Universal gradings of orders

For commutative rings, we introduce the notion of a {\em universal grading}, which can be viewed as the "largest possible grading". While not every commutative ring (or order) has a universal grading, we prove that every {\em reduced order} has a universal grading, and this grading is by a {\em finite} group. Examples of graded orders are provided by group rings of finite abelian groups over rings of integers in number fields. We generalize known properties of nilpotents, idempotents, and roots of unity in such group rings to the case of graded orders; this has applications to cryptography. Lattices play an important role in this paper; a novel aspect is that our proofs use that the additive group of any reduced order can in a natural way be equipped with a lattice structure.Comment: Added section 10; added to and rewrote introduction and abstract (new Theorem 1.4 and Examples 1.6 and 1.7

Realizing orders as group rings

An order is a commutative ring that as an abelian group is finitely generated and free. A commutative ring is reduced if it has no non-zero nilpotent elements. In this paper we use a new tool, namely, the fact that every reduced order has a universal grading, to answer questions about realizing orders as group rings. In particular, we address the Isomorphism Problem for group rings in the case where the ring is a reduced order. We prove that any non-zero reduced order $R$ can be written as a group ring in a unique ``maximal'' way, up to isomorphism. More precisely, there exist a ring $A$ and a finite abelian group $G$, both uniquely determined up to isomorphism, such that $R\cong A[G]$ as rings, and such that if $B$ is a ring and $H$ is a group, then $R\cong B[H]$ as rings if and only if there is a finite abelian group $J$ such that $B\cong A[J]$ as rings and $J\times H\cong G$ as groups. Computing $A$ and $G$ for given $R$ can be done by means of an algorithm that is not quite polynomial-time. We also give a description of the automorphism group of $R$ in terms of $A$ and $G$

Lattices with Symmetry

For large ranks, there is no good algorithm that decides whether a given lattice has an orthonormal basis. But when the lattice is given with enough symmetry, we can construct a provably deterministic polynomial-time algorithm to accomplish this, based on the work of Gentry and Szydlo. The techniques involve algorithmic algebraic number theory, analytic number theory, commutative algebra, and lattice basis reduction

Fast construction of irreducible polynomials over finite fields

International audienceWe present a randomized algorithm that on input a finite field $K$ with $q$ elements and a positive integer $d$ outputs a degree $d$ irreducible polynomial in $K[x]$. The running time is $d^{1+o(1)} \times (\log q)^{5+o(1)}$ elementary operations. The $o(1)$ in $d^{1+o(1)}$ is a function of $d$ that tends to zero when $d$ tends to infinity. And the $o(1)$ in $(\log q)^{5+o(1)}$ is a function of $q$ that tends to zero when $q$ tends to infinity. In particular, the complexity is quasi-linear in the degree $d$

Forms in odd degree extensions and selfdual normal bases

Introduction. Let K be a field. Springer has proved that an ani-sotropic quadratic form over K is also anisotropic over any odd degree extension of K (see [31], [14]). If the characteristic of K is not 2, this implies that two nonsingular quadratic forms that become isomorphic over an extension of odd degree of K are already isomorphic over

The Hidden Subgroup Problem and Eigenvalue Estimation on a Quantum Computer

A quantum computer can efficiently find the order of an element in a group, factors of composite integers, discrete logarithms, stabilisers in Abelian groups, and `hidden' or `unknown' subgroups of Abelian groups. It is already known how to phrase the first four problems as the estimation of eigenvalues of certain unitary operators. Here we show how the solution to the more general Abelian `hidden subgroup problem' can also be described and analysed as such. We then point out how certain instances of these problems can be solved with only one control qubit, or `flying qubits', instead of entire registers of control qubits.Comment: 16 pages, 3 figures, LaTeX2e, to appear in Proceedings of the 1st NASA International Conference on Quantum Computing and Quantum Communication (Springer-Verlag