Proceedings 41st Annual Symposium on Foundationsâ€¦

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 arbitraryâ€¦ Expand

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.Expand

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.Expand

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.Expand

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.Expand

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.Expand

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.Expand