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Some Improved Complexity Results for GapSVP and uSVP

In this paper, first, it is proved that finding the approximate shortest vector with length in [λ1, γλ1] could be Karp-reduced to GapSVPγ̃ where γ = γ 1 n(n+log2(γn))(n−1) . Second, it is proved that shortest vector problem itself could be reduced to GapSVP with a quite small gap. Third, we improve the complexity results of uSVP, proving uSVP could be reduced from SVP (our results are better than any known result). What’s more, we prove that the search version of uSVP could be reduced to decisional version of uSVP with almost the same gap.