Data compression with nearly uniform output

For any lossless fixed-length compression scheme operating at the optimal coding rate, it is known that the encoder output is not uniform in variational distance, which yet might be desirable in some security schemes. In the case of independent and identically distributed (i.i.d.) sources, uniformity in divergence might be achieved if a uniformly distributed sequence, called seed, of length d_n negligible compared to the message length n, is shared between the encoder and the decoder. We show that the optimal scaling of d_n that jointly ensures an optimal coding rate and a uniform encoder output in divergence, is roughly on the order of √n. We also develop a near optimal achievability scheme using invertible extractors.