328 research outputs found
A Las Vegas algorithm to solve the elliptic curve discrete logarithm problem
In this paper, we describe a new Las Vegas algorithm to solve the elliptic
curve discrete logarithm problem. The algorithm depends on a property of the
group of rational points of an elliptic curve and is thus not a generic
algorithm. The algorithm that we describe has some similarities with the most
powerful index-calculus algorithm for the discrete logarithm problem over a
finite field
A Digital Signature Scheme for Long-Term Security
In this paper we propose a signature scheme based on two intractable
problems, namely the integer factorization problem and the discrete logarithm
problem for elliptic curves. It is suitable for applications requiring
long-term security and provides a more efficient solution than the existing
ones
Minors solve the elliptic curve discrete logarithm problem
The elliptic curve discrete logarithm problem is of fundamental importance in
public-key cryptography. It is in use for a long time. Moreover, it is an
interesting challenge in computational mathematics. Its solution is supposed to
provide interesting research directions.
In this paper, we explore ways to solve the elliptic curve discrete logarithm
problem. Our results are mostly computational. However, it seems, the methods
that we develop and directions that we pursue can provide a potent attack on
this problem. This work follows our earlier work, where we tried to solve this
problem by finding a zero minor in a matrix over the same finite field on which
the elliptic curve is defined. This paper is self-contained
Identifying supersingular elliptic curves
Given an elliptic curve E over a field of positive characteristic p, we
consider how to efficiently determine whether E is ordinary or supersingular.
We analyze the complexity of several existing algorithms and then present a new
approach that exploits structural differences between ordinary and
supersingular isogeny graphs. This yields a simple algorithm that, given E and
a suitable non-residue in F_p^2, determines the supersingularity of E in O(n^3
log^2 n) time and O(n) space, where n=O(log p). Both these complexity bounds
are significant improvements over existing methods, as we demonstrate with some
practical computations.Comment: corrected a typo, 10 page
Linearizing torsion classes in the Picard group of algebraic curves over finite fields
We address the problem of computing in the group of -torsion rational
points of the jacobian variety of algebraic curves over finite fields, with a
view toward computing modular representations.Comment: To appear in Journal of Algebr
The complexity and geometry of numerically solving polynomial systems
These pages contain a short overview on the state of the art of efficient
numerical analysis methods that solve systems of multivariate polynomial
equations. We focus on the work of Steve Smale who initiated this research
framework, and on the collaboration between Stephen Smale and Michael Shub,
which set the foundations of this approach to polynomial system--solving,
culminating in the more recent advances of Carlos Beltran, Luis Miguel Pardo,
Peter Buergisser and Felipe Cucker
- …