Elliptic curve point counting

Abstract

In this thesis we present an improved algorithm for counting points on elliptic curves over finite fields. It is mainly based on Satoh-Skjernaa-Taguchi algorithm, and uses a Gaussian Normal Basis (GNB) of small type t≤4. In practice, about 42% (36% for prime N) of fields in cryptographic context (i.e., for p=2 and 160＜ N＜600) have such bases. They can be lifted from $\F_{p^N}$ to $\Z_{p^N}$ in a natural way. From the specific properties of GNBs, efficient multiplication and the Frobenius substitutions are available. Thus a fast norm computation algorithm is derived, which runs in $O(N^{2μ logN)$ with $O(N^2)$ space, where the time complexity of multiplying two n-bit objects is $O(n^μ)$. As a result, for all small characteristic p, we reduced the time complexity of the SST-algorithm from $O(N^{2μ+ 0.5})$ to $O(N^{2μ + \frac{1}{μ + 1}})$ and the space complexity still fits in $O(N^2)$.