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An improved quantum Fourier transform algorithm and applications
- Lisa Hales, Sean Hallgren
- Mathematics, Computer ScienceProceedings 41st Annual Symposium on Foundations…
- 12 November 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…
Quantum algorithms for some hidden shift problems
The hidden coset problem is defined, which generalizes the hidden shift problem and the hidden subgroup problem and provides a unified way of viewing the ability of the Fourier transform to capture subgroup and shift structure.
Polynomial-time quantum algorithms for Pell's equation and the principal ideal problem
- Sean Hallgren
- 1 March 2007
This work gives polynomial-time quantum algorithms for three problems from computational algebraic number theory, including Pell's equation, the principal ideal problem in real quadratic number fields, and the one-way function underlying the Buchmann--Williams key exchange system.
Normal subgroup reconstruction and quantum computation using group representations
It is shown that an immediate generalization of the Abelian case solution to the non-Abelian case does not efficiently solve Graph Isomorphism.
The Hidden Subgroup Problem and Quantum Computation Using Group Representations
A natural generalization of the algorithm for the abelian case to the nonabelian case is analyzed and it is shown that the algorithm determines the normal core of a hidden subgroup: in particular, normal subgroups can be determined.
Fast quantum algorithms for computing the unit group and class group of a number field
- Sean Hallgren
- Computer Science, MathematicsSTOC '05
- 22 May 2005
This work gives polynomial-time quantum algorithms for computing the unit group and class group of a number field when the number field has constant degree.
Limitations of quantum coset states for graph isomorphism
It is shown that entangled quantum measurements on at least Ω(n log n) coset states are necessary to get useful information for the case of graph isomorphism, matching an information theoretic upper bound.
Superpolynomial Speedups Based on Almost Any Quantum Circuit
It is shown that a wide class of unitary circuits can be used in place of Hadamards to obtain a O(1) vs. Ω(n) separation and a general method for amplifying quantum-classical separations is given that allows us to achieve a nO( 1)vs.nΩ(logn)separation from any dispersing circuit.
Weak Instances of PLWE
- Kirsten Eisenträger, Sean Hallgren, K. Lauter
- Mathematics, Computer ScienceSelected Areas in Cryptography
- 14 August 2014
This paper presents a new attack on the polynomial version of the Ring-LWE assumption, for certain carefully chosen number fields, and articulate the relevant properties and prove security reductions for number fields with those properties.
Supersingular Isogeny Graphs and Endomorphism Rings: Reductions and Solutions
- Kirsten Eisenträger, Sean Hallgren, K. Lauter, T. Morrison, C. Petit
- 29 April 2018
Reductions between the problem of path finding in the \(\ell \)-isogeny graph, computing maximal orders isomorphic to the endomorphism ring of a supersingular elliptic curve, and computing the end morphism ring itself are proved.