  We employ first-principles quantum field theoretical methods to investigate
the longitudinal and transverse electrical conductivities of a strongly
magnetized hot QED plasma at the leading order in coupling. The analysis
utilized the fermion damping rate in the Landau-level representation obtained
recently. In the relativistic regime, both conductivities exhibit an
approximate scaling behavior given by $\sigma_{\parallel,\perp} = T
\tilde{\sigma}_{\parallel,\perp}$, where $\tilde{\sigma}_{\parallel,\perp}$ are
functions of the dimensionless ratio $|eB|/T^2$ (here $T$ denotes the
temperature and $B$ the magnetic field). We argue that the mechanisms for the
transverse and longitudinal conductivities differ significantly, leading to a
strong suppression of the former in comparison to the latter. Additionally, we
extend our analysis to a magnetized quark-gluon plasma, although the
approximation's validity breaks down in QCD at strong coupling.
