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. \bullet 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^{ \Theta (n)} time and requires q ^{ \Theta(n /q ) } memory. In fact, we give a similar time-memory tradeoff for Discrete Gaussian sampling above the smoothing parameter. \bullet 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)}. \bullet 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.
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. \bullet A new algorithm for SVP that provides a smooth tradeoff between time complexity and memory requirement. For any positive integer 4\leq q\leq \sqrt{n , our algorithm takes q^{ 11n+o (n)} time and requires poly(n)\cdot q ^{ 16n /q ^2 } memory. This tradeoff which ranges from enumeration (q=\sqrt{n time-memory tradeoff for Discrete Gaussian sampling above the smoothing parameter. \bullet 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)}. \bullet A classical algorithm for SVP that runs in time 2^{1.73n+o(n)} time and 2^{0.5n+o(n)} space . This improves over an algorithm of [CCL18] that has the same space complexity.
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