On the construction of elliptic Chudnovsky-type algorithms for multiplication in large extensions of finite fields

We indicate a strategy in order to construct bilinear multiplication algorithms of type Chudnovsky in large extensions of any finite field. In particular, by using the symmetric version of the generalization of Randriambololona specialized on the elliptic curves, we show that it is possible to construct such algorithms with low bilinear complexity. More precisely, if we only consider the Chudnovsky-type algorithms of type symmetric elliptic, we show that the symmetric bilinear complexity of these algorithms is in $O(n(2q)^{\log_q^*(n)})$ where $n$ corresponds to the extension degree, and $\log_q^*(n)$ is the iterated logarithm. Moreover, we show that the construction of such algorithms can be done in time polynomial in $n$. Finally, applying this method we present the effective construction, step by step, of such an algorithm of multiplication in the finite field $\F_{3^{57}}$.