research-article

A quantum algorithm for computing the unit group of an arbitrary degree number field

Authors:

Kirsten Eisenträger,

Fang SongAuthors Info & Claims

STOC '14: Proceedings of the forty-sixth annual ACM symposium on Theory of computing

Pages 293 - 302

https://doi.org/10.1145/2591796.2591860

Published: 31 May 2014 Publication History

Abstract

Computing the group of units in a field of algebraic numbers is one of the central tasks of computational algebraic number theory. It is believed to be hard classically, which is of interest for cryptography. In the quantum setting, efficient algorithms were previously known for fields of constant degree. We give a quantum algorithm that is polynomial in the degree of the field and the logarithm of its discriminant. This is achieved by combining three new results. The first is a classical algorithm for computing a basis for certain ideal lattices with doubly exponentially large generators. The second shows that a Gaussian-weighted superposition of lattice points, with an appropriate encoding, can be used to provide a unique representation of a real-valued lattice. The third is an extension of the hidden subgroup problem to continuous groups and a quantum algorithm for solving the HSP over the group Rⁿ.

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Index Terms

A quantum algorithm for computing the unit group of an arbitrary degree number field
1. Mathematics of computing
  1. Mathematical analysis
    1. Numerical analysis
2. Theory of computation
  1. Design and analysis of algorithms

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cover image ACM Conferences

STOC '14: Proceedings of the forty-sixth annual ACM symposium on Theory of computing

May 2014

984 pages

ISBN:9781450327107

DOI:10.1145/2591796

Program Chair:
David Shmoys
Cornell University

Copyright © 2014 ACM.

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Published: 31 May 2014

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STOC '14: Symposium on Theory of Computing

May 31 - June 3, 2014

New York, New York

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STOC '14 Paper Acceptance Rate 91 of 319 submissions, 29%;

Overall Acceptance Rate 1,469 of 4,586 submissions, 32%

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Cited By

Lee YBoche HKutyniok G(2024)Computability of OptimizersIEEE Transactions on Information Theory10.1109/TIT.2023.334707170:4(2967-2983)Online publication date: Apr-2024
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Nie JZhu QLi MSun X(2023)Quantum Circuit Design for Integer Multiplication Based on Schönhage–Strassen AlgorithmIEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems10.1109/TCAD.2023.327930042:12(4791-4802)Online publication date: Dec-2023
https://doi.org/10.1109/TCAD.2023.3279300
Jiang KWang ALuo HLiu GYu YWang X(2023)Exploiting the Symmetry of : Randomization and the Automorphism ProblemAdvances in Cryptology – ASIACRYPT 202310.1007/978-981-99-8730-6_6(167-200)Online publication date: 4-Dec-2023
https://dl.acm.org/doi/10.1007/978-981-99-8730-6_6
Barbulescu RPoulalion A(2023)The Special Case of Cyclotomic Fields in Quantum Algorithms for Unit GroupsProgress in Cryptology - AFRICACRYPT 202310.1007/978-3-031-37679-5_10(229-251)Online publication date: 13-Jul-2023
https://doi.org/10.1007/978-3-031-37679-5_10
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https://doi.org/10.1007/978-3-031-26326-2_3
Bernard OLesavourey ANguyen TRoux-Langlois A(2023)Log-$$\mathcal {S}$$-unit Lattices Using Explicit Stickelberger Generators to Solve Approx Ideal-SVPAdvances in Cryptology – ASIACRYPT 202210.1007/978-3-031-22969-5_23(677-708)Online publication date: 25-Jan-2023
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Shim K(2022)A Survey on Post-Quantum Public-Key Signature Schemes for Secure Vehicular CommunicationsIEEE Transactions on Intelligent Transportation Systems10.1109/TITS.2021.313166823:9(14025-14042)Online publication date: Sep-2022
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Boche HPohl V(2022)On Non-Detectability of Non-Computability and the Degree of Non-Computability of Solutions of Circuit and Wave Equations on Digital ComputersIEEE Transactions on Information Theory10.1109/TIT.2022.317283768:8(5561-5578)Online publication date: Aug-2022
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Biasse JBonnetain XKirshanova ESchrottenloher ASong F(2022)Quantum algorithms for attacking hardness assumptions in classical and post‐quantum cryptographyIET Information Security10.1049/ise2.1208117:2(171-209)Online publication date: 29-Aug-2022
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Ye ZLi L(2022)Sample complexity of hidden subgroup problemTheoretical Computer Science10.1016/j.tcs.2022.04.014922:C(108-121)Online publication date: 24-Jun-2022
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