1,176 research outputs found
Two-sources Randomness Extractors for Elliptic Curves
This paper studies the task of two-sources randomness extractors for elliptic
curves defined over finite fields , where can be a prime or a binary
field. In fact, we introduce new constructions of functions over elliptic
curves which take in input two random points from two differents subgroups. In
other words, for a ginven elliptic curve defined over a finite field
and two random points and , where and are two subgroups of
, our function extracts the least significant bits of the
abscissa of the point when is a large prime, and the -first
coefficients of the asbcissa of the point when , where is a prime greater than . We show that the extracted bits
are close to uniform.
Our construction extends some interesting randomness extractors for elliptic
curves, namely those defined in \cite{op} and \cite{ciss1,ciss2}, when
. The proposed constructions can be used in any
cryptographic schemes which require extraction of random bits from two sources
over elliptic curves, namely in key exchange protole, design of strong
pseudo-random number generators, etc
Class number formulas via 2-isogenies of elliptic curves
A classical result of Dirichlet shows that certain elementary character sums
compute class numbers of quadratic imaginary number fields. We obtain analogous
relations between class numbers and a weighted character sum associated to a
2-isogeny of elliptic curves.Comment: 19 pages; To appear in the Bulletin of the London Mathematical
Societ
The Moonshine Module for Conway's Group
We exhibit an action of Conway's group---the automorphism group of the Leech
lattice---on a distinguished super vertex operator algebra, and we prove that
the associated graded trace functions are normalized principal moduli, all
having vanishing constant terms in their Fourier expansion. Thus we construct
the natural analogue of the Frenkel--Lepowsky--Meurman moonshine module for
Conway's group.
The super vertex operator algebra we consider admits a natural
characterization, in direct analogy with that conjectured to hold for the
moonshine module vertex operator algebra. It also admits a unique
canonically-twisted module, and the action of the Conway group naturally
extends. We prove a special case of generalized moonshine for the Conway group,
by showing that the graded trace functions arising from its action on the
canonically-twisted module are constant in the case of Leech lattice
automorphisms with fixed points, and are principal moduli for genus zero groups
otherwise.Comment: 54 pages including 11 pages of tables; minor revisions in v2,
submitte
Finite quotients of Z[C_n]-lattices and Tamagawa numbers of semistable abelian varieties
We investigate the behaviour of Tamagawa numbers of semistable principally
polarised abelian varieties in extensions of local fields. In view of the
Raynaud parametrisation, this translates into a purely algebraic problem
concerning the number of -invariant points on a quotient of -lattices
for varying subgroups of and integers . In
particular, we give a simple formula for the change of Tamagawa numbers in
totally ramified extensions (corresponding to varying ) and one that
computes Tamagawa numbers up to rational squares in general extensions.
As an application, we extend some of the existing results on the -parity
conjecture for Selmer groups of abelian varieties by allowing more general
local behaviour. We also give a complete classification of the behaviour of
Tamagawa numbers for semistable 2-dimensional principally polarised abelian
varieties, that is similar to the well-known one for elliptic curves. The
appendix explains how to use this classification for Jacobians of genus 2
hyperelliptic curves given by equations of the form , under some
simplifying hypotheses.Comment: Two new lemmas are added. The first describes permutation
representations, and the second describes the dependence of the B-group on
the maximal fixpoint-free invariant sublattice. Contact details and
bibliographic details have been update
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