Cyclic Isogenies for Abelian Varieties with Real Multiplication

TL;DR

This paper develops an efficient algorithm for computing explicit cyclic isogenies of abelian varieties with real multiplication over finite fields, leveraging theta functions, with applications in cryptography and discrete logarithm problems.

Contribution

It introduces a polynomial-time algorithm for cyclic isogenies of abelian varieties with real multiplication, based on Mumford's theta theory, and demonstrates its applications.

Findings

Algorithm is polynomial in field size and isogeny degree

Successfully applied to discrete logarithm problems in genus 2 and 3

Provides criteria for principal polarizability of quotients

Abstract

We study quotients of principally polarized abelian varieties with real multiplication by Galois-stable finite subgroups and describe when these quotients are principally polarizable. We use this characterization to provide an algorithm to compute explicit cyclic isogenies from kernel for abelian varieties with real multiplication over finite fields. Our algorithm is polynomial in the size of the finite field as well as the degree of the isogeny and is based on Mumford's theory of theta functions and theta embeddings. Recently, the algorithm has been successfully applied to obtain new results on the discrete logarithm problem in genus 2 as well as to study the discrete logarithm problem in genus 3.