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On Generating Solved Instances of Computational Problems (Extended Abstract)

Summary: On Generating Solved Instances of Computational Problems
(Extended Abstract)
M. Abadi \Lambda E. Allender y A. Broder \Lambda J. Feigenbaum z L. Hemachandra x
Abstract: We consider the efficient generation of solved instances of computational problems. In
particular, we consider invulnerable generators. Let S be a subset of f0; 1g \Lambda and M be a Turing
Machine that accepts S; an accepting computation w of M on input x is called a ``witness'' that
x 2 S. Informally, a program is an ff­invulnerable generator if, on input 1 n , it produces instance­
witness pairs hx; wi, with jxj = n, according to a distribution under which any polynomial­time
adversary who is given x fails to find a witness that x 2 S, with probability at least ff, for infinitely
many lengths n.
The question of which sets have invulnerable generators is intrinsically appealing theoretically,
and the results can be applied to the generation of test data for heuristic algorithms and to the
theory of zero­knowledge proof systems. The existence of invulnerable generators is closely related
to the existence of cryptographically secure one­way functions. We prove three theorems about
invulnerability. The first addresses the question of which sets in NP have invulnerable generators, if
indeed any NP sets do. The second addresses the question of how invulnerable these generators are.
Theorem (Completeness): If any set in NP has an ff­invulnerable generator, then SAT has one.
Theorem (Amplification): If S 2 NP has a fi­invulnerable generator, for some constant fi 2 (0; 1),
then S has an ff­invulnerable generator, for every constant ff 2 (0; 1).
Our third theorem on invulnerability shows that one cannot, using techniques that relativize,


Source: Abadi, Martín - Department of Computer Science, University of California at Santa Cruz


Collections: Computer Technologies and Information Sciences