Here, I write an excerpt of the following paper:
Valiant, L. G. and Vazirani, V. V. 1986. NP is as easy as detecting unique solutions. Theor. Comput. Sci. 47, 1 (Nov. 1986), 85-93. DOI= http://dx.doi.org/10.1016/0304-3975(86)90135-0
For every known NP-complete problem, the number of solutions of its instances varies over a large range, from zero to exponentially many. It is therefore natural to ask if the inherent intractability of NP-complete problem is caused by this wide variation. We give a negative answer to this question using the notion of randomized polynomial time reducibility. We show that the problems of distinguishing between instances of SAT having zero or one solution, or of finding solutions to instances of SAT having a unique solution, are as hard as SAT, under randomized reductions.
I also suggest looking at the relevant paper:
Beigel, R., Buhrman, H., and Fortnow, L. 1998. NP might not be as easy as detecting unique solutions. In Proceedings of the Thirtieth Annual ACM Symposium on theory of Computing (Dallas, Texas, United States, May 24 - 26, 1998). STOC '98. ACM, New York, NY, 203-208. DOI= http://doi.acm.org/10.1145/276698.276737