Some Consequences of Cryptographical Conjectures for S_2^1 and EF

We show that there is a pair of disjoint N P-sets, whose dis-jointness is provable in S 1 2 and which cannot be separated by a set in P=poly, if the cryptosystem RSA is secure. Further we show that factor-ing and the discrete logarithm are implicitly deenable in any extension of S 1 2 admitting an N P-deenition of primes about which it can prove that no number satisfying the deenition is composite. As a corollary we obtain that the Extended Frege (EF) proof system does not admit feasible interpolation theorem unless the RSA cryptosystem is not secure, and that an extension of EF by tautologies p (p primes), formalizing that p is not composite, as additional axioms does not admit feasible interpolation theorem unless factoring and the discrete logarithm are in P=poly. The NP 6 = coNP conjecture is equivalent to the statement that no pro-positional proof system (as deened in 6]) admits polynomial size proofs of all tautologies. However, only for few proof systems occurring in the literature explicit superpolynomial lower bounds are known; see 9] for a survey (and for important connections to bounded arithmetic). The following approach to proving lower bounds for propositional proof system was proposed in 8]. 2 Various interesting tautologies expressing some nite combinatorial principles (e.g., modular counting principles as in 8]) can be formulated in a form of an implication (x; y) ! (x; z) where x are the only atoms common to both and (equivalently we may think of refutations of ; :, as we shall, in fact, do). An interpolant for the studied earlier the question to estimate the complexity of an interpolant of an implication in terms of the size of the implication. He linked the problem to open problems in complexity theory about classes P=poly , N P , coN P , N C 1 and similar. He also singled out the class of natural proof systems, those being proof systems admitting a form of eeective interpolation (described below). He did not considered length-of-proofs questions.