MACing from an arbitrary high min-entropy shared secret
As you may know, the problem of taking a weak shared key, about which the adversary knows some information, and extracting from it a key about which the adversary has essentially no information, is often referred to as privacy amplification. Once this is done, we can use the new strong key for authentication, encryption, etc.

If interaction is allowed, Alice and Bob can extract a strong key over an inauthentic channel very efficiently, in particular there is no requirement that the min-entropy rate be at least 1/2. See "Privacy Ampliﬁcation with Asymptotically Optimal Entropy Loss" by Chandran, Kanukurthi, Ostrovsky, Reyzin, and references therein.

Their introduction mentions that non-interactive solutions (one message from Alice to Bob) exist as well (references 5, 10, 15), but these do require min-entropy rate 1/2. Depending on the application, another possibility for a one-message solution would be to use deterministic (aka seedless) extractors, which work only for certain classes of sources with some restricted structure. For instance, the adversary knows some fraction of the key's bits, but the remainder are uniformly random to him. See for example Kamp and Zuckerman, "Deterministic Extractors for Bit-Fixing Sources and Exposure-Resilient Cryptography" or Kamp, Rao, Vadhan, Zuckerman, "Deterministic Extractors for Small-Space Sources"