## Pseudorandom Generators from One-Way Functions: A Simple Construction for Any Hardness

### Thomas Holenstein

In a seminal paper, Hastad, 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 [Hol05] 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 [HILL99].

## BibTeX Citation

@inproceedings{Holens06, author = {Thomas Holenstein}, title = {Pseudorandom Generators from One-Way Functions: A Simple Construction for Any Hardness}, editor = {Shai Halevi and Tal Rabin}, booktitle = {Theory of Cryptography Conference --- TCC 2006}, pages = {443--461}, series = {Lecture Notes in Computer Science}, year = {2006}, month = {3}, publisher = {Springer-Verlag}, }