2,482 research outputs found
Discrete logarithms in curves over finite fields
A survey on algorithms for computing discrete logarithms in Jacobians of
curves over finite fields
Galois invariant smoothness basis
This text answers a question raised by Joux and the second author about the
computation of discrete logarithms in the multiplicative group of finite
fields. Given a finite residue field \bK, one looks for a smoothness basis
for \bK^* that is left invariant by automorphisms of \bK. For a broad class
of finite fields, we manage to construct models that allow such a smoothness
basis. This work aims at accelerating discrete logarithm computations in such
fields. We treat the cases of codimension one (the linear sieve) and
codimension two (the function field sieve)
Computing in Jacobians of projective curves over finite fields
We give algorithms for computing with divisors on projective curves over
finite fields, and with their Jacobians, using the algorithmic representation
of projective curves developed by Khuri-Makdisi. We show that many desirable
operations can be done efficiently in this setting: decomposing divisors into
prime divisors; computing pull-backs and push-forwards of divisors under finite
morphisms, and hence Picard and Albanese maps on Jacobians; generating
uniformly random divisors and points on Jacobians; computing Frobenius maps and
Kummer maps; and finding a basis for the -torsion of the Picard group, where
is a prime number different from the characteristic of the base field.Comment: 42 page
The Infrastructure of a Global Field of Arbitrary Unit Rank
In this paper, we show a general way to interpret the infrastructure of a
global field of arbitrary unit rank. This interpretation generalizes the prior
concepts of the giant step operation and f-representations, and makes it
possible to relate the infrastructure to the (Arakelov) divisor class group of
the global field. In the case of global function fields, we present results
that establish that effective implementation of the presented methods is indeed
possible, and we show how Shanks' baby-step giant-step method can be
generalized to this situation.Comment: Revised version. Accepted for publication in Math. Com
Discrete Logarithms in Generalized Jacobians
D\'ech\`ene has proposed generalized Jacobians as a source of groups for
public-key cryptosystems based on the hardness of the Discrete Logarithm
Problem (DLP). Her specific proposal gives rise to a group isomorphic to the
semidirect product of an elliptic curve and a multiplicative group of a finite
field. We explain why her proposal has no advantages over simply taking the
direct product of groups. We then argue that generalized Jacobians offer poorer
security and efficiency than standard Jacobians
More Discriminants with the Brezing-Weng Method
The Brezing-Weng method is a general framework to generate families of
pairing-friendly elliptic curves. Here, we introduce an improvement which can
be used to generate more curves with larger discriminants. Apart from the
number of curves this yields, it provides an easy way to avoid endomorphism
rings with small class number
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