172 research outputs found
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
A CM construction for curves of genus 2 with p-rank 1
We construct Weil numbers corresponding to genus-2 curves with -rank 1
over the finite field \F_{p^2} of elements. The corresponding curves
can be constructed using explicit CM constructions. In one of our algorithms,
the group of \F_{p^2}-valued points of the Jacobian has prime order, while
another allows for a prescribed embedding degree with respect to a subgroup of
prescribed order. The curves are defined over \F_{p^2} out of necessity: we
show that curves of -rank 1 over \F_p for large cannot be efficiently
constructed using explicit CM constructions.Comment: 19 page
Constructing suitable ordinary pairing-friendly curves: A case of elliptic curves and genus two hyperelliptic curves
One of the challenges in the designing of pairing-based cryptographic protocols is to construct suitable pairing-friendly curves: Curves which would provide e�cient implementation without compromising the security of the protocols. These curves have small embedding degree and large prime order subgroup. Random curves are likely to have large embedding degree and hence are not practical for implementation of pairing-based protocols.
In this thesis we review some mathematical background on elliptic and hyperelliptic curves in relation to the construction of pairing-friendly hyper-elliptic curves. We also present the notion of pairing-friendly curves. Furthermore, we construct new pairing-friendly elliptic curves and Jacobians of genus two hyperelliptic curves which would facilitate an efficient implementation in pairing-based protocols. We aim for curves that have smaller values than ever before reported for di�erent embedding degrees. We also discuss optimisation of computing pairing in Tate pairing and its variants. Here we show how to e�ciently multiply a point in a subgroup de�ned on a twist curve by a large cofactor. Our approach uses the theory of addition chains. We also show a new method for implementation of the computation of the hard part of the �nal exponentiation in the calculation of the Tate pairing and its varian
Cryptographic Pairings: Efficiency and DLP security
This thesis studies two important aspects of the use of pairings in cryptography, efficient
algorithms and security.
Pairings are very useful tools in cryptography, originally used for the cryptanalysis of
elliptic curve cryptography, they are now used in key exchange protocols, signature schemes
and Identity-based cryptography.
This thesis comprises of two parts: Security and Efficient Algorithms.
In Part I: Security, the security of pairing-based protocols is considered, with a thorough
examination of the Discrete Logarithm Problem (DLP) as it occurs in PBC. Results on the
relationship between the two instances of the DLP will be presented along with a discussion
about the appropriate selection of parameters to ensure particular security level.
In Part II: Efficient Algorithms, some of the computational issues which arise when using
pairings in cryptography are addressed. Pairings can be computationally expensive, so
the Pairing-Based Cryptography (PBC) research community is constantly striving to find
computational improvements for all aspects of protocols using pairings. The improvements
given in this section contribute towards more efficient methods for the computation of pairings,
and increase the efficiency of operations necessary in some pairing-based protocol
Abelian Varieties with Prescribed Embedding Degree
We present an algorithm that, on input of a CM-field , an integer ,
and a prime , constructs a -Weil number \pi \in \O_K
corresponding to an ordinary, simple abelian variety over the field \F of
elements that has an \F-rational point of order and embedding degree
with respect to . We then discuss how CM-methods over can be used to
explicitly construct .Comment: to appear in ANTS-VII
Pairings in Cryptology: efficiency, security and applications
Abstract
The study of pairings can be considered in so many di�erent ways that it
may not be useless to state in a few words the plan which has been adopted,
and the chief objects at which it has aimed. This is not an attempt to write
the whole history of the pairings in cryptology, or to detail every discovery,
but rather a general presentation motivated by the two main requirements
in cryptology; e�ciency and security.
Starting from the basic underlying mathematics, pairing maps are con-
structed and a major security issue related to the question of the minimal
embedding �eld [12]1 is resolved. This is followed by an exposition on how
to compute e�ciently the �nal exponentiation occurring in the calculation
of a pairing [124]2 and a thorough survey on the security of the discrete log-
arithm problem from both theoretical and implementational perspectives.
These two crucial cryptologic requirements being ful�lled an identity based
encryption scheme taking advantage of pairings [24]3 is introduced. Then,
perceiving the need to hash identities to points on a pairing-friendly elliptic
curve in the more general context of identity based cryptography, a new
technique to efficiently solve this practical issue is exhibited.
Unveiling pairings in cryptology involves a good understanding of both
mathematical and cryptologic principles. Therefore, although �rst pre-
sented from an abstract mathematical viewpoint, pairings are then studied
from a more practical perspective, slowly drifting away toward cryptologic
applications
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
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