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

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

$\mathcal{O}(n^8)$

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

$\mathcal{O}(n^4log^2(n))$

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 [7] 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 [6].