteristic zero, not necessarily
algebraically closed. The paper includes a generalization of Duval's Theorem on
rational Puiseux parametrizations to the multivariate case and a detailed
description of a system for multivariate algebraic power series computations.Comment: 33 pages, 2 figure
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