  Torsional oscillations of magnetized neutron stars have been well studied
since they may be excited in magnetar starquakes and relevant to the observed
quasiperiodic oscillations in the magnetar giant flares. In the crustal region
of a magnetar, the strong magnetic field can alter the equation of state and
composition of the crust due to the Landau-Rabi quantization of electron
motion. In this paper, we study this effect on the torsional oscillation modes
of neutron stars with mixed poloidal-toroidal magnetic fields in general
relativity under the Cowling approximation. Furthermore, the inner and outer
crusts are treated consistently based on the nuclear-energy density functional
theory. Depending on the magnetic-field configurations, we find that the
Landau-Rabi quantization of electron can change the frequencies of the
fundamental torsional oscillation mode of $1.4 M_\odot$ neutron star models
with a normal fluid core by about $10\%$ when the magnetic field strength at
the pole reaches the order of $10^{16}$ G. The shift can even approach $20\%$
at a field strength of $10^{15}$ G for neutron stars with a simple model of
superconducting core where the magnetic field is assumed to be expelled
completely from the core and confined only in the crust.
