Trinomial bases and Chinese remaindering for modular polynomial multiplication

Following the previous work by Bajard-Didier-Kornerup, Mc- Laughlin, Mihailescu and Bajard-Imbert-Jullien, we present an algorithm for modular polynomial multiplication that implements the Montgomery algorithm in a residue basis; here, as in Bajard et al.'s work, the mod- uli are trinomials over F2. Previous work used a second residue basis to perform the final division. In this paper, we show how to keep the same residue basis, inspired by l'Hospital rule. Additionally, applying a divide- and-conquer approach to the Chinese remaindering, we obtain improved estimates on the number of additions for some useful degree ranges.