Directed evaluation

Hoeven, Joris van der; Lecerf, Grégoire

doi:10.1016/j.jco.2020.101498

Cited by 16 publications

(19 citation statements)

References 31 publications

(64 reference statements)

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“…Our method builds on the long line of techniques known as dynamic evaluation, or the D 5 principle [6]. is is a very general technique which has since been employed for a wide range of computational problems [9,27,10,29,5,32,7,16,33]. For our purposes the idea is to start computing modulo a possibly-composite , and "split" the evaluation with an efficient GCD computation whenever we need to test for zero or perform a division.…”

Section: Algorithmic Transformation Techniquementioning

confidence: 99%

“…Like the recent directed evaluation technique proposed by van der Hoeven and Lecerf [16], we opt to first take the larger-size branch in any GCD spli ing. But unlike their method (and prior work) which is ultimately focused on recovering the correct result over the original product of fields, here the goal is only to have an answer which is consistent with what it would have been in some randomly-chosen finite field.…”

Section: Algorithmic Transformation Techniquementioning

confidence: 99%

“…To our knowledge, previous applications of the D 5 principle have considered only computational models which are deterministic and do not allow random sampling of field elements. van der Hoeven and Lecerf [16] suggest that this can be overcome by providing the deterministic computation tree with a "pool" of pre-selected random field elements, and the recent paper of Neiger, Salvy, Schost, and Villard [31] mentions this limitation and also skirts around it by providing pre-selected "random" choices as additional algorithm inputs.…”

Section: Algorithmic Transformation Techniquementioning

confidence: 99%

See 2 more Smart Citations

Random primes without primality testing

Giorgi¹,

Grenet²,

Cray³

et al. 2022

Preprint

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Numerous algorithms call for computation over the integers modulo a randomly-chosen large prime. In some cases, the quasi-cubic complexity of selecting a random prime can dominate the total running time. We propose a new variant of the classic D5 algorithm for "dynamic evaluation", applied to a randomly-chosen (composite) integer. Unlike the D5 principle which has been used in the past to compute over a direct product of fields, our method is simpler as it only requires following a single path a er any modulus splits. e transformation we propose can apply to any algorithm in the algebraic RAM model, even allowing randomization. e resulting transformed algorithm avoids any primality tests and will, with constant positive probability, have the same result as the original computation modulo a randomly-chosen prime. As an application, we demonstrate how to compute the exact number of nonzero terms in an unknown integer polynomial in quasi-linear time. We also show how the same algorithmic transformation technique can be used for computing modulo random irreducible polynomials over a finite field.

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Section: Algorithmic Transformation Techniquementioning

confidence: 99%

Section: Algorithmic Transformation Techniquementioning

confidence: 99%

Section: Algorithmic Transformation Techniquementioning

confidence: 99%

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Random primes without primality testing

Giorgi¹,

Grenet²,

Cray³

et al. 2022

Preprint

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show abstract

“…Another standard reference is [33]. Concerning complexity results, see [8] that brings fast univariate arithmetic to this multivariate context, or more recently [41]. The splitting can also be performed in an ideal-theoretic way through Gröbner bases computations, see [37].…”

Section: Previous and Related Workmentioning

confidence: 99%

Lexicographic Groebner bases of bivariate polynomials modulo a univariate one

Dahan

2020

Preprint

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Let T (x) ∈ k[x] be a monic non-constant polynomial and write R = k[x]/ T the quotient ring. Consider two bivariate polynomials a(x, y), b(x, y) ∈ R [y]. In a first part, T = p e is assumed to be the power of an irreducible polynomial p. A new algorithm that computes a minimal lexicographic Gröbner basis of the ideal a, b, p e , is introduced. A second part extends this algorithm when T is general through the "local/global" principle realized by a generalization of "dynamic evaluation", restricted so far to a polynomial T that is squarefree. The algorithm produces splittings according to the case distinction "invertible/nilpotent", extending the usual "invertible/zero" in classic dynamic evaluation. This algorithm belongs to the Euclidean family, the core being a subresultant sequence of a and b modulo T . In particular no factorization or Gröbner basis computations are necessary. The theoretical background relies on the Lazard's structural theorem for lexicographic Gröbner bases in two variables. An implementation is realized in Magma, and extensive benchmarks are provided that show clearly the benefits, sometimes huge, of this approach compared to the computation of Gröbner bases.

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“…This would make all algorithms deterministic, with a cost O(δ 1+o (1) ) instead of O˜(δ). Note also [30] for dynamic evaluation.…”

Section: Complexity Modelmentioning

confidence: 99%

Using approximate roots for irreducibility and equi-singularity issues in K[[x]][y]

Poteaux,

Weimann

2019

Preprint

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Directed evaluation

Cited by 16 publications

References 31 publications

Random primes without primality testing

Random primes without primality testing

Lexicographic Groebner bases of bivariate polynomials modulo a univariate one

Using approximate roots for irreducibility and equi-singularity issues in K[[x]][y]

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