Counting points in medium characteristic using Kedlaya's algorithm.
Counting points on elliptic curves over finite fields
We describe three algorithms to count the number of points on an elliptic curve over a finite field. The first one is very practical when the finite field is not too large ; it is based on Shanks's baby-step-giant-step strategy. The second algorithm is very efficient when the endomorphism ring of the curve is known. It exploits the natural lattice structure of this ring. The third algorithm is based on calculations with the torsion points of the elliptic curve [18]. This deterministic polynomial...
Decomposing rational numbers
Dedekind cotangent sums
Distribution de la constante d'Hermite et du plus court vecteur dans les réseaux de dimension deux
En utilisant la géométrie du demi-plan de Poincaré et des familles de disques classiques - disques de Ford, disques de Farey - nous décrivons les domaines de niveau associés à la constante d'Hermite et au plus court vecteur d'un réseau. Nous en déduisons une évaluation très précise des fonctions de répartition correspondantes, en particulier au voisinage de l'origine.
Distribution of the power generator over a residue ring for parts of the period.
Efficient computation of addition chains
The aim of this paper is to present a unifying approach to the computation of short addition chains. Our method is based upon continued fraction expansions. Most of the popular methods for the generation of addition chains, such as the binary method, the factor method, etc..., fit in our framework. However, we present new and better algorithms. We give a general upper bound for the complexity of continued fraction methods, as a function of a chosen strategy, thus the total number of operations required...
Elliptic curves with j-invariant equals 0 or 1728 over a finite prime field.
Let p be a prime number, p ≠ 2,3 and Fp the finite field with p elements. An elliptic curve E over Fp is a projective nonsingular curve of genus 1 defined over Fp. Each one of these curves has an isomorphic model given by an (Weierstrass) equation E: y2 = x3 + Ax + B, A,B ∈ Fp with D = 4A3 + 27B2 ≠ 0. The j-invariant of E is defined by j(E) = 1728·4A3/D.The aim of this note is to establish some results concerning the cardinality of the group of points on elliptic curves over Fp with j-invariants...
Equivalences between elliptic curves and real quadratic congruence function fields
In 1994, the well-known Diffie-Hellman key exchange protocol was for the first time implemented in a non-group based setting. Here, the underlying key space was the set of reduced principal ideals of a real quadratic number field. This set does not possess a group structure, but instead exhibits a so-called infrastructure. More recently, the scheme was extended to real quadratic congruence function fields, whose set of reduced principal ideals has a similar infrastructure. As always, the security...
Étude du problème du logarithme discret dans
Explicit 4-descents on an elliptic curve
Explicit computations of Hilbert modular forms on .
Extended GCD and Hermite normal form algorithms via lattice basis reduction.
Factoring and testing primes in small space
We discuss how much space is sufficient to decide whether a unary given number n is a prime. We show that O(log log n) space is sufficient for a deterministic Turing machine, if it is equipped with an additional pebble movable along the input tape, and also for an alternating machine, if the space restriction applies only to its accepting computation subtrees. In other words, the language is a prime is in pebble–DSPACE(log log n) and also in accept–ASPACE(log log n). Moreover, if the given n is...
Factoring integers with large-prime variations of the quadratic sieve.
Factoring polynomials over global fields
We prove that van Hoeij’s original algorithm to factor univariate polynomials over the rationals runs in polynomial time, as well as natural variants. In particular, our approach also yields polynomial time complexity results for bivariate polynomials over a finite field.
Factoring Polynomials with Rational Coefficients.
Factorisation sur des polynômes de degré élevé à l’aide d’un monomorphisme
Finding the eigenvalue in Elkies' algorithm.
Hurwitz and Tasoev continued fractions with long period.