  Let $q = 2^n$, and let $E / \mathbb{F}_{q^{\ell}}$ be a generalized
Galbraith--Lin--Scott (GLS) binary curve, with $\ell \ge 2$ and $(\ell, n) =
1$.We show that the GLS endomorphism on $E / \mathbb{F}_{q^{\ell}}$ induces an
efficient endomorphism on the Jacobian $J_H(\mathbb{F}_q)$ of the genus-$g$
hyperelliptic curve $H$ corresponding to the image of the GHS Weil-descent
attack applied to $E/\mathbb{F}_{q^\ell}$, and that this endomorphism yields a
factor-$n$ speedup when using standard index-calculus procedures for solving
the Discrete Logarithm Problem (DLP) on $J_H(\mathbb{F}_q)$. Our analysis is
backed up by the explicit computation of a discrete logarithm defined on a
prime-order subgroup of a GLS elliptic curve over the field
$\mathbb{F}_{2^{5\cdot 31}}$. A Magma implementation of our algorithm finds the
aforementioned discrete logarithm in about $1,035$ CPU-days.
