Although a lot of research was done in the 1980's on proving cryptosystems based on factoring, two examples being Rabin's scheme and Goldwasser-Micali-Rivest, in the last decade a very large number of papers have appeared using the Diffie-Hellman assumptions and variants.
We make two remarks on this approach. First the Diffie-Hellman assumptions may be wrong, while the factoring one may be correct. Second, the Diffie-Hellman assumption does not involve a trapdoor, but both factoring as well as RSA do. For obvious reasons it may be good to obtain a variant of the RSA assumption, for which we can give reasonable evidence that it is likely trapdoor free. We make such a proposal and discuss future lines of research on the computational complexity of this problem.
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