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We investigate various problems related to circulant graphs — finding the shortest path between two vertices, finding the shortest loop, and computing the diameter. These problems are related to shortest vector problems in a special class of lattices. We give matching upper and lower bounds on the length of the shortest loop. We claim NP-hardness results, and establish a worst-case/average-case connection for the shortest loop problem. A pseudo-polynomial time algorithm for these problems is also given. Our main tools are results and methods from the geometry of numbers.