A Quasi Quadratic Time Algorithm for Hyperelliptic Curve Point Counting

Reynald Lercier (1) , David Lubicz (1)

1 IRMAR - Institut de Recherche Mathématique de Rennes

We describe an algorithm to compute the cardinality of Jacobians of ordinary hyperelliptic curves of small genus over finite fields $F_{2^n}$ with cost $O(n^{2+o(1)})$. This algorithm is derived from ideas due to Mestre. More precisely, we state the mathematical background behind Mestre's algorithm and develop from it a variant with quasi-quadratic time complexity. Among others, we present an algorithm to find roots of a system of generalized Artin-Schreier equations and give results that we obtain with an efficient implementation. Especially, we were able to obtain the cardinality of curves of genus one, two or three in finite fields of huge size.

Domaines: Géométrie algébrique [math.AG]; Théorie des nombres [math.NT]

Reynald Lercier, David Lubicz. A Quasi Quadratic Time Algorithm for Hyperelliptic Curve Point Counting. Ramanujan Journal, 2006, 12 (3), pp.399-423. ⟨10.1007/s11139-006-0151-6⟩. ⟨hal-00456401⟩