R\'enyi divergence guarantees for hashing with linear codes

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≥ 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 lₚ sense for all integer p≥ 2.