53 research outputs found
Fast Numerical Multivariate Multipoint Evaluation
We design nearly-linear time numerical algorithms for the problem of
multivariate multipoint evaluation over the fields of rational, real and
complex numbers. We consider both \emph{exact} and \emph{approximate} versions
of the algorithm. The input to the algorithms are (1) coefficients of an
-variate polynomial with degree in each variable, and (2) points
each of whose coordinate has value bounded by one and
bit-complexity .
* Approximate version: Given additionally an accuracy parameter , the
algorithm computes rational numbers such that
for all , and has a running time of
for all and all sufficiently large .
* Exact version (when over rationals): Given additionally a bound on the
bit-complexity of all evaluations, the algorithm computes the rational numbers
, in time for all
and all sufficiently large . .
Prior to this work, a nearly-linear time algorithm for multivariate
multipoint evaluation (exact or approximate) over any infinite field appears to
be known only for the case of univariate polynomials, and was discovered in a
recent work of Moroz (FOCS 2021). In this work, we extend this result from the
univariate to the multivariate setting. However, our algorithm is based on
ideas that seem to be conceptually different from those of Moroz (FOCS 2021)
and crucially relies on a recent algorithm of Bhargava, Ghosh, Guo, Kumar &
Umans (FOCS 2022) for multivariate multipoint evaluation over finite fields,
and known efficient algorithms for the problems of rational number
reconstruction and fast Chinese remaindering in computational number theory
The complexity of class polynomial computation via floating point approximations
We analyse the complexity of computing class polynomials, that are an
important ingredient for CM constructions of elliptic curves, via complex
floating point approximations of their roots. The heart of the algorithm is the
evaluation of modular functions in several arguments. The fastest one of the
presented approaches uses a technique devised by Dupont to evaluate modular
functions by Newton iterations on an expression involving the
arithmetic-geometric mean. It runs in time for any , where
is the CM discriminant and is the degree of the class polynomial.
Another fast algorithm uses multipoint evaluation techniques known from
symbolic computation; its asymptotic complexity is worse by a factor of . Up to logarithmic factors, this running time matches the size of the
constructed polynomials. The estimate also relies on a new result concerning
the complexity of enumerating the class group of an imaginary-quadratic order
and on a rigorously proven upper bound for the height of class polynomials
Harnessing the power of GPUs for problems in real algebraic geometry
This thesis presents novel parallel algorithms to leverage the power of GPUs (Graphics Processing Units) for exact computations with polynomials having large integer coefficients. The significance of such computations, especially in real algebraic geometry, is hard to undermine. On massively-parallel architectures such as GPU, the degree of datalevel parallelism exposed by an algorithm is the main performance factor. We attain high efficiency through the use of structured matrix theory to assist the realization of relevant operations on polynomials on the graphics hardware. A detailed complexity analysis, assuming the PRAM model, also confirms that our approach achieves a substantially better parallel complexity in comparison to classical algorithms used for symbolic computations. Aside from the theoretical considerations, a large portion of this work is dedicated to the actual algorithm development and optimization techniques where we pay close attention to the specifics of the graphics hardware. As a byproduct of this work, we have developed high-throughput modular arithmetic which we expect to be useful for other GPU applications, in particular, open-key cryptography. We further discuss the algorithms for the solution of a system of polynomial equations, topology computation of algebraic curves and curve visualization which can profit to the full extent from the GPU acceleration. Extensive benchmarking on a real data demonstrates the superiority of our algorithms over several state-of-the-art approaches available to date. This thesis is written in English.Diese Arbeit beschäftigt sich mit neuen parallelen Algorithmen, die das Leistungspotenzial der Grafik-Prozessoren (GPUs) zur exakten Berechnungen mit ganzzahlige Polynomen nutzen. Solche symbolische Berechnungen sind von großer Bedeutung zur Lösung vieler Probleme aus der reellen algebraischen Geometrie. Für die effziente Implementierung eines Algorithmus auf massiv-parallelen Hardwarearchitekturen, wie z.B. GPU, ist vor allem auf eine hohe Datenparallelität zu achten. Unter Verwendung von Ergebnissen aus der strukturierten Matrix-Theorie konnten wir die entsprechenden Operationen mit Polynomen auf der Grafikkarte leicht übertragen. Außerdem zeigt eine Komplexitätanalyse im PRAM-Rechenmodell, dass die von uns entwickelten Verfahren eine deutlich bessere Komplexität aufweisen als dies für die klassischen Verfahren der Fall ist. Neben dem theoretischen Ergebnis liegt ein weiterer Schwerpunkt dieser Arbeit in der praktischen Implementierung der betrachteten Algorithmen, wobei wir auf der Besonderheiten der Grafikhardware achten. Im Rahmen dieser Arbeit haben wir hocheffiziente modulare Arithmetik entwickelt, von der wir erwarten, dass sie sich für andere GPU Anwendungen, insbesondere der Public-Key-Kryptographie, als nützlich erweisen wird. Darüber hinaus betrachten wir Algorithmen für die Lösung eines Systems von Polynomgleichungen, Topologie Berechnung der algebraischen Kurven und deren Visualisierung welche in vollem Umfang von der GPU-Leistung profitieren können. Zahlreiche Experimente belegen dass wir zur Zeit die beste Verfahren zur Verfügung stellen. Diese Dissertation ist in englischer Sprache verfasst
Faster Geometric Algorithms via Dynamic Determinant Computation
The computation of determinants or their signs is the core procedure in many
important geometric algorithms, such as convex hull, volume and point location.
