110 research outputs found
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
Fast construction of irreducible polynomials over finite fields
International audienceWe present a randomized algorithm that on input a finite field with elements and a positive integer outputs a degree irreducible polynomial in . The running time is elementary operations. The in is a function of that tends to zero when tends to infinity. And the in is a function of that tends to zero when tends to infinity. In particular, the complexity is quasi-linear in the degree
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
The number field sieve
The number field sieve is an algorithm to factor integers of the form for small positive and . The authors present a report on work in progress on this algorithm. They informally describe the algorithm, discuss several implementation related aspects, and present some of the factorizations obtained so far. They also mention some solutions to the problems encountered when generalizing the algorithm to general integers using an idea of Buhler and Pomerance. It is not unlikely that this leads to a general purpose factoring algorithm that is asymptotically substantially faster than the fastest factoring algorithms known so far, like the multiple polynomial quadratic siev
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