  Difference sets are basic combinatorial structures that have applications in
signal processing, coding theory, and cryptography. We consider the problem of
identifying a shifted version of the characteristic function of a (known)
difference set. We present a generic quantum algorithm that can be used to
tackle any hidden shift problem for any difference set in any abelian group. We
discuss special cases of this framework where the resulting quantum algorithm
is efficient. This includes: a) Paley difference sets based on quadratic
residues in finite fields, which allows to recover the shifted Legendre
function quantum algorithm, b) Hadamard difference sets, which allows to
recover the shifted bent function quantum algorithm, and c) Singer difference
sets based on finite geometries. The latter class allows us to define instances
of the dihedral hidden subgroup problem that can be efficiently solved on a
quantum computer.
