Counting Points on Elliptic Curves over Finite Fields of Small Characteristic in Quasi Quadratic Time

Abstract

Let p be a small prime and q = p ⁿ. Let E be an elliptic curve over $\mathbb{F}_q$ . 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 μ be a constant so that the product of two n-bit length integers can be carried out in O(n ^μ) bit operations, this yields an algorithm to compute the number of points on elliptic curves which reaches, at the expense of a O(n ^5/2) space complexity, a theoretical time complexity bound equal to O(n ^{max(1.19,μ)+μ+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 ^2μ) 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 computation.