Unique SAT ={$\phi$| $\phi$ has unique satisfying assignment } represents an important class of computational problems. I recall reading that P=NP iff Unique SAT is in P (I don't remember where I read it. Notice that Unique SAT is CoNP-hard and Unique SAT= SAT\ Double SAT where Double Sat={$\phi$| $\phi$ has two or more satisfying assignments})
I'm looking for an efficiently computable function $f(s_n)= \phi$ such that formula $\phi$ is uniquely satisfied by assignment $s_n$ where $n$ is the number of variables.
Also, Can $f$ be made a bijection?
Edit 1: Input is a binary string $s_n$, output is a 3SAT formula in CNF.
Basically, I'm looking for an efficient algorithm that computes $f$. Ideally, such algorithm would allows us to construct hard instances for Unique SAT.
EDIT 2: I'm seeking an algorithm that is able to generate all uniquely staisfiable instances not just the easy cases given so far.