Arithmetic over small extension fields via floating point routines

Jean-Guillaume Dumas (1)

1 LJK - Laboratoire Jean Kuntzmann

We present here an algorithm to perform the arithmetic over small extension field via numerical arithmetic. The idea is to convert the $X$-adic representation of modular polynomials, with $X$ an indeterminate, to a $q$-adic representation where $q$ is a prime power coprime with 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 floating point operators. Depending also on the number of performed numerical operations one must then convert back to the $q$-adic or $X$-adic reprentation as to mod out high residues. In this note we present a new version of both conversions. More tabulation and a way to reduce the number of divisions involved in the process are presented and compared to the original version.

Mots clés: Small extension field; implantation; floating point

Domaines: Calcul formel [cs.SC]

Jean-Guillaume Dumas. Arithmetic over small extension fields via floating point routines. 2007. ⟨hal-00173894v1⟩