  We describe the use of explicit isogenies to translate instances of the
Discrete Logarithm Problem (DLP) from Jacobians of hyperelliptic genus 3 curves
to Jacobians of non-hyperelliptic genus 3 curves, where they are vulnerable to
faster index calculus attacks. We provide explicit formulae for isogenies with
kernel isomorphic to $(\ZZ/2\ZZ)^3$ (over an algebraic closure of the base
field) for any hyperelliptic genus 3 curve over a field of characteristic not 2
or 3. These isogenies are rational for a positive fraction of all hyperelliptic
genus 3 curves defined over a finite field of characteristic $p > 3$. Subject
to reasonable assumptions, our constructions give an explicit and efficient
reduction of instances of the DLP from hyperelliptic to non-hyperelliptic
Jacobians for around 18.57% of all hyperelliptic genus 3 curves over a given
finite field. We conclude with a discussion on extending these ideas to
isogenies with more general kernels. A condensed version of this work appeared
in the proceedings of the EUROCRYPT 2008 conference.
