Rényi Divergence Guarantees for Hashing With Linear Codes
Madhura Pathegama, Alexander Barg · IEEE Transactions on Information Theory · 2025
We consider the problem of distilling uniform random bits from an unknown source with a givenp-entropy using linear hashing. As our main result, we estimate the expectedp-divergence from the uniform distribution over the ensemble of random linear codes for all integerp≥ 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 thelpsense for all integerp≥ 2.