Yes. Bourgain has a sum-product estimate for residues of a general modulus (although, the case of a composite modulus with few prime factors that covers your question was worked out prior by [Bourgain and Chang][1] to this). See:

J. Bourgain, [Sum-product theorems and exponential sum bounds in residue classes for general modulus][2].  C. R. Math. Acad. Sci. Paris 344 (2007), no. 6, 349–352 


  [1]: http://www.ams.org/mathscinet-getitem?mr=2231466
  [2]: http://www.ams.org/mathscinet-getitem?mr=2310668