374 research outputs found
Pairings on hyperelliptic curves with a real model
We analyse the efficiency of pairing computations on hyperelliptic curves given by a real model using a balanced divisor at infinity. Several optimisations are proposed and analysed. Genus two curves given by a real model arise when considering pairing friendly groups of order dividing . We compare the performance of pairings on such groups in both elliptic and hyperelliptic versions. We conclude that pairings can be efficiently computable in real models of hyperelliptic curves
The Cassels-Tate pairing on polarized abelian varieties
Let (A,\lambda) be a principally polarized abelian variety defined over a
global field k, and let \Sha(A) be its Shafarevich-Tate group. Let \Sha(A)_\nd
denote the quotient of \Sha(A) by its maximal divisible subgroup. Cassels and
Tate constructed a nondegenerate pairing \Sha(A)_\nd \times \Sha(A)_\nd
\rightarrow \Q/\Z. If A is an elliptic curve, then by a result of Cassels the
pairing is alternating. But in general it is only antisymmetric.
Using some new but equivalent definitions of the pairing, we derive general
criteria deciding whether it is alternating and whether there exists some
alternating nondegenerate pairing on \Sha(A)_\nd. These criteria are expressed
in terms of an element c \in \Sha(A)_\nd that is canonically associated to the
polarization \lambda. In the case that A is the Jacobian of some curve, a
down-to-earth version of the result allows us to determine effectively whether
\#\Sha(A) (if finite) is a square or twice a square. We then apply this to
prove that a positive proportion (in some precise sense) of all hyperelliptic
curves of even genus g \ge 2 over \Q have a Jacobian with nonsquare \#\Sha (if
finite). For example, it appears that this density is about 13% for curves of
genus 2. The proof makes use of a general result relating global and local
densities; this result can be applied in other situations.Comment: 41 pages, published versio
Computing canonical heights using arithmetic intersection theory
For several applications in the arithmetic of abelian varieties it is
important to compute canonical heights. Following Faltings and Hriljac, we show
how the canonical height on the Jacobian of a smooth projective curve can be
computed using arithmetic intersection theory on a regular model of the curve
in practice. In the case of hyperelliptic curves we present a complete
algorithm that has been implemented in Magma. Several examples are computed and
the behavior of the running time is discussed.Comment: 29 pages. Fixed typos and minor errors, restructured some sections.
Added new Example
Still Wrong Use of Pairings in Cryptography
Several pairing-based cryptographic protocols are recently proposed with a
wide variety of new novel applications including the ones in emerging
technologies like cloud computing, internet of things (IoT), e-health systems
and wearable technologies. There have been however a wide range of incorrect
use of these primitives. The paper of Galbraith, Paterson, and Smart (2006)
pointed out most of the issues related to the incorrect use of pairing-based
cryptography. However, we noticed that some recently proposed applications
still do not use these primitives correctly. This leads to unrealizable,
insecure or too inefficient designs of pairing-based protocols. We observed
that one reason is not being aware of the recent advancements on solving the
discrete logarithm problems in some groups. The main purpose of this article is
to give an understandable, informative, and the most up-to-date criteria for
the correct use of pairing-based cryptography. We thereby deliberately avoid
most of the technical details and rather give special emphasis on the
importance of the correct use of bilinear maps by realizing secure
cryptographic protocols. We list a collection of some recent papers having
wrong security assumptions or realizability/efficiency issues. Finally, we give
a compact and an up-to-date recipe of the correct use of pairings.Comment: 25 page
Generalized Jacobians and explicit descents
We develop a cohomological description of various explicit descents in terms
of generalized Jacobians, generalizing the known description for hyperelliptic
curves. Specifically, given an integer dividing the degree of some reduced
effective divisor on a curve , we show that multiplication by
on the generalized Jacobian factors through an isogeny
whose kernel is
naturally the dual of the Galois module
. By geometric class
field theory, this corresponds to an abelian covering of of exponent
unramified outside . The -coverings of parameterized
by explicit descents are the maximal unramified subcoverings of the -forms
of this ramified covering. We present applications of this to the computation
of Mordell-Weil groups of Jacobians.Comment: to appear in Math. Com
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