Abstract
We show that there is a pair of disjoint NP sets, whose disjointness is provable in S 12 and which cannot be separated by a set in P/poly, if the cryptosystem RSA is secure. Further we show that factoring and the discrete logarithm are implicitly definable in any extension of S 12 admitting an NP -definition of primes about which it can prove that no number satisfying the definition is composite.
As a corollary we obtain that the Extended Frege (EF) proof system does not admit a feasible interpolation theorem unless the RSA cryptosystem is not secure, and that an extension of EF by tautologies τ 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.
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© 1995 Springer-Verlag Berlin Heidelberg
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Krajíček, J., Pudlák, P. (1995). Some consequences of cryptographical conjectures for S 12 and EF. In: Leivant, D. (eds) Logic and Computational Complexity. LCC 1994. Lecture Notes in Computer Science, vol 960. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-60178-3_86
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DOI: https://doi.org/10.1007/3-540-60178-3_86
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