In this paper, we present a simple attack on LWE and Ring LWE encryption schemes used directly as Key Encapsulation Mechanisms (KEMs). This attack could work due to the fact that a key mismatch in a KEM is accessible to an adversary. Our method clearly indicates that any LWE or RLWE (or any similar type of construction) encryption directly used as KEM can be broken by modifying our attack method according to the respective cases.

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In this paper, we consider the problem of approximating the longest path in a grid graph that possesses a square-free 2-factor. By (1) analyzing several characteristics of four types of cross cells and (2) presenting three cycle-merging operations and one extension operation, we propose an algorithm that finds in an n-vertex grid graph G, a long path of length at least 5/6 n + 2. Our approximation algorithm runs in quadratic time. In addition to its theoretical value, our work has also an interesting application in image-guided maze generation.

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We construct a 2-dimensional complex manifold$X$which is the increasing union of proper subdomains that are biholomorphic to ℂ^{2}, but$X$is not Stein.