# Q-adic Transform revisited

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Abstract : We present an algorithm to perform a simultaneous modular reduction of several residues. This algorithm is applied fast modular polynomial multiplication. The idea is to convert the $X$-adic representation of modular polynomials, with $X$ an indeterminate, to a $q$-adic representation where $q$ is an integer larger than the field characteristic. With some control on the different involved sizes it is then possible to perform some of the $q$-adic arithmetic directly with machine integers or floating points. Depending also on the number of performed numerical operations one can then convert back to the $q$-adic or $X$-adic representation and eventually mod out high residues. In this note we present a new version of both conversions: more tabulations and a way to reduce the number of divisions involved in the process are presented. The polynomial multiplication is then applied to arithmetic in small finite field extensions.
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https://hal.archives-ouvertes.fr/hal-00173894
Contributor : Jean-Guillaume Dumas <>
Submitted on : Monday, June 23, 2008 - 9:59:36 AM
Last modification on : Thursday, July 4, 2019 - 9:54:02 AM
Long-term archiving on : Friday, November 25, 2016 - 9:50:41 PM

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Jean-Guillaume Dumas. Q-adic Transform revisited. ISSAC '08 - 21st International Symposium on Symbolic and Algebraic Computation 2008, Jul 2008, Hagenberg, Austria. pp.63-70, ⟨10.1145/1390768.1390780⟩. ⟨hal-00173894v6⟩

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