  We propose several constructions for the original multiplication algorithm of
D.V. and G.V. Chudnovsky in order to improve its scalar complexity. We
highlight the set of generic strategies who underlay the optimization of the
scalar complexity, according to parameterizable criteria. As an example, we
apply this analysis to the construction of type elliptic Chudnovsky$^2$
multiplication algorithms for small extensions. As a case study, we
significantly improve the Baum-Shokrollahi construction for multiplication in
$\mathbb F_{256}/\mathbb F_4$.
