Resource Bounded Unprovability O F Computational Lower Bounds (part 1)

This paper shows that the proof complexity (minimum computational complexity of proving formally or asymptotically) of \P 6=NP" is super-polynomial-time with respect to a theory T , which is a consistent extension of Peano Arithmetic (PA), and PTM-!-consistent, where the PTM-!-consistency is a polynomial-time Turing machine (PTM) version of !consistency. In other words, to prove \P 6=NP" (by any technique) requires super-polynomialtime computational power over T . This result is a kind of generalization of the result of \Natural Proofs" by Razborov and Rudich [20], who showed that to prove \P 6=NP" by a class of techniques called \Natural" implies computational power that can break a typical cryptographic primitive, a pseudo-random generator. This result implies that P6=NP is formally unproven in PTM-!-consistent theory T . We also show that to prove the independence of P vs NP from T by proving the PTM-!-consistency of T requires super-polynomial-time computational power. This seems to be related to the results of Ben-David and Halevi [4] and Kurz, O'Donnell and Royer [16], who showed that to prove the independence of P vs NP from PA using any currently known mathematical paradigm implies an extremely-close-topolynomial time algorithm that can solve NP-complete problems. Based on this result, we show that the security of any computational cryptographic scheme is unprovable in the standard setting of modern cryptography, where an adversary is modeled as a polynomial-time Turing machine.