We show how to solve a polynomial equation (mod N ) of degree k in a single variable z, as long as there is a solution smaller than “Ik. We give two applications to RSA encryption with exponent 3. First, knowledge of all the ciphertext and 2/3 of the plaintext bits for a single message reveals that message. Second, if messages are padded with truly random padding and then encrypted with an expo...

A method to solve the problem f(x) = 0 efficiently on any n-dimensional domain Ω under very broad hypoteses is proposed. The position of the root of f , assumed unique, is found by computing the center of mass of an Ω-shaped object having a singular mass density. It is shown that although the mass of the object is infinite, the position of its center of mass can be computed exactly and correspo...

Univariate polynomial root-finding is both classical and important for modern computing. Frequently one seeks just the real roots of a polynomial with real coefficients. They can be approximated at a low computational cost if the polynomial has no nonreal roots, but typically nonreal roots are much more numerous than the real ones. We dramatically accelerate the known algorithms in this case by...

In this paper we consider the application of polynomial root-finding methods to the solution of the tridiagonal matrix eigenproblem. All considered solvers are based on evaluating the Newton correction. We show that the use of scaled three-term recurrence relations complemented with error free transformations yields some compensated schemes which significantly improve the accuracy of computed r...

The problem of finding low rank m × m matrices in a real affine subspace of dimension n has many applications in information and systems theory, where low rank is synonymous of structure and parcimony. We design a symbolic computation algorithm to solve this problem efficiently, exactly and rigorously: the input are the rational coefficients of the matrices spanning the affine subspace as well ...

We show that all algebraic-geometric codes possess a succinct representation that allows for the list decoding algorithms of [15, 7] to run in polynomial time. We do this by presenting a root-finding algorithm for univariate polynomials over function fields when their coefficients lie in finite-dimensional linear spaces, and proving that there is a polynomial size representation given which the...

A family of regula falsi methods for finding a root ξ of a nonlinear equation f(x)=0 in the interval [a,b] is studied in this paper. The Illinois, Pegasus, Anderson & Björk and more nine new proposed methods have been tested on a series of examples. The new methods, inspired on Pegasus procedure, are pedagogically important algorithms. The numerical results empirically show that some of new met...

A new iterative method of the fourth order for solving nonlinear equations of the form f(x) = 0 is derived. The comparison with other existing methods is performed regarding the computational efficiency and numerical examples. The fourth-order derivative free method for the simultaneous calculation of all polynomial zeros, arising from the proposed method, is also studied. AMS Mathematics Subje...

Matrix methods are increasingly popular for polynomial root-finding. The idea is to approximate the roots as the eigenvalues of the companion or generalized companion matrix associated with an input polynomial. The algorithms also solve secular equation. QR algorithm is the most customary method for eigen-solving, but we explore the inverse Rayleigh quotient iteration instead, which turns out t...

Keywords--Determinat ion of polynomial zeros, Simultaneous iterative methods, Convergence analysis, Accelerated convergence, R-order of convergence. 1. I N T R O D U C T I O N The problem of determining polynomial zeros has a great impor tance in theory and practice (for instance, in the theory of control systems, digital signal processing, s tabil i ty of systems, analysis of transfer function...