  Random illumination is proposed to enforce absolute uniqueness and resolve
all types of ambiguity, trivial or nontrivial, from phase retrieval. Almost
sure irreducibility is proved for any complex-valued object of a full rank
support. While the new irreducibility result can be viewed as a probabilistic
version of the classical result by Bruck, Sodin and Hayes, it provides a novel
perspective and an effective method for phase retrieval.
  In particular, almost sure uniqueness, up to a global phase, is proved for
complex-valued objects under general two-point conditions. Under a tight sector
constraint absolute uniqueness is proved to hold with probability exponentially
close to unity as the object sparsity increases. Under a magnitude constraint
with random amplitude illumination, uniqueness modulo global phase is proved to
hold with probability exponentially close to unity as object sparsity
increases. For general complex-valued objects without any constraint, almost
sure uniqueness up to global phase is established with two sets of Fourier
magnitude data under two independent illuminations. Numerical experiments
suggest that random illumination essentially alleviates most, if not all,
numerical problems commonly associated with the standard phasing algorithms.
