A key exchange system based on real quadratic fields
CRYPTO '89 Proceedings on Advances in cryptology
A course in computational algebraic number theory
A course in computational algebraic number theory
SIAM Journal on Computing
On the Power of Quantum Computation
SIAM Journal on Computing
Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer
SIAM Journal on Computing
Normal subgroup reconstruction and quantum computation using group representations
STOC '00 Proceedings of the thirty-second annual ACM symposium on Theory of computing
Efficient quantum algorithms for some instances of the non-Abelian hidden subgroup problem
Proceedings of the thirteenth annual ACM symposium on Parallel algorithms and architectures
Quantum algorithms for solvable groups
STOC '01 Proceedings of the thirty-third annual ACM symposium on Theory of computing
Quantum mechanical algorithms for the nonabelian hidden subgroup problem
STOC '01 Proceedings of the thirty-third annual ACM symposium on Theory of computing
Why haven't more quantum algorithms been found?
Journal of the ACM (JACM)
Quantum algorithms for some hidden shift problems
SODA '03 Proceedings of the fourteenth annual ACM-SIAM symposium on Discrete algorithms
Asymptotically Fast Discrete Logarithms in Quadratic Number Fields
ANTS-IV Proceedings of the 4th International Symposium on Algorithmic Number Theory
Hidden translation and orbit coset in quantum computing
Proceedings of the thirty-fifth annual ACM symposium on Theory of computing
The power of basis selection in fourier sampling: hidden subgroup problems in affine groups
SODA '04 Proceedings of the fifteenth annual ACM-SIAM symposium on Discrete algorithms
Solving Norm Form Equations over Number Fields
CAI '09 Proceedings of the 3rd International Conference on Algebraic Informatics
Quantum random walks - new method for designing quantum algorithms
SOFSEM'08 Proceedings of the 34th conference on Current trends in theory and practice of computer science
New developments in quantum algorithms
MFCS'10 Proceedings of the 35th international conference on Mathematical foundations of computer science
On the Impossibility of a Quantum Sieve Algorithm for Graph Isomorphism
SIAM Journal on Computing
Quantum Computation and the Evaluation of Tensor Networks
SIAM Journal on Computing
Quantum measurements for hidden subgroup problems with optimal sample complexity
Quantum Information & Computation
Classical cryptographic protocols in a quantum world
CRYPTO'11 Proceedings of the 31st annual conference on Advances in cryptology
Quipper: a scalable quantum programming language
Proceedings of the 34th ACM SIGPLAN conference on Programming language design and implementation
Quantum phase estimation with arbitrary constant-precision phase shift operators
Quantum Information & Computation
Quantum rotations: a case study in static and dynamic machine-code generation for quantum computers
Proceedings of the 40th Annual International Symposium on Computer Architecture
An introduction to quantum programming in quipper
RC'13 Proceedings of the 5th international conference on Reversible Computation
Quantum algorithms for one-dimensional infrastructures
Quantum Information & Computation
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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.