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In a seminal paper, Håstad, Impagliazzo, Levin, and Luby showed that pseudorandom generators exist if and only if one-way functions exist. The construction they propose to obtain a pseudorandom generator from an n-bit one-way function uses O(n 8) random bits in the input (which is the most important complexity measure of such a construction). In this work we study how much this can be reduced if the one-way function satisfies a stronger security requirement. For example, we show how to obtain a pseudorandom generator which satisfies a standard notion of security using only O(n 4 log 2(n)) bits of randomness if a one-way function with exponential security is given, i.e., a one-way function for which no polynomial time algorithm has probability higher than 2 -cn in inverting for some constant c. Using the uniform variant of Impagliazzo's hard-core lemma given in  our constructions and proofs are self-contained within this paper, and as a special case of our main theorem, we give the first explicit description of the most efficient construction from . © Springer-Verlag Berlin Heidelberg 2006.
Holenstein, T. (2006). Pseudorandom generators from one-way functions: A simple construction for any hardness. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 3876 LNCS, pp. 443–461). https://doi.org/10.1007/11681878_23