Quantum algorithm for a generalized hidden shift problem

Consider the following generalized hidden shift problem: given a function f
on {0,...,M-1} x Z_N satisfying f(b,x)=f(b+1,x+s) for b=0,1,...,M-2, find the
unknown shift s in Z_N. For M=N, this problem is an instance of the abelian
hidden subgroup problem, which can be solved efficiently on a quantum computer,
whereas for M=2, it is equivalent to the dihedral hidden subgroup problem, for
which no efficient algorithm is known. For any fixed positive epsilon, we give
an efficient (i.e., poly(log N)) quantum algorithm for this problem provided M
> N^epsilon. The algorithm is based on the "pretty good measurement" and uses
H. Lenstra's (classical) algorithm for integer programming as a subroutine.