  Using the non-perturbative renormalization technique, we calculate the
renormalization factors for quark bilinear operators made of overlap fermions
on the lattice. The background gauge field is generated by the JLQCD and TWQCD
collaborations including dynamical effects of two or 2+1 flavors of light
quarks on a 16$^3\times$32 or 16$^3\times$48 lattice at lattice spacing around
0.1 fm. By reducing the quark mass close to the chiral limit, where the finite
volume system enters the so-called $\epsilon$-regime, the unwanted effect of
spontaneous chiral symmetry breaking on the renormalization factors is
suppressed. On the lattices in the conventional $p$-regime, this effect is
precisely subtracted by separately calculating the contributions from the
chiral condensate.
