Results 1 to 10 of about 94 (71)
Schoof's algorithm and isogeny cycles
, 1994 The heart of Schoof's algorithm for computing the cardinality m of an elliptic curve over a finite field is the computation of m modulo small primes l. Elkies and Atkin have designed practical improvements to the basic algorithm, that make use of “good” primes l. We show how to use powers of good primes in an efficient way.
François Morain, Jean-Marc Couveignes
openaire +4 more sources
Binary quadratic forms, elliptic curves and Schoof's algorithm [PDF]
, 2015 In this thesis, I show that the representation of prime integers by reduced binary quadratic forms of given discriminant can be obtained in polynomial complexity using Schoof's algorithm for counting the number of points of elliptic curves over finite fields.
Pintore, Federico
openaire +2 more sources
Schoofův algoritmus pro Weierstrassovy křivky
, 2023 Schoof's algorithm is the starting point for the most efficient methods for determining the number of rational points on an elliptic curve defined over a finite field. The goal of this thesis is to introduce the subject of elliptic curves, with the emphasis on Weierstrass curves over a finite field, to describe Schoof's algorithm and its time ...
Zvoníček, Václav
openaire +2 more sources
Fast algorithms for computing the eigenvalue in the Schoof-Elkies-Atkin algorithm [PDF]
Proceedings of the 2006 international symposium on Symbolic and algebraic computation, 2006 The Schoof-Elkies-Atkin algorithm is the only known method for counting the number of points of an elliptic curve defined over a finite field of large characteristic. Several practical and asymptotical improvements for the phase called eigenvalue computation are proposed.
Gaudry, Pierrick, Morain, François
openaire +3 more sources
Computing the eigenvalue in the schoof-elkies-atkin algorithm using abelian lifts [PDF]
Proceedings of the 2007 international symposium on Symbolic and algebraic computation, 2007 The Schoof-Elkies-Atkin algorithm is the best known method for counting the number of points of an elliptic curve defined over a finite field of large characteristic. We use Abelian properties of division polynomials to design a fast theoretical and practical algorithm for nding the eigenvalue.
Éric Schost +2 more
openaire +1 more source
Remarks on the Schoof-Elkies-Atkin algorithm [PDF]
Mathematics of Computation, 1998 Schoof’s algorithm computes the number m m of points on an elliptic curve E E defined over a finite field F q {\Bbb F}_q . Schoof determines m m modulo small primes ℓ \ell using the characteristic equation of ...
openaire +2 more sources
Counting points on hyperelliptic curves over finite fields [PDF]
, 2000 International audienceWe describe some algorithms for computing the cardinality of hyperelliptic curves and their Jacobians over finite fields. They include several methods for obtaining the result modulo small primes and prime powers, in particular an ...
D.G. Cantor +21 more
core +3 more sources
Efficient Implementation of Schoof’s Algorithm [PDF]
, 1998 Schoof's algorithm is used to find a secure elliptic curve for cryptosystems, as it can compute the number of rational points on a randomly selected elliptic curve defined over a finite field. By realizing efficient combination of several improvements, such as Atkin-Elkies's method, the isogeny cycles method, and trial search by match-and-sort ...
Masayuki Noro +3 more
openaire +2 more sources
On the distribution of orders of Frobenius action on $\ell$-torsion of abelian surfaces [PDF]
, 2020 The computation of the order of Frobenius action on the $\ell$-torsion is a part of Schoof-Elkies-Atkin algorithm for point counting on an elliptic curve $E$ over a finite field $\mathbb{F}_q$.
Kolesnikov, Nikita, Novoselov, Semyon
core +3 more sources
Efficient Implementation of Schoof’s Algorithm in Case of Characteristic 2 [PDF]
, 2000 In order to choose a secure elliptic curve for Elliptic Curve Cryptosystems, it is necessary to count the order of a randomly selected elliptic curve. Schoof’s algorithm and its variants by Elkies and Atkin are known as efficient methods to count the orders of elliptic curves.
Kazuhiro Yokoyama +2 more
openaire +2 more sources