Counting Points on an Abelian Variety over a Finite Field

Abstract

Matsuo, Chao and Tsujii [16] have proposed an algorithm for counting the number of points on the Jacobian variety of a hyperelliptic curve over a finite field. The Matsuo-Chao-Tsujii algorithm is an improvement of the ‘baby-step-giant-step’ part of the Gaudry-Harley scheme. This scheme consists of two parts: firstly to compute the residue modulo a positive integer m of the order of a given Jacobian variety, and then to search for the actual order by a square-root algorithm. In this paper, following the Matsuo-Chao-Tsujii algorithm, we propose an improvement of the square-root algorithm part in the Gaudry-Harley scheme by optimizing the use of the residue modulo m of the characteristic polynomial of the Frobenius endomorphism of an Abelian variety. It turns out that the computational complexity is \(O \left( q^{\frac{4g -- 2 + i^{2} -- i}{8}} / m^{\frac{i + 1}{2}} \right)\), where i is an integer in the range 1 ≤ i ≤ g. We will show that for each g and each finite field \(\mathbb{F}_q\) of q=p n elements, there exists an i which gives rise to the optimum complexity among all three corresponding algorithms.