Nearly optimal refinement of real roots of a univariate polynomial

Computing the Supremum of the Real Roots of a Parametric Univariate Polynomial ( extended abstract )

Given a bivariate polynomial p(W,X) we aim at computing the supremum of the real values x such that there exists a real value w satisfying p(w, x) = 0. We allow the coefficients of p to depend on real parameters. Our approach relies on the notion of border polynomial and takes advantage of triangular decomposition techniques. We report on the implementation of our algorithm and illustrate its e...

New method for bounding the roots of a univariate polynomial

We present a new algorithm for computing upper bounds for the maximum positive real root of a univariate polynomial. The algorithm improves complexity and accuracy of current methods. These improvements do impact in the performance of methods for root isolation, which are the first step (and most expensive, in terms of computational effort) executed by current methods for computing the real roo...

Nearly Optimal Algorithms for Univariate Polynomial Factorization and Rootfinding II: Computing a Basic Annulus for Splitting

A Polynomial Algorithm for Optimal Univariate Microaggregation

Microaggregation is a technique used by statistical agencies to limit disclosure of sensitive microdata. Noting that no polynomial algorithms are known to microaggregate optimally, Domingo-Ferrer and Mateo-Sanz have presented heuristic microaggregation methods. This paper is the first to present an efficient polynomial algorithm for optimal univariate microaggregation. Optimal partitions are sh...

Rounding LLL: Finding Faster Small Roots of Univariate Polynomial Congruences

In a seminal work at EUROCRYPT ’96, Coppersmith showed how to find all small roots of a univariate polynomial congruence in polynomial time: this has found many applications in public-key cryptanalysis and in a few security proofs. However, the running time of the algorithm is a high-degree polynomial, which limits experiments: the bottleneck is an LLL reduction of a high-dimensional matrix wit...