22 research outputs found
Distortion maps for genus two curves
Distortion maps are a useful tool for pairing based cryptography. Compared
with elliptic curves, the case of hyperelliptic curves of genus g > 1 is more
complicated since the full torsion subgroup has rank 2g. In this paper we prove
that distortion maps always exist for supersingular curves of genus g>1 and we
construct distortion maps in genus 2 (for embedding degrees 4,5,6 and 12).Comment: 16 page
Algorithms and cryptographic protocols using elliptic curves
En els darrers anys, la criptografia amb corbes el.lĂptiques ha
adquirit una importĂ ncia creixent, fins a arribar a formar part en
la actualitat de diferents estĂ ndards industrials. Tot i que s'han
dissenyat variants amb corbes el.lĂptiques de criptosistemes
clĂ ssics, com el RSA, el seu mĂ xim interĂšs rau en la seva
aplicaciĂł en criptosistemes basats en el Problema del Logaritme
Discret, com els de tipus ElGamal. En aquest cas, els
criptosistemes el.lĂptics garanteixen la mateixa seguretat que els
construĂŻts sobre el grup multiplicatiu d'un cos finit primer, perĂČ
amb longituds de clau molt menor.
Mostrarem, doncs, les bones propietats d'aquests criptosistemes,
aixĂ com els requeriments bĂ sics per a que una corba
sigui criptogrĂ ficament Ăștil, estretament relacionat amb la seva
cardinalitat. Revisarem alguns mĂštodes que permetin descartar
corbes no criptogrĂ ficament Ăștils, aixĂ com altres que permetin
obtenir corbes bones a partir d'una de donada. Finalment,
descriurem algunes aplicacions, com sĂłn el seu Ășs en Targes
Intel.ligents i sistemes RFID, per concloure amb alguns avenços
recents en aquest camp.The relevance of elliptic curve cryptography has grown in recent
years, and today represents a cornerstone in many industrial
standards. Although elliptic curve variants of classical
cryptosystems such as RSA exist, the full potential of elliptic
curve cryptography is displayed in cryptosystems based on the
Discrete Logarithm Problem, such as ElGamal. For these, elliptic
curve cryptosystems guarantee the same security levels as their
finite field analogues, with the additional advantage of using
significantly smaller key sizes.
In this report we show the positive properties of elliptic curve
cryptosystems, and the requirements a curve must meet to be
useful in this context, closely related to the number of points.
We survey methods to discard cryptographically uninteresting
curves as well as methods to obtain other useful curves from
a given one. We then describe some real world applications
such as Smart Cards and RFID systems and conclude with a
snapshot of recent developments in the field