Deterministic root finding over finite fields using Graeffe transforms

Grenet, Bruno; Hoeven, Joris van der; Lecerf, Grégoire

doi:10.1007/s00200-015-0280-5

Cited by 19 publications

(22 citation statements)

References 46 publications

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“…This result was already known in the linear case: Bostan et al (2005) gave the first proof and then Grenet, Hoeven, and Lecerf (2015) gave a simpler one. In the non-linear case, Lercier and Sirvent (2008) obtained a weaker bound: they showed that an approximation modulo (p κ , t n ) can be computed from approximations modulo (p κ+O(log(n) 2 ) , t n ) of g and h. A preliminary version of the present work appeared in Vaccon's PhD thesis (Vaccon 2015).…”

Section: Resultsmentioning

confidence: 63%

On p-Adic Differential Equations with Separation of Variables

Lairez

Vaccon

2016

Proceedings of the ACM on International Symposium on Symbolic and Algebraic Computation

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Several algorithms in computer algebra involve the computation of a power series solution of a given ordinary differential equation. Over finite fields, the problem is often lifted in an approximate p-adic setting to be well-posed. This raises precision concerns: how much precision do we need on the input to compute the output accurately? In the case of ordinary differential equations with separation of variables, we make use of the recent technique of differential precision to obtain optimal bounds on the stability of the Newton iteration. The results apply, for example, to algorithms for manipulating algebraic numbers over finite fields, for computing isogenies between elliptic curves or for deterministically finding roots of polynomials in finite fields. The new bounds lead to significant speedups in practice.

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Section: Resultsmentioning

confidence: 63%

On p-Adic Differential Equations with Separation of Variables

Lairez

Vaccon

2016

Proceedings of the ACM on International Symposium on Symbolic and Algebraic Computation

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“…This larger prime starts to reveal the interest in our modified version of Cantor-Zassenhaus with χ > 2. The speed-up measured with the modified version even more increases with larger primes: With p = 7 · 2 120 + 1, it reaches a factor 1.6 in size 2 14 .…”

Section: Timingsmentioning

confidence: 91%

“…One may then find the roots of h m−1 among the π m -th roots of the roots of h m , and, by induction, the roots of h i−1 are to be found among the π i -th roots of the roots of h i . This is the starting point of the efficient deterministic algorithm designed in [14]. But in order to make it much more efficient than Cantor-Zassenhaus' algorithm we introduce and analyze two types of randomizations in the next sections.…”

Section: Overview Of Our Methodsmentioning

confidence: 99%

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Randomized Root Finding over Finite FFT-fields using Tangent Graeffe Transforms

Grenet

Hoeven

Lecerf

2015

Proceedings of the 2015 ACM International Symposium on Symbolic and Algebraic Computation

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International audienceConsider a finite field Fq whose multiplicative group has smooth cardinality. We study the problem of computing all roots of a polynomial that splits over Fq, which was one of the bottlenecks for fast sparse interpolation in practice. We revisit and slightly improve existing algorithms and then present new randomized ones based on the Graeffe transform. We report on our implementation in the MATHEMAGIX computer algebra system, confirming that our ideas gain by a factor ten at least in practice, for sufficiently large inputs

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Implementing the Tangent Graeffe Root Finding Method

Hoeven¹,

Monagan²

2020

Lecture Notes in Computer Science

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The tangent Graeffe method has been developed for the efficient computation of single roots of polynomials over finite fields with multiplicative groups of smooth order. It is a key ingredient of sparse interpolation using geometric progressions, in the case when blackbox evaluations are comparatively cheap. In this paper, we improve the complexity of the method by a constant factor and we report on a new implementation of the method and a first parallel implementation.Note: This paper received funding from NSERC (Canada) and "Agence de l'innovation de défense" (France). Note: This document has been written using GNU T E X macs [13].

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Deterministic root finding over finite fields using Graeffe transforms

Cited by 19 publications

References 46 publications

On p-Adic Differential Equations with Separation of Variables

On p-Adic Differential Equations with Separation of Variables

Randomized Root Finding over Finite FFT-fields using Tangent Graeffe Transforms

Implementing the Tangent Graeffe Root Finding Method

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