275 research outputs found
Fast norm computation in smooth-degree Abelian number fields
This paper presents a fast method to compute algebraic norms of integral elements of smooth-degree cyclotomic fields, and, more generally, smooth-degree Galois number fields with commutative Galois groups. The typical scenario arising in -unit searches (for, e.g., class-group computation) is computing a -bit norm of an element of weight in a degree- field; this method then uses bit operations.
An operation count was already known in two easier special cases: norms from power-of-2 cyclotomic fields via towers of power-of-2 cyclotomic subfields, and norms from multiquadratic fields via towers of multiquadratic subfields. This paper handles more general Abelian fields by identifying tower-compatible integral bases supporting fast multiplication; in particular, there is a synergy between tower-compatible Gauss-period integral bases and a fast-multiplication idea from Rader.
As a baseline, this paper also analyzes various standard norm-computation techniques that apply to arbitrary number fields, concluding that all of these techniques use at least bit operations in the same scenario, even with fast subroutines for continued fractions and for complex FFTs. Compared to this baseline, algorithms dedicated to smooth-degree Abelian fields find each norm times faster, and finish norm computations inside -unit searches times faster
Attacks on the Search-RLWE problem with small errors
The Ring Learning-With-Errors (RLWE) problem shows great promise for
post-quantum cryptography and homomorphic encryption. We describe a new attack
on the non-dual search RLWE problem with small error widths, using ring
homomorphisms to finite fields and the chi-squared statistical test. In
particular, we identify a "subfield vulnerability" (Section 5.2) and give a new
attack which finds this vulnerability by mapping to a finite field extension
and detecting non-uniformity with respect to the number of elements in the
subfield. We use this attack to give examples of vulnerable RLWE instances in
Galois number fields. We also extend the well-known search-to-decision
reduction result to Galois fields with any unramified prime modulus q,
regardless of the residue degree f of q, and we use this in our attacks. The
time complexity of our attack is O(nq2f), where n is the degree of K and f is
the residue degree of q in K. We also show an attack on the non-dual (resp.
dual) RLWE problem with narrow error distributions in prime cyclotomic rings
when the modulus is a ramified prime (resp. any integer). We demonstrate the
attacks in practice by finding many vulnerable instances and successfully
attacking them. We include the code for all attacks
アルシュ ノ アーベルタイ ノ キンテイ ニツイテ
Let K be an abeilian field over the rationals Q and let Z_K be the ring of integers of K.K is said to be monogenic when there exists an element θ of Z_K with Z_K = Z[θ]. In thiscase θ is said to be a generator of Z_K. Hasse proposed for any given field,
On the trace map between absolutely abelian number fields of equal conductor
Let L/K be an extension of absolutely abelian number fields of equal
conductor, n. The image of the ring of integers of L under the trace map from L
to K is an ideal in the ring of integers in K. We compute the absolute norm of
this ideal exactly for any such L/K, thereby sharpening an earlier result of
Kurt Girstmair. Furthermore, we define an "adjusted trace map" that allows the
proof of Leopoldt's Theorem to be reduced to the cyclotomic case.Comment: 11 pages, 1 figure, uses xypic and amscd. Completely revised version.
To appear in Acta Arithmetic
Ramification groups and Artin conductors of radical extensions of the rationals
We compute the higher ramification groups and the Artin conductors of radical
extensions of the rationals. As an application, we give formulas for their
discriminant (using the conductor-discriminant formula). The interest in such
number fields is due to the fact that they are among the simplest non-abelian
extensions of the rationals (and so not classified by Class Field Theory). We
show that this extensions have non integer jumps in the superior ramification
groups, contrarily to the case of abelian extensions (as prescribed by
Hasse-Arf theorem).Comment: 29 pages, to be published on the Journal de Theorie de Nombres de
Bourdeau
Monogenic period equations are cyclotomic polynomials
We study monogeneity in period equations, psi(e)(x), the auxiliary equations introduced by Gauss to solve cyclotomic polynomials by radicals. All monogenic psi(e)(x) of degrees 4 = 4, we conjecture all monogenic period equations to be cyclotomic polynomials. Totally real period equations are of interest in applications of quadratic discrete-time dynamical systems
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