Theorem 8.2 [CR92] The threshold r2 exists and is r2 = 1. Proof: We proved one direction of Theorem 8.2 in the last lecture, by showing that Pr[φ is satisfiable]→ 1 for r ≤ (1− ). We now prove the other direction. Assume m = (1 + )n. Define t = n. (We’ll see the reason for this choice later.) The approach we’ll take is to define a collection of sets of clauses that are not satisfiable, and then...

We characterize the structure of maximum-size sum-free subsets of a random subset of an Abelian group G. In particular, we determine the threshold above which, with high probability as |G| → ∞, each such subset is contained in some maximum-size sum-free subset of G, whenever q divides |G| for some (fixed) prime q with q ≡ 2 (mod 3). Moreover, in the special case G = Z2n, we determine the sharp ...

This paper presents a deterministic and efficient algorithm for online facility location. The algorithm is based on a simple hierarchical partitioning and is extremely simple to implement. It also applies to a variety of models, i.e., models where the facilities can be placed anywhere in the region, or only at customer sites, or only at fixed locations. The paper shows that the algorithm is O(l...

We introduce software for the generation of instances of the LWE and Ring-LWE problems, allowing both the generation of generic instances and also particular instances closely-related to those arising from cryptomania proposals in the literature. Our goal is to allow researchers to attack different instances in order to assess the practical hardness of LWE and Ring-LWE. This will in turn give i...

We construct a lattice-based predicate encryption scheme for multi-dimensional range and multidimensional subset queries. Our scheme is selectively secure and weakly attribute-hiding, and its security is based on the standard learning with errors (LWE) assumption. Multi-dimensional range and subset queries capture many interesting applications pertaining to searching on encrypted data. To the b...

A case study of clinical practice in children with glue ear is presented. The case is part of a larger project, funded by the North Thames Research and Development Programme, that sought to explore the part played by clinicians in the implementation of research and development into practice in two areas: adult asthma and glue ear in children. What is striking about this case is the differences ...

In this paper we present a new attack on the polynomial version of the Ring-LWE assumption, for certain carefully chosen number fields. This variant of RLWE, introduced in [BV11] and called the PLWE assumption, is known to be as hard as the RLWE assumption for 2-power cyclotomic number fields, and for cyclotomic number fields in general with a small cost in terms of error growth. For general nu...

Let G be a finite abelian group, let 0 < α < 1, and let A ⊆ G be a random set of size |G|. We let μ(A) = max B,C:|B|=|C|=|A| |{(a, b, c) ∈ A×B × C : a = b+ c}|. The issue is to determine upper bounds on μ(A) that hold with high probability over the random choice of A. Mennink and Preneel [4] conjecture that μ(A) should be close to |A| (up to possible logarithmic factors in |G|) for α ≤ 1/2 and ...

We report a case of an elderly patient with a limited form of Wegener granulomatosis, which simulated the clinical and imaging features of organizing pneumonia. Here we call attention to this atypical case presentation that eloquently illustrates the many faces of Wegener granulomatosis.

We show that several versions of Floyd and Rivest’s algorithm Select for finding the kth smallest of n elements require at most n+min{k, n− k}+ o(n) comparisons on average and with high probability. This rectifies the analysis of Floyd and Rivest, and extends it to the case of nondistinct elements. Our computational results confirm that Select may be the best algorithm in practice.