Polynomial-Time Quantum Algorithms for Pell’s Equation and the Principal Ideal Problem

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 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 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