Hierarchy of general invariants for bivariate LPDOs

We study invariants under gauge transformations of linear partial differential operators on two variables. Using results of BK-factorization, we construct hierarchy of general invariants for operators of an arbitrary order. Properties of general invariants are studied and some examples are presented. We also show that classical Laplace invariants correspond to some particular cases of general invariants.Comment: to appear in J. "Theor.Math.Phys." in May 200

Monopolelike probes for quantitative magnetic force microscopy: calibration and application

A local magnetization measurement was performed with a Magnetic Force Microscope (MFM) to determine magnetization in domains of an exchange coupled [Co/Pt]/Co/Ru multilayer with predominant perpendicular anisotropy. The quantitative MFM measurements were conducted with an iron filled carbon nanotube tip, which is shown to behave like a monopole. As a result we determined an additional in-plane magnetization component of the multilayer, which is explained by estimating the effective permeability of the sample within the \mu*-method.Comment: 3 pages, 3 figure

Relatively Prime Polynomials and Nonsingular Hankel Matrices over Finite Fields

The probability for two monic polynomials of a positive degree n with coefficients in the finite field F_q to be relatively prime turns out to be identical with the probability for an n x n Hankel matrix over F_q to be nonsingular. Motivated by this, we give an explicit map from pairs of coprime polynomials to nonsingular Hankel matrices that explains this connection. A basic tool used here is the classical notion of Bezoutian of two polynomials. Moreover, we give simpler and direct proofs of the general formulae for the number of m-tuples of relatively prime polynomials over F_q of given degrees and for the number of n x n Hankel matrices over F_q of a given rankComment: 10 pages; to appear in the Journal of Combinatorial Theory, Series

Revisit Sparse Polynomial Interpolation based on Randomized Kronecker Substitution

In this paper, a new reduction based interpolation algorithm for black-box multivariate polynomials over finite fields is given. The method is based on two main ingredients. A new Monte Carlo method is given to reduce black-box multivariate polynomial interpolation to black-box univariate polynomial interpolation over any ring. The reduction algorithm leads to multivariate interpolation algorithms with better or the same complexities most cases when combining with various univariate interpolation algorithms. We also propose a modified univariate Ben-or and Tiwarri algorithm over the finite field, which has better total complexity than the Lagrange interpolation algorithm. Combining our reduction method and the modified univariate Ben-or and Tiwarri algorithm, we give a Monte Carlo multivariate interpolation algorithm, which has better total complexity in most cases for sparse interpolation of black-box polynomial over finite fields

A kilobit hidden SNFS discrete logarithm computation

We perform a special number field sieve discrete logarithm computation in a 1024-bit prime field. To our knowledge, this is the first kilobit-sized discrete logarithm computation ever reported for prime fields. This computation took a little over two months of calendar time on an academic cluster using the open-source CADO-NFS software. Our chosen prime $p$ looks random, and $p--1$ has a 160-bit prime factor, in line with recommended parameters for the Digital Signature Algorithm. However, our p has been trapdoored in such a way that the special number field sieve can be used to compute discrete logarithms in $\mathbb{F}\_p^*$ , yet detecting that p has this trapdoor seems out of reach. Twenty-five years ago, there was considerable controversy around the possibility of back-doored parameters for DSA. Our computations show that trapdoored primes are entirely feasible with current computing technology. We also describe special number field sieve discrete log computations carried out for multiple weak primes found in use in the wild. As can be expected from a trapdoor mechanism which we say is hard to detect, our research did not reveal any trapdoored prime in wide use. The only way for a user to defend against a hypothetical trapdoor of this kind is to require verifiably random primes

The computation of the degree of an approximate greatest common divisor of two Bernstein polynomials

This paper considers the computation of the degree t of an approximate greatest common divisor d(y) of two Bernstein polynomials f(y) and g(y), which are of degrees m and n respectively. The value of t is computed from the QR decomposition of the Sylvester resultant matrix S(f, g) and its subresultant matrices Sk(f, g), k = 2, . . . , min(m, n), where S1(f, g) = S(f, g). It is shown that the computation of t is significantly more complicated than its equivalent for two power basis polynomials because (a) Sk(f, g) can be written in several forms that differ in the complexity of the computation of their entries, (b) different forms of Sk(f, g) may yield different values of t, and (c) the binomial terms in the entries of Sk(f, g) may cause the ratio of its entry of maximum magnitude to its entry of minimum magnitude to be large, which may lead to numerical problems. It is shown that the QR decomposition and singular value decomposition (SVD) of the Sylvester matrix and its subresultant matrices yield better results than the SVD of the B´ezout matrix, and that f(y) and g(y) must be processed before computations are performed on these resultant and subresultant matrices in order to obtain good results

Fast construction of irreducible polynomials over finite fields

International audienceWe present a randomized algorithm that on input a finite field $K$ with $q$ elements and a positive integer $d$ outputs a degree $d$ irreducible polynomial in $K[x]$. The running time is $d^{1+o(1)} \times (\log q)^{5+o(1)}$ elementary operations. The $o(1)$ in $d^{1+o(1)}$ is a function of $d$ that tends to zero when $d$ tends to infinity. And the $o(1)$ in $(\log q)^{5+o(1)}$ is a function of $q$ that tends to zero when $q$ tends to infinity. In particular, the complexity is quasi-linear in the degree $d$

Discrete Logarithm in GF(2809) with FFS

International audienceThe year 2013 has seen several major complexity advances for the discrete logarithm problem in multiplicative groups of small- characteristic finite fields. These outmatch, asymptotically, the Function Field Sieve (FFS) approach, which was so far the most efficient algorithm known for this task. Yet, on the practical side, it is not clear whether the new algorithms are uniformly better than FFS. This article presents the state of the art with regard to the FFS algorithm, and reports data from a record-sized discrete logarithm computation in a prime-degree extension field

A non-linear structure-preserving matrix method for the computation of the coefficients of an approximate greatest common divisor of two Bernstein polynomials

This paper describes a non-linear structure-preserving ma trix method for the com- putation of the coefficients of an approximate greatest commo n divisor (AGCD) of degree t of two Bernstein polynomials f ( y ) and g ( y ). This method is applied to a modified form S t ( f, g ) Q t of the t th subresultant matrix S t ( f, g ) of the Sylvester resultant matrix S ( f, g ) of f ( y ) and g ( y ), where Q t is a diagonal matrix of com- binatorial terms. This modified subresultant matrix has sig nificant computational advantages with respect to the standard subresultant matri x S t ( f, g ), and it yields better results for AGCD computations. It is shown that f ( y ) and g ( y ) must be pro- cessed by three operations before S t ( f, g ) Q t is formed, and the consequence of these operations is the introduction of two parameters, α and θ , such that the entries of S t ( f, g ) Q t are non-linear functions of α, θ and the coefficients of f ( y ) and g ( y ). The values of α and θ are optimised, and it is shown that these optimal values allo w an AGCD that has a small error, and a structured low rank approxi mation of S ( f, g ), to be computed