As the dimension of the computation space grows, a higher percentage of the
total computation time is consumed by these computations. In this paper we
study the sequences of determinants that appear in geometric algorithms. The
computation of a single determinant is accelerated by using the information
from the previous computations in that sequence.
We propose two dynamic determinant algorithms with quadratic arithmetic
complexity when employed in convex hull and volume computations, and with
linear arithmetic complexity when used in point location problems. We implement
the proposed algorithms and perform an extensive experimental analysis. On one
hand, our analysis serves as a performance study of state-of-the-art
determinant algorithms and implementations. On the other hand, we demonstrate
the supremacy of our methods over state-of-the-art implementations of
determinant and geometric algorithms. Our experimental results include a 20 and
78 times speed-up in volume and point location computations in dimension 6 and
11 respectively.Comment: 29 pages, 8 figures, 3 table
Sparse Polynomial Interpolation and Testing
Interpolation is the process of learning an unknown polynomial f from some set of its evaluations. We consider the interpolation of a sparse polynomial, i.e., where f is comprised of a small, bounded number of terms. Sparse interpolation dates back to work in the late 18th century by the French mathematician Gaspard de Prony, and was revitalized in the 1980s due to advancements by Ben-Or and Tiwari, Blahut, and Zippel, amongst others. Sparse interpolation has applications to learning theory, signal processing, error-correcting codes, and symbolic computation. Closely related to sparse interpolation are two decision problems. Sparse polynomial identity testing is the problem of testing whether a sparse polynomial f is zero from its evaluations. Sparsity testing is the problem of testing whether f is in fact sparse.
We present effective probabilistic algebraic algorithms for the interpolation and testing of sparse polynomials. These algorithms assume black-box evaluation access, whereby the algorithm may specify the evaluation points. We measure algorithmic costs with respect to the number and types of queries to a black-box oracle.
Building on previous work by Garg–Schost and Giesbrecht–Roche, we present two methods for the interpolation of a sparse polynomial modelled by a straight-line program (SLP): a sequence of arithmetic instructions. We present probabilistic algorithms for the sparse interpolation of an SLP, with cost softly-linear in the sparsity of the interpolant: its number of nonzero terms. As an application of these techniques, we give a multiplication algorithm for sparse polynomials, with cost that is sensitive to the size of the output.
Multivariate interpolation reduces to univariate interpolation by way of Kronecker substitu- tion, which maps an n-variate polynomial f to a univariate image with degree exponential in n. We present an alternative method of randomized Kronecker substitutions, whereby one can more efficiently reconstruct a sparse interpolant f from multiple univariate images of considerably reduced degree.
In error-correcting interpolation, we suppose that some bounded number of evaluations may be erroneous. We present an algorithm for error-correcting interpolation of polynomials that are sparse under the Chebyshev basis. In addition we give a method which reduces sparse Chebyshev-basis interpolation to monomial-basis interpolation.
Lastly, we study the class of Boolean functions that admit a sparse Fourier representation. We give an analysis of Levin’s Sparse Fourier Transform algorithm for such functions. Moreover, we give a new algorithm for testing whether a Boolean function is Fourier-sparse. This method reduces sparsity testing to homomorphism testing, which in turn may be solved by the Blum–Luby–Rubinfeld linearity test
Simultaneous Diagonalization of Incomplete Matrices and Applications
We consider the problem of recovering the entries of diagonal matrices
for from multiple "incomplete" samples
of the form , where and are unknown matrices of low rank. We
devise practical algorithms for this problem depending on the ranks of and
. This problem finds its motivation in cryptanalysis: we show how to
significantly improve previous algorithms for solving the approximate common
divisor problem and breaking CLT13 cryptographic multilinear maps.Comment: 16 page
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