# Improved (Provable) Algorithms for the Shortest Vector Problem via Bounded Distance Decoding

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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