  We exhibit a probabilistic algorithm which solves a polynomial system over
the rationals defined by a reduced regular sequence. Its bit complexity is
roughly quadratic in the B\'ezout number of the system and linear in its bit
size. Our algorithm solves the input system modulo a prime number p and applies
p-adic lifting. For this purpose, we establish a number of results on the bit
length of a "lucky" prime p, namely one for which the reduction of the input
system modulo p preserves certain fundamental geometric and algebraic
properties of the original system. These results rely on the analysis of Chow
forms associated to the set of solutions of the input system and effective
arithmetic Nullstellens\"atze.
