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IMPROVED CLASSICAL AND QUANTUM ALGORITHMS FOR THE SHORTEST VECTOR PROBLEM VIA BOUNDED DISTANCE DECODING.

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Titel:	IMPROVED CLASSICAL AND QUANTUM ALGORITHMS FOR THE SHORTEST VECTOR PROBLEM VIA BOUNDED DISTANCE DECODING.
Autoren:	AGGARWAL, DIVESH¹ dcsdiva@nus.edu.sg, YANLIN CHEN² yanlin@cwi.nl, KUMARS, RAJENDRA³ rjndr2503@gmail.com, YIXIN SHEN⁴ yixin.shen@inria.fr
Quelle:	SIAM Journal on Computing. 2025, Vol. 54 Issue 2, p233-278. 46p.
Schlagwörter:	TIME complexity, RANDOM access memory, QUANTUM computing, ISOMORPHISM (Mathematics), *ALGORITHMS
Abstract:	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: (1) A new algorithm for SVP that provides a smooth tradeoff between time complexity and memory requirement. For any positive integer 4 ≤ q ≤ √(n) our algorithm takes q 13n + o(n) time and requires poly(n) q16n/q² memory. This tradeoff, which ranges from enumeration (q = √n) to sieving (q constant), is a consequence of a new time-memory tradeoff for discrete Gaussian sampling above the smoothing parameter. (2) A quantum algorithm for SVP that runs in time 20.95n + o(n) and requires 20.5n + o(n) classical memory and poly(n) qubits. In a quantum random access memory (QRAM) model, this algorithm takes only 20.835n + o(n) time and requires a QRAM of size 20.293n + o(n), poly(n) qubits and 20.5n classical space. This improves over the previously fastest classical (which is also the fastest quantum) algorithm due to [D. Aggarwal et al., Solving the shortest vector problem in 2n time using discrete Gaussian sampling: Extended abstract, in Proceedings of the Forty-Seventh Annual ACM on Symposium on Theory of Computing (STOC), 2015, pp. 733-742] that has a time and space complexity 2n + o(n) (3) A classical algorithm for SVP that runs in time 21.669n + o(n) time and 20.5n + o(n) space. This improves over an algorithm of [Y. Chen, K. Chung, and C. Lai, Quantum Inf. Comput., 18 (2018), pp. 285-306] that has the same space complexity. The time complexity of our classical and quantum algorithms are obtained using a known upper bound on a quantity related to the lattice kissing number, which is 20.402n. We conjecture that for most lattices this quantity is a 2o(n). Assuming that this is the case, our classical algorithm runs in time 21.292 + o(n), our quantum algorithm runs in time 20.75n + o(n) and our quantum algorithm in a QRAM model runs in time 20.667n + o(n) . As a direct application of our result, using the reduction in [L. Ducas, Des. Codes. Cryptogr., 922024), pp. 909-916], we obtain a provable quantum algorithm for the lattice isomorphism problem in the case of the trivial lattice Zn (ZLIP) that runs in time 20.417n + o(n) . Our algorithm requires a QRAM of size 20.147n + o(n), poly(n) qubits and 20.25n classical space. [ABSTRACT FROM AUTHOR]
Datenbank:	Academic Search Index

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Beschreibung
Abstract:	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: (1) A new algorithm for SVP that provides a smooth tradeoff between time complexity and memory requirement. For any positive integer 4 ≤ q ≤ √(n) our algorithm takes q 13n + o(n) time and requires poly(n) q16n/q² memory. This tradeoff, which ranges from enumeration (q = √n) to sieving (q constant), is a consequence of a new time-memory tradeoff for discrete Gaussian sampling above the smoothing parameter. (2) A quantum algorithm for SVP that runs in time 20.95n + o(n) and requires 20.5n + o(n) classical memory and poly(n) qubits. In a quantum random access memory (QRAM) model, this algorithm takes only 20.835n + o(n) time and requires a QRAM of size 20.293n + o(n), poly(n) qubits and 20.5n classical space. This improves over the previously fastest classical (which is also the fastest quantum) algorithm due to [D. Aggarwal et al., Solving the shortest vector problem in 2n time using discrete Gaussian sampling: Extended abstract, in Proceedings of the Forty-Seventh Annual ACM on Symposium on Theory of Computing (STOC), 2015, pp. 733-742] that has a time and space complexity 2n + o(n) (3) A classical algorithm for SVP that runs in time 21.669n + o(n) time and 20.5n + o(n) space. This improves over an algorithm of [Y. Chen, K. Chung, and C. Lai, Quantum Inf. Comput., 18 (2018), pp. 285-306] that has the same space complexity. The time complexity of our classical and quantum algorithms are obtained using a known upper bound on a quantity related to the lattice kissing number, which is 20.402n. We conjecture that for most lattices this quantity is a 2o(n). Assuming that this is the case, our classical algorithm runs in time 21.292 + o(n), our quantum algorithm runs in time 20.75n + o(n) and our quantum algorithm in a QRAM model runs in time 20.667n + o(n) . As a direct application of our result, using the reduction in [L. Ducas, Des. Codes. Cryptogr., 922024), pp. 909-916], we obtain a provable quantum algorithm for the lattice isomorphism problem in the case of the trivial lattice Zn (ZLIP) that runs in time 20.417n + o(n) . Our algorithm requires a QRAM of size 20.147n + o(n), poly(n) qubits and 20.25n classical space. [ABSTRACT FROM AUTHOR]
ISSN:	00975397
DOI:	10.1137/22M1486959