Problem Definition
Associated with each number field is a finite abelian group called the class group. The order of the class group is called the class number. Computing the class number and the structure of the class group of a number field are among the main tasks in computational algebraic number theory [3].
A number field F can be defined as a subfield of the complex numbers \( \mathbb{C} \) which is generated over the rational numbers \( \mathbb{Q} \) by an algebraic number, i. e. \( F = \mathbb{Q} (\theta) \) where θ is the root of a polynomial with rational coefficients. The ring of integers \( \mathcal{O} \) of F is the subset consisting of all elements that are roots of monic polynomials with integer coefficients. The ring \( \mathcal{O} \subseteq F \) can be thought of as a generalization of \( \mathbb{Z} \), the ring of integers in \( \mathbb{Q} \). In particular, one can ask whether \( \mathcal{O} \) is a principal ideal domain and whether elements in \( \mathcal{O} \)...
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Buchmann, J.: A subexponential algorithm for the determination of class groups and regulators of algebraic number fields. In: Goldstein, C. (ed.) Séminaire de Théorie des Nombres, Paris 1988–1989, Progress in Mathematics, vol. 91, pp. 27–41. Birkhäuser (1990)
Buchmann, J.A., Williams, H.C.: A key exchange system based on real quadratic fields (extended abstract). In: Brassard, G. (ed.) Advances in Cryptology–CRYPTO '89. Lecture Notes in Computer Science, vol. 435, 20–24 Aug 1989, pp. 335–343. Springer (1990)
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Hallgren, S.: Fast quantum algorithms for computing the unit group and class group of a number field. In: Proceedings of the 37th ACM Symposium on Theory of Computing. (2005)
Hamdy, S., Maurer, M.: Feige-fiat-shamir identification based on real quadratic fields, Tech. Report TI-23/99. Technische Universität Darmstadt, Fachbereich Informatik. http://www.informatik.tu-darmstadt.de/TI/Veroeffentlichung/TR/ (1999)
Schmidt, A., Vollmer, U.: Polynomial time quantum algorithm for the computation of the unit group of a number field. In: Proceedings of the 37th ACM Symposium on Theory of Computing. (2005)
Thiel, C.: On the complexity of some problems in algorithmic algebraic number theory, Ph. D. thesis. Universität des Saarlandes, Saarbrücken, Germany (1995)
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Hallgren, S. (2008). Quantum Algorithms for Class Group of a Number Field. In: Kao, MY. (eds) Encyclopedia of Algorithms. Springer, Boston, MA. https://doi.org/10.1007/978-0-387-30162-4_310
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