Sean Hallgren
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.
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Index Terms
- Polynomial-time quantum algorithms for Pell's equation and the principal ideal problem
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Reviews
Harvey Cohn
This is a comprehensive paper on how the (as yet unbuilt) quantum computers would be applied to widely known classical problems of number theory, such as Pell’s equation and class structure of a (real quadratic) field. It extends the scope of the basic work of Shor [1], originally applied to primality and factorization. Quantum algorithms are refined particularly for finding the order of a subgroup, and the role of the generalized Riemann hypothesis (GRH) is fully considered. Online Computing Reviews Service
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