A Primitive for Proving the Security of Every Bit andAbout Universal Hash Functions & Hard CorePredicates ?

In 1999, J. H astad and M. NN aslund 13] could prove that every bit is a hard core of the RSA function. From this work we extract an abstract theorem about the hidden number problem which can be used to prove that every bit is a hard core of many speciic cryptographic functions. Applications are RSA, ElGamal, Rabin, a modiied Diie-Hellman function, Pailler's cryptosystem, the Diie-Hellman function for elliptic curves and discrete exponentiation. So far all in the literature known general constructions of hard core predicates for any one-way function (for example the famous Goldreich-Levin Bit 12]) are based on some set of universal hash functions (UHF). The natural open question was if there may be a nice connection between UHF and hard core predicates. We present an example providing a negative answer to that question. Furthermore, as an alternative to the Goldreich-Levin Bit, we give a new and eecient construction of a hard core predicate of any one-way function that is based on the hidden number problem.