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Some Consequences of Cryptographical Conjectures for S12 and EF

We show that there is a pair of disjoint NP sets, whose disjointness is provable in S 2 1 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 2 1 admitting an NP -definition of primes about which it can prove that no number satisfying the definition is composite.