75 research outputs found
ALGEBRAIC GEOMETRY CODES OVER ABELIAN SURFACES CONTAINING NO ABSOLUTELY IRREDUCIBLE CURVES OF LOW GENUS
We provide a theoretical study of Algebraic Geometry codes constructed from abelian surfaces defined over finite fields. We give a general bound on their minimum distance and we investigate how this estimation can be sharpened under the assumption that the abelian surface does not contain low genus curves. This approach naturally leads us to consider Weil restrictions of elliptic curves and abelian surfaces which do not admit a principal polarization
Heuristics on pairing-friendly abelian varieties
We discuss heuristic asymptotic formulae for the number of pairing-friendly
abelian varieties over prime fields, generalizing previous work of one of the
authors arXiv:math1107.0307Comment: Pages 6-7 rewritten, other minor changes mad
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Finite Fields: Theory and Applications
Finite fields are the focal point of many interesting geometric, algorithmic and combinatorial problems. The workshop was devoted to progress on these questions, with an eye also on the important applications of finite field techniques in cryptography, error correcting codes, and random number generation
Curves, Jacobians, and Cryptography
The main purpose of this paper is to give an overview over the theory of
abelian varieties, with main focus on Jacobian varieties of curves reaching
from well-known results till to latest developments and their usage in
cryptography. In the first part we provide the necessary mathematical
background on abelian varieties, their torsion points, Honda-Tate theory,
Galois representations, with emphasis on Jacobian varieties and hyperelliptic
Jacobians. In the second part we focus on applications of abelian varieties on
cryptography and treating separately, elliptic curve cryptography, genus 2 and
3 cryptography, including Diffie-Hellman Key Exchange, index calculus in Picard
groups, isogenies of Jacobians via correspondences and applications to discrete
logarithms. Several open problems and new directions are suggested.Comment: 66 page
Quantization, Classical and Quantum Field Theory and Theta - Functions
In the abelian case (the subject of several beautiful books) fixing some
combinatorial structure (so called theta structure of level k) one obtains a
special basis in the space of sections of canonical polarization powers over
the jacobians. These sections can be presented as holomorphic functions on the
"abelian Schottky space". This fact provides various applications of these
concrete analytic formulas to the integrable systems, classical mechanics and
PDE's. Our practical goal is to do the same in the non abelian case that is to
give an answer to the Beauville's question. In future we hope to extend this
digest to a mathematical mohograph with title "VBAC".Comment: To Igor Rostislavovich Shafarevich on his 80th birthday (will be
published by CRS, Canada
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