On bivariate Hensel and its parallelization

2018

Our goal is to develop a high-performance code for factoring a multivariate polynomial in n variables with integer coefficients which is polynomial time in the sparse case and efficient in the dense case. Maple, Magma, Macsyma, Singular and Mathematica all implement Wang’s multivariate Hensel lifting, which, for sparse polynomials, can be exponential in n. Wang’s algorithm is also highly sequential. In this work we reorganize multivariate Hensel lifting to facilitate a highperformance parallel implementation. We identify multivariate polynomial evaluation and bivariate Hensel lifting as two core components. We have also developed a library of algorithms for polynomial arithmetic which allow us to assign each core an independent task with all the memory it needs in advance so that memory management is eliminated and all important operations operate on dense arrays of 64 bit integers. We have implemented our algorithm and library using Cilk C for the case of two monic factors. We disc...

Proceedings of the 2004 international symposium on Symbolic and algebraic computation - ISSAC '04, 2004

Many polynomial factorization algorithms rely on Hensel lifting and factor recombination. For bivariate polynomials we show that lifting the factors up to a precision linear in the total degree of the polynomial to be factored is sufficient to deduce the recombination by linear algebra, using trace recombination. Then, the total cost of the lifting and the recombination stage is subquadratic in the size of the dense representation of the input polynomial. Lifting is often the practical bottleneck of this method: we propose an algorithm based on a faster multi-moduli computation for univariate polynomials and show that it saves a constant factor compared to the classical multifactor lifting algorithm.

ACM Communications in Computer Algebra, 2019

Table of Contents iv List of Tables vi List of Figures vii List of Algorithms viii

Journal of Symbolic Computation, 2020

The standard approach to factor a multivariate polynomial in Z[x1, x2,. .. , xn] is to factor a univariate image in Z[x1] then recover the multivariate factors from their univariate images using a process known as multivariate Hensel lifting. Wang's multivariate Hensel lifting recovers the variables one at a time. It is currently implemented in many computer algebra systems, including Maple, Magma and Singular. When the factors are sparse, Wang's approach can be exponential in the number of variables n. To address this, sparse Hensel lifting was introduced by Zippel and then improved by Kaltofen. Recently, Monagan & Tuncer introduced a new approach which uses sparse polynomial interpolation to solve the multivariate polynomial diophantine equations that arise inside Hensel lifting in random polynomial time. This approach is shown to be practical and faster than Zippel's and Kaltofen's algorithms and faster than Wang's algorithm for non-zero evaluation points. In this work we first present a complete description of the sparse interpolation used by Monagan & Tuncer and show that it runs in random polynomial time. Next we study what happens to the sparsity of multivariate polynomials when the variables are successively evaluated at numbers. We determine the expected number of remaining terms. We use this result to revisit and correct the complexity analysis of Zippel's original sparse interpolation. Next we present an average case complexity analysis of our approach. We have implemented our algorithm in Maple with some sub-algorithms implemented in C. We present some experimental data comparing our approach with Wang's method for both sparse and dense factors. The data shows that our method is always competitive with Wang's method and faster when Wang's method is exponential in n.

ACM Communications in Computer Algebra

Table of Contents iv List of Tables vi List of Figures vii List of Algorithms viii

IEEE Transactions on Signal Processing, 1992

The initial value problem arising in the recursive evaluation of a 2-D polynomial at equispaced points is treated in detail; the results facilitate efficient implementation of Bose's recursive algorithm. A comparison has been made of the computational complexity with that involved in a direct computation, and some general observations have been made for an alternative scheme proposed by Nie and Unbehauen.

Proceedings of the 2009 international symposium on Symbolic and algebraic computation - ISSAC '09, 2009

We present an efficient algorithm for factoring a multivariate polynomial f ∈ L[x1,. .. , xv] where L is an algebraic function field with k ≥ 0 parameters t1,. .. , t k and r ≥ 0 field extensions. Our algorithm uses Hensel lifting and extends the EEZ algorithm of Wang which was designed for factorization over Q. We also give a multivariate p-adic lifting algorithm which uses sparse interpolation. This enables us to avoid using poor bounds on the size of the integer coefficients in the factorization of f when using Hensel lifting. We have implemented our algorithm in Maple 13. We provide timings demonstrating the efficiency of our algorithm.

Siam Journal on Computing, 1990

The authors consider the problem of reconstructing (i.e., interpolating) a t-sparse multivariate polynomial given a black box which will produce the value of the polynomial for any value of the arguments. It is shown that, if the polynomial has coefficients in a finite field GF[q] and the black box can evaluate the polynomial in the field GF[qr2g,tnt+37], where n is the number of variables, then there is an algorithm to interpolate the polynomial in O(log (nt)) boolean parallel time and O(n2t log nt) processors.

2012

This paper aims to develop and analyze an effective parallel algorithm for multiplying polynomials and power series with integer coefficients. Such operations are of fundamental importance when generating parameters for public key cryptosystems, whereas their effective implementation translates directly into the speed of such algorithms in practical applications. The algorithm has been designed specifically to accelerate the process of generating modular polynomials, but due to its good numerical properties it may surely be used to solve other computational problems as well. The basic idea behind this new method was to adapt it to parallel computing. Nowadays, it is a very important property, as it allows to fully exploit the computing power offered by modern processors. The combination of the Chinese Remainder Theorem and the Fast Fourier Transform made it possible to develop a highly effective multiplication method. Under certain conditions, it is asymptotically faster than the al...

Parallel computing, 1997

The performance of the Knuth-Eve algorithm for data parallel evaluation of univariate polynomials of degree 8, 16 and 32 has been systematically compared with that of the classical Newton-Homer algorithm using three vector processors and three processor arrays. Significant performance improvements have been observed beyond a critical problem size, as predicted by theory. The practical implications of this result for large-scale computations involving function approximation are outlined. 0 1997 Elsevier Science B.V.