We depart from our approximation of 2000 all root radii a polynomial, which has readily extended Schönhage’s efficient algorithm 1982 for single radius. revisit this extension, advance it, based on simple but novel idea, and yield significant practical acceleration the known near optimal subdivision algorithms complex real root-finding user’s choice. achieve by means saving exclusion tests Tayl...

There has always been a close relationship between mathematics and the sciences (astronomy, chemistry, engineering, physics, etc.). When applying mathematics to study some physical phenomenon, it is often the case that we arrive at a mathematical model (an equation or set of equations) of the phenomenon that cannot be conveniently solved with exact formulas. The following classical example from...

In this paper, we rst show that there are several equivalent keys for t + 1 chosen plaintexts if the degree of the reduced cipher is t?1. This is against the claim by Jakobsen and Knudsen. We also derive an upper bound on the number of equivalent last round keys for t + 1 chosen plaintexts. We further show an eecient method which nds all the equivalent keys by using Rabin's root nding algorithm...

Root isolation of univariate polynomials is one of the fundamental problems in computational algebra. It aims to find disjoint regions on the real line or complex plane, each containing a single root of a given polynomial, such that the union of the regions comprises all roots. For root solving over the field of complex numbers, numerical methods are the de facto standard. They are known to be ...