Valiant & Vazirani proved SAT is reducible to UNIQUE SAT under randomized probabilistic reductions in polynomial time.  Calabro et al. showed that UNIQUE k-SAT is as hard as k-SAT. Now the question is, if someone shows that UNIQUE k-SAT is in P,  does it imply P=NP?ReferencesL. G. Valiant and V. V. Vazirani, NP is as easy as detecting unique solutions. Theoretical Computer Science 47:85–93, 1986. (PDF on ScienceDirect.)C. Calabro, R. Impagliazzo, V. Kabanets and R. Paturi, The Complexity of Unique k-SAT: An Isolation Lemma for k-CNFs.  Journal of Computer and System Sciences 74(3):386–393, 2008. (PDF at ACM Digital Library; free PDF.)
If one shows that UNIQUE k-SAT is in P, does it imply P=NP?
complexity theory;satisfiability;p vs np
This is still an open question; UP is not known to be equivalent to NP.  In the paper NP Might Not Be As Easy As Detecting Unique Solutions, Beigel, Burhman and Fortnow construct an oracle under which P contains UP but P is still not equivalent to NP.