Constructing a Pseudorandom Generator Requires an Almost Linear Number of Calls

TL;DR

This paper proves that constructing a pseudorandom generator from a one-way function requires nearly linear calls, establishing a fundamental lower bound that matches existing upper bounds, even for regular functions.

Contribution

It establishes a tight lower bound of Omega(n/log(n)) calls for black-box constructions of pseudorandom generators from one-way functions, confirming the optimality of known methods.

Findings

Lower bound of Omega(n/log(n)) calls for pseudorandom generator construction

Bound holds even for regular one-way functions

Matches the best known upper bound by Goldreich, Krawczyk, and Luby

Abstract

We show that a black-box construction of a pseudorandom generator from a one-way function needs to make Omega(n/log(n)) calls to the underlying one-way function. The bound even holds if the one-way function is guaranteed to be regular. In this case it matches the best known construction due to Goldreich, Krawczyk, and Luby (SIAM J. Comp. 22, 1993), which uses O(n/log(n)) calls.