Efficient Identification of Equivalences in Dynamic Graphs and Pedigree Structures

We propose a new framework for designing test and query functions for complex structures that vary across a given parameter such as genetic marker position. The operations we are interested in include equality testing, set operations, isolating unique states, duplication counting, or finding equivalence classes under identifiability constraints. A motivating application is locating equivalence classes in identity-by-descent (IBD) graphs, graph structures in pedigree analysis that change over genetic marker location. The nodes of these graphs are unlabeled and identified only by their connecting edges, a constraint easily handled by our approach. The general framework introduced is powerful enough to build a range of testing functions for IBD graphs, dynamic populations, and other structures using a minimal set of operations. The theoretical and algorithmic properties of our approach are analyzed and proved. Computational results on several simulations demonstrate the effectiveness of our approach.Comment: Code for paper available at http://www.stat.washington.edu/~hoytak/code/hashreduc

Verifying proofs in constant depth

In this paper we initiate the study of proof systems where verification of proofs proceeds by NC circuits. We investigate the question which languages admit proof systems in this very restricted model. Formulated alternatively, we ask which languages can be enumerated by NC functions. Our results show that the answer to this problem is not determined by the complexity of the language. On the one hand, we construct NC proof systems for a variety of languages ranging from regular to NP-complete. On the other hand, we show by combinatorial methods that even easy regular languages such as Exact-OR do not admit NC proof systems. We also present a general construction of proof systems for regular languages with strongly connected NFA's

An All-But-One Entropic Uncertainty Relation, and Application to Password-based Identification

Entropic uncertainty relations are quantitative characterizations of Heisenberg's uncertainty principle, which make use of an entropy measure to quantify uncertainty. In quantum cryptography, they are often used as convenient tools in security proofs. We propose a new entropic uncertainty relation. It is the first such uncertainty relation that lower bounds the uncertainty in the measurement outcome for all but one choice for the measurement from an arbitrarily large (but specifically chosen) set of possible measurements, and, at the same time, uses the min-entropy as entropy measure, rather than the Shannon entropy. This makes it especially suited for quantum cryptography. As application, we propose a new quantum identification scheme in the bounded quantum storage model. It makes use of our new uncertainty relation at the core of its security proof. In contrast to the original quantum identification scheme proposed by Damg{\aa}rd et al., our new scheme also offers some security in case the bounded quantum storage assumption fails hold. Specifically, our scheme remains secure against an adversary that has unbounded storage capabilities but is restricted to non-adaptive single-qubit operations. The scheme by Damg{\aa}rd et al., on the other hand, completely breaks down under such an attack.Comment: 33 pages, v