Let p be a fixed small prime. We give an algorithm with preprocessing to compute the j-invariant of the canonical lift of a given ordinary elliptic curve E/Fq (q = pN, j(E) ∉ Fp2) modulo pN/2+O(1) in O(N2μ+1/μ+1) bit operations (assuming the time complexity of multiplying two n-bit objects is O(nμ) using O(N2) memory, not including preprocessing. This is faster than the algorithm of Vercauteren et al.  by a factor of Nμ/μ+1. Let K be the unramified extension field of degree N over Qp. We also develop an algorithm to compute NK/Qp (x) mod pN/2+O(1) with O(N2μ+0.5) bit operations and O(N2) memory when x∈K satisfies certain conditions, which are always satisfied when applied to our point counting algorithm. As a result, we get an O(N 2μ+0.5) time, O(N2) memory algorithm for counting the Fq-rational points on E/Fq, which turns out to be very fast in practice for cryptographic size elliptic curves. © 2002 Elsevier Science (USA). All rights reserved.
Satoh, T., Skjernaa, B., & Taguchi, Y. (2003, January). Fast computation of canonical lifts of elliptic curves and its application to point counting. Finite Fields and Their Applications. https://doi.org/10.1016/S1071-5797(02)00013-8