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ACM Symposium on Theory of Computing (STOC)

Polynomial time quantum algorithm for the computation of the unit group of a number field

Arthur Schmidt, Ulrich Vollmer

Published May 22, 2005• 43 citations
Publisher

Abstract

We present a quantum algorithm for the computation of the irrational period lattice of a function on Z which is periodic in a relaxed sense. This algorithm is applied to compute the unit group of finite extensions of Q. Execution time for fixed field degree over Q is polynomial in the discriminant of the field. Our algorithms generalize and improve upon Hallgren's work [Hal02] for the one-dimensional case corresponding to real-quadratic fields.

DOI
10.1145/1060590.1060661

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Computing the unit group and class group of a number field are two of the main tasks in computational algebraic number theory. Factoring integers reduces to solving Pell's equation, which is a special case of computing the unit group, but a reduction in the other direction is not known and appears more difficult. We give polynomial-time quantum algorithms for computing the unit group and class group when the number field has constant degree.

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We present a quantum algorithm for the computation of the irrational period lattice of a function on Z which is periodic in a relaxed sense. This algorithm is applied to compute the unit group of finite extensions of Q. Execution time for fixed field degree over Q is polynomial in the discriminant of the field. Our algorithms generalize and improve upon Hallgren's work [Hal02] for the one-dimensional case corresponding to real-quadratic fields.
Computing the unit group and solving the principal ideal problem for a number field are two of the main tasks in computational algebraic number theory. This paper proposes efficient quantum algorithms for these two problems when the number field has constant degree. We improve these algorithms proposed by Hallgren by using a period function which is not one-to-one on its fundamental period. Furthermore, given access to a function which encodes the lattice, a new method to compute the basis of an unknown real-valued lattice is presented.
Computing the group of units in a field of algebraic numbers is one of the central tasks of computational algebraic number theory. It is believed to be hard classically, which is of interest for cryptography. In the quantum setting, efficient algorithms were previously known for fields of constant degree. We give a quantum algorithm that is polynomial in the degree of the field and the logarithm of its discriminant. This is achieved by combining three new results. The first is a classical algorithm for computing a basis for certain ideal lattices with doubly exponentially large generators. The second shows that a Gaussian-weighted superposition of lattice points, with an appropriate encoding, can be used to provide a unique representation of a real-valued lattice. The third is an extension of the hidden subgroup problem to continuous groups and a quantum algorithm for solving the HSP over the group Rn.
This paper gives polynomial time quantum algorithms for computing the ideal class group (CGP) under the Generalized Riemann Hypothesis and solving the principal ideal problem (PIP) in number fields of arbitrary degree. These are are fundamental problems in number theory and they are connected to many unproven conjectures in both analytic and algebraic number theory. Previously the best known algorithms by Hallgren [20] only allowed to solve these problems in quantum polynomial time for number fields of constant degree. In a recent breakthrough, Eisenträger et al. [11] showed how to compute the unit group in arbitrary fields, thus opening the way to the resolution of CGP and PIP in the general case. For example, Biasse and Song [3] pointed out how to directly apply this result to solve PIP in classes of cyclotomic fields of arbitrary degree. The methods we introduce in this paper run in quantum polynomial time in arbitrary classes of number fields. They can be applied to solve other problems in computational number theory as well including computing the ray class group and solving relative norm equations. They are also useful for ongoing cryptanalysis of cryptographic schemes based on ideal lattices [5, 10]. Our algorithms generalize the quantum algorithm for computing the (ordinary) unit group [11]. We first show that CGP and PIP reduce naturally to the computation of S-unit groups, which is another fundamental problem in number theory. Then we show an efficient quantum reduction from computing S-units to the continuous hidden subgroup problem introduced in [11]. This step is our main technical contribution, which involves careful analysis of the metrical properties of lattices to prove the correctness of the reduction. In addition, we show how to convert the output into an exact compact representation, which is convenient for further algebraic manipulations.
Infrastructures are group-like objects that make their appearance in arithmetic geometry in the study of computational problems related to number fields and function fields over finite fields. The most prominent computational tasks of infrastructures are the computation of the circumference of the infrastructure and the generalized discrete logarithms. Both these problems are not known to have efficient classical algorithms for an arbitrary infrastructure. Our main contributions are polynomial time quantum algorithms for one-dimensional infrastructures that satisfy certain conditions. For instance, these conditions are always fulfilled for infrastructures obtained from number fields and function fields, both of unit rank one. Since quadratic number fields give rise to such infrastructures, this algorithm can be used to solve Pell's equation and the principal ideal problem. In this sense we generalize Hallgren's quantum algorithms for quadratic number fields, while also providing a polynomial speedup over them. Our more general approach shows that these quantum algorithms can also be applied to infrastructures obtained from complex cubic and totally complex quartic number fields. Our improved way of analyzing the performance makes it possible to show that these algorithms succeed with constant probability independent of the problem size. In contrast, the lower bound on the success probability due to Hallgren decreases as the fourth power of the logarithm of the circumference. Our analysis also shows that fewer qubits are required. We also contribute to the study of infrastructures, and show how to compute efficiently within infrastructures.
Infrastructures are group-like objects that make their appearance in arithmetic geometry in the study of computational problems related to number fields and function fields over finite fields. The most prominent computational tasks of infrastructures are the computation of the circumference of the infrastructure and the generalized discrete logarithms. Both these problems are not known to have efficient classical algorithms for an arbitrary infrastructure. Our main contributions are polynomial time quantum algorithms for one-dimensional infrastructures that satisfy certain conditions. For instance, these conditions are always fulfilled for infrastructures obtained from number fields and function fields, both of unit rank one. Since quadratic number fields give rise to such infrastructures, this algorithm can be used to solve Pell's equation and the principal ideal problem. In this sense we generalize Hallgren's quantum algorithms for quadratic number fields, while also providing a polynomial speedup over them. Our more general approach shows that these quantum algorithms can also be applied to infrastructures obtained from complex cubic and totally complex quartic number fields. Our improved way of analyzing the performance makes it possible to show that these algorithms succeed with constant probability independent of the problem size. In contrast, the lower bound on the success probability due to Hallgren decreases as the fourth power of the logarithm of the circumference. Our analysis also shows that fewer qubits are required. We also contribute to the study of infrastructures, and show how to compute efficiently within infrastructures.