For the 3-Satisfiability Problem (3SAT), we propose three algorithms 
for generating its positive instance (and its solution) randomly.
We design these algorithms so that they produce, with high 
probability, a {\em unique solution 3SAT instances}, i.e., a 3SAT 
instances with only one satisfying assignment.
For our first algorithm, we proved theoretically that the algorithm 
yields a unique solution 3SAT instance with high probability if the 
number of clauses $m$ is larger than $(7/3)n\ln n+O(n)$; furthermore, 
we also proved that $(7/3)n\ln n$ clauses are necessary.
Then by modifying this algorithm, we obtain two algorithms that need 
only $O(n)$ clauses to yield a unique solution 3SAT instance with high 
probability.
Hardness against of generated instances against standard heuristics is 
also investigated experimentally.