Non-Uniform Attacks Against Pseudoentropy

De, Trevisan and Tulsiani [CRYPTO 2010] show that every distribution over n-bit strings which has constant statistical distance to uniform (e.g., the output of a pseudorandom generator mapping n-1 to n bit strings), can be distinguished from the uniform distribution with advantage epsilon by a circuit of size O( 2^n epsilon^2). We generalize this result, showing that a distribution which has less than k bits of min-entropy, can be distinguished from any distribution with k bits of delta-smooth min-entropy with advantage epsilon by a circuit of size O(2^k epsilon^2/delta^2). As a special case, this implies that any distribution with support at most 2^k (e.g., the output of a pseudoentropy generator mapping k to n bit strings) can be distinguished from any given distribution with min-entropy k+1 with advantage epsilon by a circuit of size O(2^k epsilon^2). Our result thus shows that pseudoentropy distributions face basically the same non-uniform attacks as pseudorandom distributions.