Generating Hard, Certified Elements of NP-Complete Sets

We investigate the question of whether it is easier to generate certified members of NP sets than it is to certify membership in those sets. Informally, a randomized program is an "invulnerable generator" of a set S in NP if, on input n, it produces pairs (x,w), where x is an element of S, w is a witness that x is in S, and |x|=n, according to a distribution under which any randomized polynomial-time adversary who is given x but not w fails with high probability to find any witness that x is in S. 

We present three results about the existence of efficient, invulnerable generators for NP sets and the degree of invulnerability that is achievable.