Elliptic curve point counting over finite fields with Gaussian normal basis

Abstract

In this paper, we present the GNB-aided MSST algorithm for the curves over finite fields that have a Gaussian normal basis of type $t \le 2$. It is based on the MSST algorithm proposed by P. Gaudry [3] at ASIACRYPT 2002. For those fields, we combine the lifting phase of the MSST algorithm and the norm computation algorithm in [6]. So the time complexity of the MSST is reduced from $O(N^{2\mu + 0.5})$ to $O(N^{2\mu + 1/ (\mu + 1)})$ and it runs faster than any other algorithms in our case.