On the Power of the Randomized Iterate∗ Draft

We consider two of the most fundamental theorems in Cryptography. The first, due to H̊astad, Impagliazzo, Levin and Luby (STOC ’89, STOC ’90, SIAM J. on Computing ’99), is that pseudorandom generators can be constructed from any one-way function. The second, due to Yao (FOCS ’82), states that the existence of weak one-way functions (i.e. functions on which every efficient algorithm fails to invert with some noticeable probability) implies the existence of full fledged one-way functions. These powerful plausibility results shape our understanding of hardness and randomness in Cryptography. Unfortunately, the reductions given by H̊astad et al. and Yao, are not as security preserving as one may desire. The main reason for the security deterioration is the input blow up in both of these constructions. For example, given one-way functions on n bits one obtains by to H̊astad et al. pseudorandom generators with seed length Ω(n). This work revisits a technique that we call the randomized iterate, introduced by Goldreich, Krawczyk and Luby (SIAM J. on Computing ’93). This technique was used by Goldreich et al. to give a construction of pseudorandom generators from regular one-way functions. We simplify and strengthen this technique in order to obtain a similar reduction where the seed length of the resulting generators is as short as O(n log n) rather than Ω(n) achieved by Goldreich et al. Our technique has the potential of implying seed-length O(n), and the only bottleneck for such a result is the parameters of current generators against space bounded computations. We give a reduction with similar parameters for security amplification of regular one-way functions. This improves upon the reduction of Goldreich, Impagliazzo, Levin, Venkatesan and Zuckerman (FOCS ’90) in that the reduction does not need to know the regularity parameter of the functions (in terms of security, the two reductions are incomparable). In addition we use the randomized iterate to show a construction of a pseudorandom generator based on an exponentially hard oneway function that has seed length of only O(n2). This improves a recent result of Holenstein (TCC ’06) that shows a construction with seed length O(n5) based on such one-way functions. Finally, we show that the randomized iterate may even be useful in the general context of H̊astad et al. In Particular, we use the randomized iterate to replace the basic building block of the H̊astad et al. construction. Interestingly, this modification improves efficiency by an n factor and reduces the seed length to O(n7) (which also implies improvement in the security of the construction).