A quantum algorithm for computing the unit group of an arbitrary degree number field

Abstract

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 *

Efficient quantum algorithms for computing class groups and solving the principal ideal problem in arbitrary degree number fields

“…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.…”

“…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.…”

“…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.…”

“…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 approach of [20] relies on the resolution of the HSP in a bounded and discretized approximation of R m , which does not seem to apply when the degree of the fields grows to infinity. In a recent breakthrough, Eisenträger, Hallgren, Kitaev and Song [11] described a polynomial time algorithm for computing the unit group in classes of number fields of arbitrary degree. One of the main tools they developed is a continuous HSP definition on R m and an efficient quantum algorithm solving it.…”