  The coupling between the magnetic field and the gas during the collapsing
phase of star-forming cores is strongly affected by the dust size distribution,
which is expected to evolve. We aim to investigate the influence of key
parameters on the evolution of the dust distribution as well as on the magnetic
resistivities during the protostellar collapse. We perform collapsing single
zone simulations with shark. The code computes the evolution of the dust
distribution, accounting for different grain growth and destruction processes.
It also computes the magnetic resistivities. We find that the dust distribution
significantly evolves during the protostellar collapse, shaping the magnetic
resistivities. The peak size of the distribution, the population of small
grains and consequently the magnetic resistivities are controlled by both
coagulation and fragmentation rates. Under standard assumptions, the small
grains coagulate very early as they collide by ambipolar drift, yielding
magnetic resistivities orders of magnitude away from the non-evolving dust
case. In particular, the ambipolar resistivity \eta_AD is very high prior to
nH=10^10 cm^-3, and as a consequence magnetic braking should be ineffective. In
this case, large size protoplanetary disks should result, which is inconsistent
with recent observations. To alleviate this tension, we identify mechanisms to
reduce the ambipolar resistivity during the protostellar collapse. The most
promising are namely: electrostatic repulsion and grain-grain erosion. The
evolution of the magnetic resistivities during the protostellar collapse and
consequently the shape of the magnetic field in the early life of the
protoplanetary disk strongly depends on the possibility to repopulate the small
grains or to prevent their early coagulation. Therefore, it is crucial to
better constrain the collision outcomes and the dust grain elastic properties.
