Fast Computation of Hyperelliptic Curve Isogenies in Odd Characteristic

Abstract

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Computing isogenies between jacobians of hyperelliptic curves of arbitrary genus via differential equations

“…In this section, we sketch the proof of the main theorem by showing that the Newton iteration given in Equation (3) can be executed with quasi-linear time complexity to give the desired polynomial in Theorem 1. The precision analysis has been already studied in [Eid20]. Throughout this section, the letter p refers to a fixed odd prime number and the letter K refers to a fixed unramified extension of Q p of degree d and k its residue field.…”

“…In order to have optimal algorithms for computing isogenies, in particular those which are defined over finite fields, several approaches have been suggested. One of them consists in reducing the problem to the computation of a solution of a nonlinear differential equation [Elk97,CE15], possibly after having lifted the problem to the p-adics [LS08, LV16,CEL20,Eid20]. In this work, we focus on p-adic algorithms that compute the explicit form of a rational representation of an isogeny between Jacobians of hyperelliptic curves for fields of odd characteristic.…”

“…, ℓ)-isogeny I : J 1 → J defined over k and we are interested in computing one of its rational representations. Let us recall briefly the definition of a rational representation and how we compute it (see [Eid20] for more details). Let P be a Weierstrass point on C 1 and j P : C 1 → J 1 the Jacobi map with origin P .…”

“…In this article, we revisit the algorithm of [Eid20] and manage to lower its complexity in g and make it quasi-linear as well. For this, we work directly on the first Mumford coordinate…”

“…Although, they exhibit acceptable running time in practice, their theoretical complexity has not been well studied yet and experiments show that they become much slower when the genus gets higher. Actually, in many cases, we have observed that the algorithms of [Eid20] perform better in practice even if their theoretical complexity in g is not optimal. Consequently, even though the algorithms designed in the present paper use Kedlaya-Umans algorithm [KU11] as a subroutine and then could be difficult to implement in an optimized way, they appear as attractive alternatives for the computation of ℓ-division polynomials on Jacobians of hyperelliptic curves.…”