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On the Quantum Complexity of the Continuous Hidden Subgroup Problem

  • Koen de BoerEmail author
  • Léo Ducas
  • Serge FehrEmail author
Conference paper
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Part of the Lecture Notes in Computer Science book series (LNCS, volume 12106)

Abstract

The Hidden Subgroup Problem (HSP) aims at capturing all problems that are susceptible to be solvable in quantum polynomial time following the blueprints of Shor’s celebrated algorithm. Successful solutions to this problems over various commutative groups allow to efficiently perform number-theoretic tasks such as factoring or finding discrete logarithms.

The latest successful generalization (Eisenträger et al. STOC 2014) considers the problem of finding a full-rank lattice as the hidden subgroup of the continuous vector space \(\mathbb {R}^m\), even for large dimensions m. It unlocked new cryptanalytic algorithms (Biasse-Song SODA 2016, Cramer et al. EUROCRYPT 2016 and 2017), in particular to find mildly short vectors in ideal lattices.

The cryptanalytic relevance of such a problem raises the question of a more refined and quantitative complexity analysis. In the light of the increasing physical difficulty of maintaining a large entanglement of qubits, the degree of concern may be different whether the above algorithm requires only linearly many qubits or a much larger polynomial amount of qubits.

This is the question we start addressing with this work. We propose a detailed analysis of (a variation of) the aforementioned HSP algorithm, and conclude on its complexity as a function of all the relevant parameters. Our modular analysis is tailored to support the optimization of future specialization to cases of cryptanalytic interests. We suggest a few ideas in this direction.

Keywords

Quantum algorithm Hidden subgroup Period finding Fourier transform Cryptanalysis 
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Notes

Acknowledgments

We would like to thank Stacey Jeffery, Oded Regev and Ronald de Wolf for helpful discussions on the topic of this article.

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Copyright information

© International Association for Cryptologic Research 2020

Authors and Affiliations

  1. 1.Cryptology GroupCentrum Wiskunde & Informatica (CWI)AmsterdamThe Netherlands
  2. 2.Mathematical InstituteLeiden UniversityLeidenThe Netherlands

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