3,380 research outputs found
Algebraic Problems Equivalent to Beating Exponent 3/2 for Polynomial Factorization over Finite Fields
The fastest known algorithm for factoring univariate polynomials over finite
fields is the Kedlaya-Umans (fast modular composition) implementation of the
Kaltofen-Shoup algorithm. It is randomized and takes time to factor polynomials of degree over the finite field
with elements. A significant open problem is if the
exponent can be improved. We study a collection of algebraic problems and
establish a web of reductions between them. A consequence is that an algorithm
for any one of these problems with exponent better than would yield an
algorithm for polynomial factorization with exponent better than
On the complexity of computing with zero-dimensional triangular sets
We study the complexity of some fundamental operations for triangular sets in
dimension zero. Using Las-Vegas algorithms, we prove that one can perform such
operations as change of order, equiprojectable decomposition, or quasi-inverse
computation with a cost that is essentially that of modular composition. Over
an abstract field, this leads to a subquadratic cost (with respect to the
degree of the underlying algebraic set). Over a finite field, in a boolean RAM
model, we obtain a quasi-linear running time using Kedlaya and Umans' algorithm
for modular composition. Conversely, we also show how to reduce the problem of
modular composition to change of order for triangular sets, so that all these
problems are essentially equivalent. Our algorithms are implemented in Maple;
we present some experimental results
Fast algorithms for computing isogenies between ordinary elliptic curves in small characteristic
The problem of computing an explicit isogeny between two given elliptic
curves over F_q, originally motivated by point counting, has recently awaken
new interest in the cryptology community thanks to the works of Teske and
Rostovstev & Stolbunov.
While the large characteristic case is well understood, only suboptimal
algorithms are known in small characteristic; they are due to Couveignes,
Lercier, Lercier & Joux and Lercier & Sirvent. In this paper we discuss the
differences between them and run some comparative experiments. We also present
the first complete implementation of Couveignes' second algorithm and present
improvements that make it the algorithm having the best asymptotic complexity
in the degree of the isogeny.Comment: 21 pages, 6 figures, 1 table. Submitted to J. Number Theor
Fast algorithms for computing isogenies between elliptic curves
We survey algorithms for computing isogenies between elliptic curves defined
over a field of characteristic either 0 or a large prime. We introduce a new
algorithm that computes an isogeny of degree ( different from the
characteristic) in time quasi-linear with respect to . This is based in
particular on fast algorithms for power series expansion of the Weierstrass
-function and related functions
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