Most Juntas Saturate the Hardcore Lemma
Vinayak M. Kumar
TL;DR
This work addresses the quantitative tightness of Impagliazzo's hardcore lemma, which asserts the existence of a dense hardcore subset on which all small circuits fail to compute a target function with nontrivial accuracy. The authors present a simpler and more general argument showing that a random junta witnesses the lemma's tightness across almost all parameter regimes, avoiding the need for delta-smoothness assumptions in the distribution and achieving tight degradation bounds via a 4-wise uniform generator. A key technical contribution is a distribution-free reduction that translates first-k-bit hardness, obtained via a Shannon-style argument, into an $n$-bit hardness via a four-wise uniform construction, yielding a circuit of size $O\left(\frac{\gamma^2 2^n}{\log(\gamma^2 2^n)} + n\right)$ that $\gamma$-approximates the function over any distribution. The paper also clarifies the regimes where the random-junta tightness matches or falls behind prior results (e.g., BHKT24) and leverages low-depth, locally computable 4-wise uniform generators, including Spielman codes, to realize the constructions. Overall, the results deepen our understanding of hardness amplification and the role of juntas in circuit lower bounds, with implications for pseudorandomness and cryptography.
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.