Randomized root finding over finite FFT-fields using tangent Graeffe transforms

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Abstract

Consider 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.

Original languageEnglish
Title of host publicationISSAC 2015 - Proceedings of the 2015 ACM International Symposium on Symbolic and Algebraic Computation
PublisherAssociation for Computing Machinery
Pages197-204
Number of pages8
ISBN (Electronic)9781450334358
DOIs
Publication statusPublished - 24 Jun 2015
Event40th ACM International Symposium on Symbolic and Algebraic Computation, ISSAC 2015 - Bath, United Kingdom
Duration: 6 Jul 20159 Jul 2015

Publication series

NameProceedings of the International Symposium on Symbolic and Algebraic Computation, ISSAC
Volume2015-June

Conference

Conference40th ACM International Symposium on Symbolic and Algebraic Computation, ISSAC 2015
Country/TerritoryUnited Kingdom
CityBath
Period6/07/159/07/15

Keywords

  • Algorithm
  • Finite fields
  • Mathemagix
  • Polynomial root finding

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