Translating the Discrete Logarithm Problem on Jacobians of Genus 3 Hyperelliptic Curves with (ℓ ,ℓ ,ℓ ) -Isogenies

We give an algorithm to compute (ℓ ,ℓ ,ℓ ) -isogenies from the Jacobians of genus three hyperelliptic curves to the Jacobians of non-hyperelliptic curves over a finite field of characteristic different from 2 in time Õ(ℓ ^3) , where ℓ is an odd prime which is coprime to the characteristic. An important application is to reduce the discrete logarithm problem in the Jacobian of a hyperelliptic curve to the corresponding problem in the Jacobian of a non-hyperelliptic curve.