Computing the Canonical Lift of Genus 2 Curves in Odd Characteristics

Damien Robert (1, 2, 3) , Abdoulaye Maiga (2)

1 LFANT - Lithe and fast algorithmic number theory
2 IMB - Institut de Mathématiques de Bordeaux
3 CANARI - Analyse cryptographique et arithmétique

Let A/Fq be an ordinary abelian surface. We explain how to use the Siegel modular polynomials, and if available the Hilbert modular polynomials to compute the canonical lift of A. As an application, if q = p n , we show how to use the canonical lift to count the number of points on A in quasi-quadratic time Õ(n 2), this is a direct extension of Satoh's original algorithm for elliptic curves. We give a detailed description with the necessary optimizations for an efficient implementation.

Keywords: Abelian variety; Arithmetic invariants of genus 2 curves; Modular polynomials; Canonical lift; Point counting

Domains: Symbolic Computation [cs.SC]; Number Theory [math.NT]

Damien Robert, Abdoulaye Maiga. Computing the Canonical Lift of Genus 2 Curves in Odd Characteristics. 2022. ⟨hal-03738314⟩