The STOC 2008 paper "Fast Integer Multiplication using Modular Arithmetic" by De et al http://arxiv.org/abs/0801.1416 shows how to use $p$-adic numbers instead of $\mathbb C$ used in Furer's multiplication algorithm. Given that Furer's algorithm is not practical for less than a few thousand bits (is there a good estimate of its breakeven point vs Toom-Cook and other algorithms?), I wonder if the De et al algorithm has a lower breakeven point, is easier to implement, and is more practical in general.

Pointers to follow-up work and/or attempts at implementing the STOC 2008 algorithm would be appreciated.

Igor