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

@inproceedings{Schmidt2005PolynomialTQ, title={Polynomial time quantum algorithm for the computation of the unit group of a number field}, author={Arthur Schmidt and Ulrich Vollmer}, booktitle={STOC '05}, year={2005} }

We present a quantum algorithm for the computation of the irrational period lattice of a function on Zn 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 [9] for the one-dimensional case corresponding to real-quadratic fields.

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## 46 Citations

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## References

SHOWING 1-10 OF 20 REFERENCES

Quantum measurements and the Abelian Stabilizer Problem

- Computer Science, MathematicsElectron. Colloquium Comput. Complex.
- 1996

A polynomial quantum algorithm for the Abelian stabilizer problem which includes both factoring and the discrete logarithm is presented, based on a procedure for measuring an eigenvalue of a unitary operator.

A subexponential algorithm for the determination of class groups and regulators of algebraic number fields

- Mathematics
- 1990

A new probabilistic algorithm for the determination of class groups and regulators of an algebraic number eld F is presented. Heuristic evidence is given which shows that the expected running time of…

Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer

- Computer Science, MathematicsSIAM Rev.
- 1999

Efficient randomized algorithms are given for factoring integers and finding discrete logarithms, two problems which are generally thought to be hard on a classical computer and have been used as the basis of several proposed cryptosystems.

On the period length of the generalized Lagrange algorithm

- Mathematics
- 1987

Abstract The generalized Lagrange algorithm is a number geometric generalization of Lagrange's continued fraction method for computing fundamental unit and class number of real quadratic number…

Quantum Cryptanalysis of Hidden Linear Functions (Extended Abstract)

- Mathematics, Computer ScienceCRYPTO
- 1995

It is shown that any cryptosystem based on what is referred to as a ‘hidden linear form’ can be broken in quantum polynomial time and the notion of ‘junk bits’ is introduced which are helpful when performing classical computations that are not injective.

An improved quantum Fourier transform algorithm and applications

- Mathematics, Computer ScienceProceedings 41st Annual Symposium on Foundations of Computer Science
- 2000

We give an algorithm for approximating the quantum Fourier transform over an arbitrary Z/sub p/ which requires only O(n log n) steps where n=log p to achieve an approximation to within an arbitrary…

The Hidden Subgroup Problem and Eigenvalue Estimation on a Quantum Computer

- Mathematics, PhysicsQCQC
- 1998

It is pointed out how certain instances of Abelian hidden subgroup problems can be solved with only one control qubit, or flying qubits, instead of entire registers of control qubits.

Quantum algorithms and the fourier transform

- Mathematics
- 2004

The search for practical problems that can be solved exponentially faster on a quantum computer is one of the primary goals in research into algorithms for quantum computers. It has long been known…

On the computation of units and class numbers by a generalization of Lagrange's algorithm

- Mathematics
- 1987

Based on a number geometric interpretation of the continued fraction algorithm in real quadratic fields an algorithm is developed by which one can compute a system of fundamental units of any order O…

CRYPTOGRAPHY BASED ON NUMBER FIELDS WITH LARGE REGULATOR

- Mathematics
- 2000

OLLER Abstract. We explain a variant of the Fiat-Shamir identication and sig- nature protocol which is based on the intractability of computing generators of principal ideals in algebraic number…