Improved Classical and Quantum Algorithms for Subset-Sum

Bonnetain, Xavier; Bricout, Rémi; Schrottenloher, André; Shen, Yixin

doi:10.1007/978-3-030-64834-3_22

Cited by 29 publications

(7 citation statements)

References 29 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Bernstein et al [BJLM13] improved on [HJ10] using quantum walks, and their algorithm (again under some hypotheses) runs in time O(2 0.242n ). Further quantum improvements were made in this context by [HM18] and [BBSS20]. However, we would like to emphasize that the algorithms in all these papers work for random inputs generated from some distributions.…”

Section: Related Workmentioning

confidence: 99%

Classical and Quantum Algorithms for Variants of Subset-Sum via Dynamic Programming

Allcock¹,

Hamoudi²,

Joux³

et al. 2021

Preprint

View full text Add to dashboard Cite

Subset-Sum is an NP-complete problem where one must decide if a multiset of n integers contains a subset whose elements sum to a target value m. The best known classical and quantum algorithms run in time O(2 n/2 ) and O(2 n/3 ), respectively, based on the wellknown meet-in-the-middle technique. Here we introduce a novel dynamic programming data structure with applications to Subset-Sum and a number of variants, including Equal-Sums (where one seeks two disjoint subsets with the same sum), 2-Subset-Sum (a relaxed version of Subset-Sum where each item in the input set can be used twice in the summation), and Shifted-Sums, a generalization of both of these variants, where one seeks two disjoint subsets whose sums differ by some specified value.Given any modulus p, our data structure can be constructed in time O(np), after which queries can be made in time O(n) to the lists of subsets summing to a same value modulo p. We use this data structure to give new O(2 n/2 ) and O(2 n/3 ) classical and quantum algorithms for Subset-Sum, not based on the meet-in-the-middle method. We then use the data structure in combination with variable time amplitude amplification and a quantum pair finding algorithm, extending quantum element distinctness and claw finding algorithms to the multiple solutions case, to give an O(2 0.504n ) quantum algorithm for Shifted-Sums, an improvement on the best known O(2 0.773n ) classical running time. We also study Pigeonhole Equal-Sums and Pigeonhole Modular Equal-Sums, variants of Equal-Sums where the existence of a solution is guaranteed by the pigeonhole principle. For the former problem we give classical and quantum algorithms with running time O(2 n/2 ) and O(2 2n/5 ), respectively. For the more general modular problem we give a classical algorithm which also runs in time O(2 n/2 ).

show abstract

Section: Related Workmentioning

confidence: 99%

Classical and Quantum Algorithms for Variants of Subset-Sum via Dynamic Programming

Allcock¹,

Hamoudi²,

Joux³

et al. 2021

Preprint

View full text Add to dashboard Cite

show abstract

“…Quantum walks are a powerful algorithmic tool which has been used to provide state-of-the-art algorithms for various important problems in post-quantum cryptography, such as the shortest vector problem (SVP) via lattice sieving [8], the subset sum problem [4], information set decoding [19], etc.…”

Section: Introductionmentioning

confidence: 99%

Finding many Collisions via Reusable Quantum Walks

Bonnetain¹,

Chailloux²,

Schrottenloher³

et al. 2022

Preprint

Self Cite

View full text Add to dashboard Cite

Given a random function f with domain [2 n ] and codomain [2 m ], with m ≥ n, a collision of f is a pair of distinct inputs with the same image. Collision finding is an ubiquitous problem in cryptanalysis, and it has been well studied using both classical and quantum algorithms. Indeed, the quantum query complexity of the problem is well known to be Θ(2 m/3 ), and matching algorithms are known for any value of m. The situation becomes different when one is looking for multiple collision pairs. Here, for 2 k collisions, a query lower bound of Θ(2 (2k+m)/3 ) was shown by Liu and Zhandry (EUROCRYPT 2019). A matching algorithm is known, but only for relatively small values of m, when many collisions exist. In this paper, we improve the algorithms for this problem and, in particular, extend the range of admissible parameters where the lower bound is met. Our new method relies on a chained quantum walk algorithm, which might be of independent interest. It allows to extract multiple solutions of an MNRS-style quantum walk, without having to recompute it entirely: after finding and outputting a solution, the current state is reused as the initial state of another walk. As an application, we improve the quantum sieving algorithms for the shortest vector problem (SVP), with a complexity of 2 0.2563d+o(d) instead of the previous 2 0.2570d+o(d) .

show abstract

“…In [33], the authors used a classical and quantum hybrid algorithm to analyze the SSP, and obtained a runtime of O(2 0.218n ). Their primary strategy was a merge-and-filter operation that required the help of classical memory.…”

Section: Introductionmentioning

confidence: 99%

Quantum algorithm and experimental demonstration for the subset sum problem

Zheng

Zhu

Xue

et al. 2021

Sci. China Inf. Sci.

View full text Add to dashboard Cite

To solve the subset sum problem, a well-known nondeterministic polynomial-time complete problem that is widely used in encryption and resource scheduling, we propose a feasible quantum algorithm that utilizes fewer qubits to encode and achieves quadratic speedup. Specifically, this algorithm combines an amplitude amplification algorithm with quantum phase estimation, and requires n + t + 1 qubits and O(2 (0.5+o(1))n ) operations to obtain the solution, where n is the number of elements, and t is the number of qubits used to store the eigenvalues. To verify the performance of the algorithm, we simulate the algorithm with the online quantum simulator of IBM named ibmq simulator using Qiskit and then run it on two IBM quantum computers called ibmq santiago and ibmq bogota. The experimental results indicate that compared with the brute force algorithm, the proposed algorithm results in quadratic acceleration for the problem of a set S with four elements and two subsets whose sum equals target w. Using the iterator twice, we obtain success probabilities of 0.940 ± 0.004, 0.751 ± 0.040, and 0.665 ± 0.060 on the simulator, ibmq santiago, and ibmq bogota, respectively, and the fidelity between the theoretical and experimental quantum states is calculated to be 0.944 ± 0.002, 0.753 ± 0.017, and 0.657 ± 0.028, respectively. If the error rates of the experimental quantum logic gates can be reduced, the success probabilities of the proposed algorithm on real quantum devices can be further improved.

show abstract

Improved Classical and Quantum Algorithms for Subset-Sum

Cited by 29 publications

References 29 publications

Classical and Quantum Algorithms for Variants of Subset-Sum via Dynamic Programming

Classical and Quantum Algorithms for Variants of Subset-Sum via Dynamic Programming

Finding many Collisions via Reusable Quantum Walks

Quantum algorithm and experimental demonstration for the subset sum problem

Contact Info

Product

Resources

About