20 research outputs found
Computing functions on Jacobians and their quotients
We show how to efficiently compute functions on jacobian varieties and their
quotients. We deduce a quasi-optimal algorithm to compute isogenies
between jacobians of genus two curves
Minimal Hopf-Galois Structures on Separable Field Extensions
In Hopf-Galois theory, every -Hopf-Galois structure on a field extension
gives rise to an injective map from the set of -sub-Hopf
algebras of into the intermediate fields of . Recent papers on the
failure of the surjectivity of reveal that there exist many
Hopf-Galois structures for which there are many more subfields than sub-Hopf
algebras. This paper surveys and illustrates group-theoretical methods to
determine -Hopf-Galois structures on finite separable extensions in the
extreme situation when has only two sub-Hopf algebras.Comment: 10 page
On finite field arithmetic in characteristic
We are interested in extending normal bases of
to bases of
which allow fast arithmetic in
. This question has been recently studied by Thomson and
Weir in case is equal to . We construct efficient extended bases in case
is equal to and . We also give conditions under which Thomson-Weir
construction can be combined with ours.Comment: 22 page
On finite field arithmetic in characteristic 2
We are interested in extending normal bases of F 2 n /F 2 to bases of F 2 nd /F 2 which allow fast arithmetic in F 2 nd. This question has been recently studied by Thomson and Weir in case d is equal to 2. We construct efficient extended bases in case d is equal to 3 and 4. We also give conditions under which Thomson-Weir construction can be combined with ours
A faster pseudo-primality test
We propose a pseudo-primality test using cyclic extensions of . For every positive integer , this test achieves the
security of Miller-Rabin tests at the cost of Miller-Rabin
tests.Comment: Published in Rendiconti del Circolo Matematico di Palermo Journal,
Springe
NORMAL BASES USING 1-DIMENSIONAL ALGEBRAIC GROUPS
This paper surveys and illustrates geometric methods for constructing normal bases allowing efficient finite field arithmetic. These bases are constructed using the additive group, the multiplicative group and the Lucas torus. We describe algorithms with quasi-linear complexity to multiply two elements given in each one of the bases.
On finite field arithmetic in characteristic 2
We are interested in extending normal bases of F 2 n /F 2 to bases of F 2 nd /F 2 which allow fast arithmetic in F 2 nd. This question has been recently studied by Thomson and Weir in case d is equal to 2. We construct efficient extended bases in case d is equal to 3 and 4. We also give conditions under which Thomson-Weir construction can be combined with ours