  We propose a Recursive Polynomial Generic Construction (RPGC) of
multiplication algorithms in any finite field $\mathbb{F}_{q^n}$ based on the
method of D.V. and G.V. Chudnovsky specialized on the projective line. They are
usual polynomial interpolation algorithms in small extensions and the Karatsuba
algorithm is seen as a particular case of this construction. Using an explicit
family of such algorithms, we show that their bilinear complexity is
quasi-linear with respect to the extension degree n, and we give a uniform
bound for this complexity. We also prove that the construction of these
algorithms is deterministic and can be done in polynomial time. We give an
asymptotic bound for the complexity of their construction.
