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Consider key agreement by two parties who start out knowing a common secret (which we refer to as "pass-string", a generalization of "password"), but face two complications: (1) the pass-string may come from a low-entropy distribution, and (2) the two parties' copies of the pass-string may have some noise, and thus not match exactly. We provide the first efficient and general solutions to this problem that enable, for example, key agreement based on commonly used biometrics such as iris scans. The problem of key agreement with each of these complications individually has been well studied in literature. Key agreement from low-entropy shared pass-strings is achieved by password-authenticated key exchange (PAKE), and key agreement from noisy but highentropy shared pass-strings is achieved by information-reconciliation protocols as long as the two secrets are "close enough." However, the problem of key agreement from noisy low-entropy pass-strings has never been studied. We introduce (universally composable) fuzzy password-authenticated key exchange (fPAKE), which solves exactly this problem. fPAKE does not have any entropy requirements for the pass-strings, and enables secure key agreement as long as the two pass-strings are "close" for some notion of closeness. We also give two constructions. The first construction achieves our fPAKE definition for any (efficiently computable) notion of closeness, including those that could not be handled before even in the high-entropy setting. It uses Yao's garbled circuits in a way that is only two times more costly than their use against semi-honest adversaries, but that guarantees security against malicious adversaries. The second construction is more efficient, but achieves our fPAKE definition only for pass-strings with low Hamming distance. It builds on very simple primitives: robust secret sharing and PAKE.

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This paper investigates pattern avoidance in linear extensions of particular partially ordered sets (posets). Since the problem of enumerating pattern-avoiding linear extensions of posets without any additional restrictions is a very hard one, we focus on the class of posets called combs. A comb consists of a fully ordered spine and several fully ordered teeth, where the first element of each tooth coincides with a corresponding element of the spine. We consider two natural assignments of integers to elements of the combs; we refer to the resulting integer posets as type-α and type-β combs. In this paper, we enumerate the linear extensions of type-α and type-β combs that avoid some of the length-3 patterns w ∈ S 3 . Most notably, we shown the number of linear extensions of type-β combs that avoid 312 to be the same as the number 1 st+1 s(t+1) s of (t + 1)-ary trees on s nodes, where t is the length of each tooth and s is the length of the comb spine or, equivalently, the number of its teeth. We also investigate the enumeration of linear extensions of type-α and type-β combs that avoid multiple length-3 patterns simultaneously.