Security preserving amplification of hardness

The task of transforming a weak one-way function (which may be easily inverted on all but a polynomial fraction of the range) into a strong one-way function (which can be easily inverted only on a negligible function of the range) is considered. The previously known transformation does not preserve the security (i.e. the running time of the inverting algorithm) within any polynomial. Its resulting function, F(x), applies the weak one-way function to many small (of length |x|θ, θ<1) pieces of the input. Consequently, the function can be inverted for reasonable input lengths by exhaustive search. Random walks on constructive expanders are used to transform any regular (e.g. one-to-one) weak one-way function into a strong one, while preserving security. The resulting function, F(x), applies the weak one-way f to strings of length Θ(|x|). The security-preserving constructions yield efficient pseudorandom generators and signatures based on any regular one-way function