22 research outputs found
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
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
The probability that the number of points on the Jacobian of a genus 2 curve is prime
In 2000, Galbraith and McKee heuristically derived a formula that estimates
the probability that a randomly chosen elliptic curve over a fixed finite prime
field has a prime number of rational points. We show how their heuristics can
be generalized to Jacobians of curves of higher genus. We then elaborate this
in genus 2 and study various related issues, such as the probability of
cyclicity and the probability of primality of the number of points on the curve
itself. Finally, we discuss the asymptotic behavior as the genus tends to
infinity.Comment: Minor edits, 37 pages. To appear in Proceedings of the London
Mathematical Societ
On the distribution of Atkin and Elkies primes for reductions of elliptic curves on average
For an elliptic curve E/Q without complex multiplication we study the
distribution of Atkin and Elkies primes l, on average, over all good reductions
of E modulo primes p. We show that, under the Generalised Riemann Hypothesis,
for almost all primes p there are enough small Elkies primes l to ensure that
the Schoof-Elkies-Atkin point-counting algorithm runs in (log p)^(4+o(1))
expected time.Comment: 20 pages, to appear in LMS J. Comput. Mat