424 research outputs found
Easy decision-Diffie-Hellman groups
The decision-Diffie-Hellman problem (DDH) is a central computational problem
in cryptography. It is known that the Weil and Tate pairings can be used to
solve many DDH problems on elliptic curves. Distortion maps are an important
tool for solving DDH problems using pairings and it is known that distortion
maps exist for all supersingular elliptic curves. We present an algorithm to
construct suitable distortion maps. The algorithm is efficient on the curves
usable in practice, and hence all DDH problems on these curves are easy. We
also discuss the issue of which DDH problems on ordinary curves are easy
A Gluing Lemma And Overconvergent Modular Forms
We prove a gluing lemma for sections of line bundles on a rigid analytic
variety. We apply the lemma, in conjunction with a result of Buzzard's, to give
a proof of (a generalization) of Coleman's theorem which states that
overconvergent modular forms of small slope are classical. The proof is
"geometric" in nature, and is suitable for generalization to other PEL Shimura
varieties
On the modularity of supersingular elliptic curves over certain totally real number fields
We study generalisations to totally real fields of methods originating with
Wiles and Taylor-Wiles. In view of the results of Skinner-Wiles on elliptic
curves with ordinary reduction, we focus here on the case of supersingular
reduction. Combining these, we then obtain some partial results on the
modularity problem for semistable elliptic curves, and end by giving some
applications of our results, for example proving the modularity of all
semistable elliptic curves over .Comment: 36 pages (revised version of 2002 preprint
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