### Abstract

We give polynomial-time quantum algorithms for three problems from computational algebraic number theory. The first is Pell's equation. Given a positive nonsquare integer d, Pell's equation is x^{2} - dy^{2} = 1 and the goal is to find its integer solutions. Factoring integers reduces to finding integer solutions of Pell's equation, but a reduction in the other direction is not known and appears more difficult. The second problem we solve is the principal ideal problem in real quadratic number fields. This problem, which is at least as hard as solving Pell's equation, is the one-way function underlying the Buchmann - Williams key exchange system, which is therefore broken by our quantum algorithm. Finally, assuming the generalized Riemann hypothesis, this algorithm can be used to compute the class group of a real quadratic number field.

Original language | English (US) |
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Article number | 1206039 |

Journal | Journal of the ACM |

Volume | 54 |

Issue number | 1 |

DOIs | |

State | Published - Mar 1 2007 |

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### All Science Journal Classification (ASJC) codes

- Hardware and Architecture
- Information Systems
- Computer Graphics and Computer-Aided Design
- Software
- Theoretical Computer Science
- Computational Theory and Mathematics

### Cite this

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**Polynomial-time quantum algorithms for Pell's equation and the principal ideal problem.** / Hallgren, Sean.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Polynomial-time quantum algorithms for Pell's equation and the principal ideal problem

AU - Hallgren, Sean

PY - 2007/3/1

Y1 - 2007/3/1

N2 - We give polynomial-time quantum algorithms for three problems from computational algebraic number theory. The first is Pell's equation. Given a positive nonsquare integer d, Pell's equation is x2 - dy2 = 1 and the goal is to find its integer solutions. Factoring integers reduces to finding integer solutions of Pell's equation, but a reduction in the other direction is not known and appears more difficult. The second problem we solve is the principal ideal problem in real quadratic number fields. This problem, which is at least as hard as solving Pell's equation, is the one-way function underlying the Buchmann - Williams key exchange system, which is therefore broken by our quantum algorithm. Finally, assuming the generalized Riemann hypothesis, this algorithm can be used to compute the class group of a real quadratic number field.

AB - We give polynomial-time quantum algorithms for three problems from computational algebraic number theory. The first is Pell's equation. Given a positive nonsquare integer d, Pell's equation is x2 - dy2 = 1 and the goal is to find its integer solutions. Factoring integers reduces to finding integer solutions of Pell's equation, but a reduction in the other direction is not known and appears more difficult. The second problem we solve is the principal ideal problem in real quadratic number fields. This problem, which is at least as hard as solving Pell's equation, is the one-way function underlying the Buchmann - Williams key exchange system, which is therefore broken by our quantum algorithm. Finally, assuming the generalized Riemann hypothesis, this algorithm can be used to compute the class group of a real quadratic number field.

UR - http://www.scopus.com/inward/record.url?scp=33947288591&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33947288591&partnerID=8YFLogxK

U2 - 10.1145/1206035.1206039

DO - 10.1145/1206035.1206039

M3 - Article

VL - 54

JO - Journal of the ACM

JF - Journal of the ACM

SN - 0004-5411

IS - 1

M1 - 1206039

ER -