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

Research output: Contribution to journalArticle

53 Citations (Scopus)

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 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.

Original languageEnglish (US)
Article number1206039
JournalJournal of the ACM
Volume54
Issue number1
DOIs
StatePublished - Mar 1 2007

Fingerprint

Pell's equation
Quantum Algorithms
Polynomial-time Algorithm
Polynomials
Integer
Quadratic field
Number field
Number theory
One-way Function
Key Exchange
Riemann hypothesis
Class Group
Algebraic number
Factoring

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.

In: Journal of the ACM, Vol. 54, No. 1, 1206039, 01.03.2007.

Research output: Contribution to journalArticle

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