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

Abstract

We give an algorithm to compute $$(\\ell ,\\ell ,\\ell )$$ ( 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 $$\\tilde{O}(\\ell ^3)$$ O ~ ( ℓ 3 ) , where $$\\ell $$ 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.