Counting Points on Elliptic Curves over Finite Fields of Small Characteristic in Quasi Quadratic Time
Résumé
Let $p$ be a small prime and $q=p^n$. Let $E$ be an elliptic curve
over $F_q$. We propose an algorithm which computes without any
preprocessing the $j$-invariant of the canonical lift of $E$ with
the cost of $O(\log n)$ times the cost needed to compute a power of
the lift of the Frobenius. Let $\mu$ be a constant so that the
product of two $n$-bit length integers can be carried out in
$O(n^\mu)$ bit operations, this yields an algorithm to compute the
number of points on elliptic curves which reaches, at the expense of
a $O(n^{\frac{5}{2}})$ space complexity, a theoretical time
complexity bound equal to $O(n^{\max(1.19, \mu)+\mu+\frac{1}{2}}\log
n)$. When the field has got a Gaussian Normal Basis of small type,
we obtain furthermore an algorithm with $O(\log(n)n^{2\mu})$ time
and $O(n^2)$ space complexities. From a practical viewpoint, the
corresponding algorithm is particularly well suited for
implementations. We outline this by a 100002-bit