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

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We give polynomial-time quantum algorithms for two problems from computational algebraic number theory. The first is Pell's equation. Given a positive non-square 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 is the principal ideal problem in real quadratic number fields. Solving this problem is at least as hard as solving Pell's equation, and is the basis of a cryptosystem which is broken by our algorithm.

Original languageEnglish (US)
Pages (from-to)653-658
Number of pages6
JournalConference Proceedings of the Annual ACM Symposium on Theory of Computing
Publication statusPublished - Sep 23 2002
EventProceedings of the 34th Annual ACM Symposium on Theory of Computing - Montreal, Que., Canada
Duration: May 19 2002May 21 2002


All Science Journal Classification (ASJC) codes

  • Software

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