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.

翻译：考虑一个对于规模为$s$的电路具有中等难度的函数。当$s$足够大时，Impagliazzo的硬核引理保证存在一个常数密度的输入子集，在该子集上同一函数对于规模$s'<\!\!<s$的电路具有极高的难度。Blanc、Hayderi、Koch和Tan [FOCS 2024]最近证明，在某些参数范围内，该引理中从$s$到$s'$的性能衰减在量级上是紧致的。我们通过证明随机junta以高概率见证了硬核引理的紧致性，为几乎所有相关参数范围提供了更简单且更通用的证明。