Improved (Provable) Algorithms for the Shortest Vector Problem via Bounded Distance Decoding
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The most important computational problem on lattices is the Shortest Vector Problem (SVP). In this paper we present new algorithms that improve the state-of-the-art for provable classical/quantum algorithms for SVP. We present the following results. ∙ A new algorithm for SVP that provides a smooth tradeoff between time complexity and memory requirement. For any positive integer q>1, our algorithm takes q^Θ(n) time and requires q^Θ(n/q) memory. In fact, we give a similar time-memory tradeoff for Discrete Gaussian sampling above the smoothing parameter. ∙ A quantum algorithm that runs in time 2^0.9532n+o(n) and requires 2^0.5n+o(n) classical memory and poly(n) qubits. This improves over the previously fastest classical (which is also the fastest quantum) algorithm due to [ADRS15] that has a time and space complexity 2^n+o(n). ∙ A classical algorithm for SVP that runs in time 2^1.73n+o(n) time and 2^0.5n+o(n) space and improves over an algorithm from [CCL18] that has the same space complexity.
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