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By applying Grover's quantum search algorithm to the lattice algorithms of Micciancio and Voulgaris, Nguyen and Vidick, Wang et al., and Pujol and Stehl\'{e}, we obtain improved asymptotic quantum results for solving the shortest vector problem. With quantum computers we can provably find a shortest vector in time 21.799n+o(n), improving upon the classical time complexity of 22.465n+o(n) of Pujol and Stehl\'{e} and the 22n+o(n) of Micciancio and Voulgaris, while heuristically we expect to find a shortest vector in time 20.312n+o(n), improving upon the classical time complexity of 20.384n+o(n) of Wang et al. These quantum complexities will be an important guide for the selection of parameters for post-quantum cryptosystems based on the hardness of the shortest vector problem
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