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

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 find...

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Bibliographic Details
Published in:Journal of the ACM 2007-03, Vol.54 (1), p.1-19
Main Author: HALLGREN, Sean
Format: Article
Language:English
Subjects:
Algebra
Algorithmics. Computability. Computer arithmetics
Algorithms
Applied sciences
Computation
Computer science
control theory
systems
Computers
Cryptography
Exact sciences and technology
Exchange
Hypotheses
Integers
Laboratories
Mathematical analysis
Mathematical models
Number theory
Polynomials
Principal components analysis
Quantum computing
Studies
Theoretical computing
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Summary: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.
ISSN:0004-5411
1557-735X
DOI:10.1145/1206035.1206039