Interactive proof systems: Provers that never fail and random selection

An interactive proof system with Perfect Completeness (resp. Perfect Soundness) for a language L is an interactive proof (for L) in which for every x ∈ L (resp. x ∉ L) the verifier always accepts (resp. always rejects). Zachos and Fuerer showed that any language having a bounded interactive proof has one with perfect completeness. We extend their result and show that any language having a (possibly unbounded) interactive proof system has one with perfect completeness. On the other hand, only languages in NP have interactive proofs with perfect soundness. We present two proofs of the main result. One proof extends Lautemann's proof that BPP is in the polynomial-time hierarchy. The other proof, uses a new protocol for proving approximately lower bounds and "random selection". The problem of random selection consists of a verifier selecting at random, with uniform probability distribution, an element from an arbitrary set held by the prover. Previous protocols known for approximate lower bound do not solve the random selection problem. Interestingly, random selection can be implemented by an unbounded Arthur-Merlin game but can not be implemented by a two-iteration game.

[1]  Andrew Chi-Chih Yao,et al.  Theory and application of trapdoor functions , 1982, 23rd Annual Symposium on Foundations of Computer Science (sfcs 1982).

[2]  Manuel Blum,et al.  How to generate cryptographically strong sequences of pseudo random bits , 1982, 23rd Annual Symposium on Foundations of Computer Science (sfcs 1982).

[3]  Andrew Chi-Chih Yao,et al.  Theory and Applications of Trapdoor Functions (Extended Abstract) , 1982, FOCS.

[4]  Clemens Lautemann,et al.  BPP and the Polynomial Hierarchy , 1983, Inf. Process. Lett..

[5]  Michael Sipser,et al.  A complexity theoretic approach to randomness , 1983, STOC.

[6]  László Babai,et al.  Trading group theory for randomness , 1985, STOC '85.

[7]  Silvio Micali,et al.  The knowledge complexity of interactive proof-systems , 1985, STOC '85.

[8]  Shafi Goldwasser,et al.  On the Power of Interaction , 1986, FOCS.

[9]  Shafi Goldwasser,et al.  Private coins versus public coins in interactive proof systems , 1986, STOC '86.

[10]  Stathis Zachos,et al.  Probabilistic Quantifiers, Adversaries, and Complexity Classes: An Overview , 1986, SCT.

[11]  Silvio Micali,et al.  Proofs that yield nothing but their validity and a methodology of cryptographic protocol design , 1986, 27th Annual Symposium on Foundations of Computer Science (sfcs 1986).

[12]  Stathis Zachos,et al.  Does co-NP Have Short Interactive Proofs? , 1987, Inf. Process. Lett..

[13]  Stathis Zachos,et al.  Probabalistic Quantifiers vs. Distrustful Adversaries , 1987, FSTTCS.

[14]  Uwe Schöning.  Probabilistic Complexity Classes and Lowness , 1989, J. Comput. Syst. Sci..

[15]  Lance Fortnow,et al.  The Complexity of Perfect Zero-Knowledge , 1987, Proceeding Structure in Complexity Theory.