On p-adic point counting algorithms for elliptic curves over finite fields

Abstract

Let p be a prime and let q:= pN. L et E be an elliptic curve over Fq. We are interested in efficient algorithms to compute the order of the group E(Fq) of Fq-rational points of E. An l-adic algorithm, known as the SEA algorithm, computes #E(Fq) with O((log q)4+ε) bit operations (with fast arithmetic) and O((log q)2) memory. In this article, we survey recent advances in p-adic algorithms. For a fixed small p, the computational complexity of the known fastest p-adic point counting algorithm is O(N3+ε) in time and O(N2) in space. If we accept some precomputation depending only on p and N or a certain restriction on N, the time complexity is reduced to O(N2.5+ε) still with O(N2) space requirement.