87 research outputs found
Normal Elliptic Bases and Torus-Based Cryptography
We consider representations of algebraic tori over finite fields.
We make use of normal elliptic bases to show that, for infinitely many
squarefree integers and infinitely many values of , we can encode
torus elements, to a small fixed overhead and to -tuples of
elements, in quasi-linear time in .
This improves upon previously known algorithms, which all have a
quasi-quadratic complexity. As a result, the cost of the encoding phase is now
negligible in Diffie-Hellman cryptographic schemes
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)
Explicit Galois obstruction and descent for hyperelliptic curves with tamely cyclic reduced automorphism group
This paper is devoted to the explicit description of the Galois descent
obstruction for hyperelliptic curves of arbitrary genus whose reduced
automorphism group is cyclic of order coprime to the characteristic of their
ground field. Along the way, we obtain an arithmetic criterion for the
existence of a hyperelliptic descent.
The obstruction is described by the so-called arithmetic dihedral invariants
of the curves in question. If it vanishes, then the use of these invariants
also allows the explicit determination of a model over the field of moduli; if
not, then one obtains a hyperelliptic model over a degree 2 extension of this
field.Comment: 35 pages; improve the readability of the pape
The geometry of some parameterizations and encodings
We explore parameterizations by radicals of low genera algebraic curves. We
prove that for a prime power that is large enough and prime to , a fixed
positive proportion of all genus 2 curves over the field with elements can
be parameterized by -radicals. This results in the existence of a
deterministic encoding into these curves when is congruent to modulo
. We extend this construction to parameterizations by -radicals for
small odd integers , and make it explicit for
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