Non-malleable extractors and symmetric key cryptography from weak secrets

We study the question of basing symmetric key cryptography on weak secrets. In this setting, Alice and Bob share an n-bit secret W, which might not be uniformly random, but the adversary has at least k bits of uncertainty about it (formalized using conditional min-entropy). Since standard symmetric-key primitives require uniformly random secret keys, we would like to construct an authenticated key agree- ment protocol in which Alice and Bob use W to agree on a nearly uniform key R, by communicating over a public channel controlled by an active adversary Eve. We study this question in the information theoretic setting where the attacker is computationally unbounded. We show that single- round (i.e. one message) protocols do not work when k ≤ n/2 , and require poor parameters even when n/2 < k « n. On the other hand, for arbitrary values of k, we design a communication e±cient two-round (challenge-response) protocol extracting nearly k random bits. This dramatically improves the prior construction of Renner and Wolf [32], which requires θ(λ±log(n)) rounds where λ is the security parameter. Our solution takes a new approach by studying and constructing \non-malleable" seeded randomness extractors - if an attacker sees a random seed X and comes up with an arbitrarily related seed X, then we bound the relationship between R' = Ext(W;X) and R0 = Ext(W;X'). We also extend our two-round key agreement protocol to the ""fuzzy" setting, where Alice and Bob share "close" (but not equal) secrets WA and WB, and to the Bounded Retrieval Model (BRM) where the size of the secret W is huge.