Strong Chain Rules for Min-Entropy under Few Bits Spoiled

It is well established that the notion of min-entropy fails to satisfy the chain rule of the form H(X,Y ) = H(X|Y )+ H(Y ), known for Shannon Entropy. The lack of a chain rule causes a lot of technical difficulties, particularly in cryptography where the chain rule would be a natural way to analyze how min-entropy is split among smaller blocks. Such problems arise for example when constructing extractors and dispersers. We show that any sequence of variables exhibits a very strong strong block-source structure (conditional distributions of blocks are nearly flat) when we spoil few correlated bits. This implies, conditioned on the spoiled bits, that splitting-recombination properties hold. In particular, we have many nice properties that minentropy doesn’t obey in general, for example strong chain rules, “information can’t hurt” inequalities, equivalences of average and worst-case conditional entropy definitions and others. Quantitatively, for any sequence X1, . . . , Xt of random variables over an alphabet X we prove that, when conditioned on m = t · O(log log |X | + log log(1/ǫ) + log t) bits of auxiliary information, all conditional distributions of the form Xi|X<i are ǫ-close to be nearly flat (only a constant factor away). The argument is combinatorial (based on simplex coverings). This result may be used as a generic tool for exhibiting blocksource structures. We demonstrate this by reproving the fundamental converter due to Nisan and Zuckermann (J. Computer and System Sciences, 1996), which shows that sampling blocks from a min-entropy source roughly preserves the entropy rate. Our bound implies, only by straightforward chain rules, an additive loss of o(1) (for sufficiently many samples), which qualitatively meets the first tighter analysis of this problem due to Vadhan (CRYPTO’03), obtained by large deviation techniques. Keywords—chain rule, min-entropy, spoiling knowledge, block sources, local extractors