Problems that are NP-complete under randomized or P/poly reductions. In this question, we appear to have identified a natural problem that is NP-complete under randomized reductions, but possibly not under deterministic reductions (although this depends on which unproven assumptions in number theory are true). Are there any other such problems known? Are there any natural problems that are NP-complete under P/poly reductions, but not known to be under P reductions?
As suggested by Peter, I converted my comment into an answer. 

Valiant-Vazirani Theorem states that if Unique SAT $\in P$ then $NP=RP$. To prove their theorem they showed that the promise problem Unique SAT is $NP$-hard under randomized reductions.

[1] Valiant, Leslie; Vazirani, Vijay. "NP is as easy as detecting unique solutions", Theoretical Computer Science, 47: 85–93