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**2007/428**

**Title:**

Isogenies and the Discrete Logarithm Problem on Jacobians of Genus 3 Hyperelliptic Curves

**Authors:**
Benjamin Smith

**Abstract:**

We describe the use of explicit isogenies to reduce Discrete Logarithm Problems (DLPs) on Jacobians of hyperelliptic genus~3 curves to Jacobians of non-hyperelliptic genus~3 curves, which are vulnerable to faster index calculus attacks. We provide algorithms which compute an isogeny with kernel isomorphic to (\mathbb{Z}/2\mathbb{Z})^3 for any hyperelliptic genus~3 curve. These algorithms provide a rational isogeny for a positive fraction of all hyperelliptic genus~3 curves defined over a finite field of characteristic p > 3. Subject to reasonable assumptions, our algorithms provide an explicit and efficient reduction from hyperelliptic DLPs to non-hyperelliptic DLPs for around 18.57\% of all hyperelliptic genus~3 curves over a given finite field.

**ePrint:**
https://eprint.iacr.org/2007/428

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