Q-adic Transform revisited

We present an algorithm to perform a simultaneous modular reduction of several residues. This enables to compress polynomials into integers and perform several modular operations with machine integer 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 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 qadic 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 and linear algebra in small finite field extensions...

• Type: Journal
• Year: 2007
• Where: CORR
• Authors: Jean-Guillaume Dumas
• Keywords: CORR 2007, Education, Polynomial, Simultaneous Modular Reduction, X-adic Representation