Most Juntas Saturate the Hardcore Lemma

TL;DR

This paper provides a simpler, more general proof demonstrating the tightness of Impagliazzo's hardcore lemma for functions mildly hard for size-$s$ circuits, using random juntas to witness this tightness across most parameters.

Contribution

The authors present a streamlined and broadly applicable proof that confirms the tightness of the hardcore lemma, leveraging random juntas to illustrate this in nearly all relevant parameter regimes.

Findings

The tightness of the hardcore lemma is demonstrated in most parameter regimes.

Random juntas serve as witnesses for the lemma's tightness.

The proof simplifies previous approaches and extends their applicability.

Abstract

Consider a function that is mildly hard for size-$s$ circuits. For sufficiently large $s$, Impagliazzo's hardcore lemma guarantees a constant-density subset of inputs on which the same function is extremely hard for circuits of size $s'<\!\!<s$. Blanc, Hayderi, Koch, and Tan [FOCS 2024] recently showed that the degradation from $s$ to $s'$ in this lemma is quantitatively tight in certain parameter regimes. We give a simpler and more general proof of this result in almost all parameter regimes of interest by showing that a random junta witnesses the tightness of the hardcore lemma with high probability.