Universal Covertness for Discrete Memoryless Sources
share
Consider a sequence X^n of length n emitted by a Discrete Memoryless Source (DMS) with unknown distribution p_X. The objective is to construct a lossless source code that maps X^n to a sequence Y^m of length m that is indistinguishable, in terms of Kullback-Leibler divergence, from a sequence emitted by another DMS with known distribution p_Y. The main result is the existence of a coding scheme that performs this task with an optimal ratio m/n equal to H(X)/H(Y), the ratio of the Shannon entropies of the two distributions, as n goes to infinity. The coding scheme overcomes the challenges created by the lack of knowledge about p_X by relying on a sufficiently fine estimation of H(X), followed by an appropriately designed type-based source coding that jointly performs source resolvability and universal lossless source coding. The result recovers and extends previous results that either assume p_X or p_Y uniform, or p_X known. The price paid for these generalizations is the use of common randomness with vanishing rate, whose length roughly scales as the square root of n. By allowing common randomness strictly larger than the square root of n but still negligible compared to n, a constructive low-complexity encoding and decoding counterpart to the main result is also provided for binary sources by means of polar codes.
READ FULL TEXT