A course in computational algebraic number theory
A course in computational algebraic number theory
Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer
SIAM Journal on Computing
Computing ray class groups, conductors and discriminants
Mathematics of Computation
Algorithmic methods for finitely generated Abelian groups
Journal of Symbolic Computation - Special issue on computational algebra and number theory: proceedings of the second Magma conference
Fast quantum algorithms for computing the unit group and class group of a number field
Proceedings of the thirty-seventh annual ACM symposium on Theory of computing
Polynomial time quantum algorithm for the computation of the unit group of a number field
Proceedings of the thirty-seventh annual ACM symposium on Theory of computing
Lattices that admit logarithmic worst-case to average-case connection factors
Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
Algorithmic Number Theory. Lattices, Number Fields, Curves and Cryptography
Algorithmic Number Theory. Lattices, Number Fields, Curves and Cryptography
Polynomial-time theory of matrix groups
Proceedings of the forty-first annual ACM symposium on Theory of computing
Constructions of codes from number fields
IEEE Transactions on Information Theory
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This paper analyzes the complexity of problems from class field theory. Class field theory can be used to show the existence of infinite families of number fields with constant root discriminant. Such families have been proposed for use in lattice-based cryptography and for constructing error-correcting codes. Little is known about the complexity of computing them. We show that computing the ray class group and computing certain subfields of Hilbert class fields efficiently reduce to known computationally difficult problems. These include computing the unit group and class group, the principal ideal problem, factoring, and discrete log. As a consequence, efficient quantum algorithms for these problems exist in constant degree number fields.