Hypercube orientations with only two in-degrees

We consider the problem of orienting the edges of the $n$-dimensional hypercube so only two different in-degrees $a$ and $b$ occur. We show that this can be done, for two specified in-degrees, if and only if an obvious necessary condition holds. Namely, there exist non-negative integers $s$ and $t$ so that $s+t=2^n$ and $as+bt=n2^{n-1}$. This is connected to a question arising from constructing a strategy for a "hat puzzle."Comment: 9 pages, 4 figure

5PM: Secure Pattern Matching

In this paper we consider the problem of secure pattern matching that allows single-character wildcards and substring matching in the malicious (stand-alone) setting. Our protocol, called 5PM, is executed between two parties: Server, holding a text of length $n$, and Client, holding a pattern of length $m$ to be matched against the text, where our notion of matching is more general and includes non-binary alphabets, non-binary Hamming distance and non-binary substring matching. 5PM is the first secure expressive pattern matching protocol designed to optimize round complexity by carefully specifying the entire protocol round by round. In the malicious model, 5PM requires $O((m+n)k^2)$ bandwidth and $O(m+n)$ encryptions, where $m$ is the pattern length and $n$ is the text length. Further, 5PM can hide pattern size with no asymptotic additional costs in either computation or bandwidth. Finally, 5PM requires only two rounds of communication in the honest-but-curious model and eight rounds in the malicious model. Our techniques reduce pattern matching and generalized Hamming distance problems to a novel linear algebra formulation that allows for generic solutions based on any additively homomorphic encryption. We believe our efficient algebraic techniques are of independent interest