We consider the problem of distilling uniform random bits from an unknown source with a given $p$-entropy using linear hashing. As our main result, we estimate the expected $p$-divergence from the uniform distribution over the ensemble of random linear codes for all integer $p\ge 2$. The proof relies on analyzing how additive noise, determined by a random element of the code from the ensemble, acts on the source distribution. This action leads to the transformation of the source distribution into an approximately uniform one, a process commonly referred to as distribution smoothing. We also show that hashing with Reed-Muller matrices reaches intrinsic randomness of memoryless Bernoulli sources in the $l_p$ sense for all integer $p\ge 2$.

翻译：我们考虑了利用线性哈希从具有给定$p$熵的未知信源中提取均匀随机比特的问题。作为主要结果，我们估计了在所有整数$p\ge 2$情况下，随机线性编码系综与均匀分布之间的期望$p$散度。证明依赖于分析由系综中编码的随机元素决定的加性噪声如何作用于信源分布。这种作用导致信源分布转化为近似均匀分布，这一过程通常被称为分布平滑。我们还证明，对于所有整数$p\ge 2$，使用Reed-Muller矩阵进行哈希在$l_p$意义上能够达到无记忆伯努利信源的内在随机性。