  Motivated by the X-ray crystallography technology to determine the atomic
structure of biological molecules, we study the crystallographic phase
retrieval problem, arguably the leading and hardest phase retrieval setup. This
problem entails recovering a K-sparse signal of length N from its Fourier
magnitude or, equivalently, from its periodic auto-correlation. Specifically,
this work focuses on the fundamental question of uniqueness: what is the
maximal sparsity level K/N that allows unique mapping between a signal and its
Fourier magnitude, up to intrinsic symmetries. We design a systemic
computational technique to affirm uniqueness for any specific pair (K,N), and
establish the following conjecture: the Fourier magnitude determines a generic
signal uniquely, up to intrinsic symmetries, as long as K<=N/2. Based on
group-theoretic considerations and an additional computational technique, we
formulate a second conjecture: if K<N/2, then for any signal the set of
solutions to the crystallographic phase retrieval problem has measure zero in
the set of all signals with a given Fourier magnitude. Together, these
conjectures constitute the first attempt to establish a mathematical theory for
the crystallographic phase retrieval problem.
