Q-adic Floating-point Transform revisited: arithmetic over small extension field via floating point routines

We present an algorithm to perform arithmetic operations over small extension field via numerical routines. 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 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 floating point operators. 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.