Related papers: Computing endomorphism rings of elliptic curves un…
We present an algorithm for computing curves and families of curves of prescribed degree and geometric genus on real rational surfaces.
We present families of (hyper)elliptic curve which admit an efficient deterministic encoding function.
We present a new approach to handling the case of Atkin primes in Schoof's algorithm for counting points on elliptic curves over finite fields. Our approach is based on the theory of polynomially cyclic algebras, which we recall as far as…
We provide an explicit and algorithmic version of a theorem of Momose classifying isogenies of prime degree of elliptic curves over number fields, which we implement in Sage and PARI/GP. Combining this algorithm with recent work of…
We establish asymptotic lower bounds for the number of elliptic curves over $\mathbb{Q}$ with prescribed entanglement of division fields, ordered by naive height. Such elliptic curves are obtained as $1$-parameter families arising from…
We propose an algorithm for computing an isogeny between two elliptic curves $E_1,E_2$ defined over a finite field such that there is an imaginary quadratic order $\mathcal{O}$ satisfying $\mathcal{O}\simeq \operatorname{End}(E_i)$ for $i =…
We present a deterministic algorithm that computes the zeta function of a nonsupersingular elliptic curve E over a finite field with p^n elements in time quasi-quadratic in n. An older algorithm having the same time complexity uses the…
Graphical models and factor analysis are well-established tools in multivariate statistics. While these models can be both linked to structures exhibited by covariance and precision matrices, they are generally not jointly leveraged within…
We survey some aspects of the theory of elliptic surfaces and give some results aimed at determining the Picard number of such a surface. For the surfaces considered, this will be equivalent to determining the Mordell-Weil rank of an…
We present an algorithm for computing the integral closure of a reduced ring that is finitely generated over a finite field.
An important open problem in supersingular isogeny-based cryptography is to produce, without a trusted authority, concrete examples of "hard supersingular curves" that is, equations for supersingular curves for which computing the…
The problem of finding elliptical shapes in an image will be considered. We discuss the solution which uses cross-entropy clustering. The proposed method allows the search for ellipses with predefined sizes and position in the space.…
We relate the endomorphism rings of certain $D$-elliptic sheaves of finite characteristic to hereditary orders in central division algebras over function fields.
A new approach to the analytic theory of difference equations with rational and elliptic coefficients is proposed. It is based on the construction of canonical meromorphic solutions which are analytical along "thick paths". The concept of…
We introduce a new iterative method for computing solutions of elliptic equations with random rapidly oscillating coefficients. Similarly to a multigrid method, each step of the iteration involves different computations meant to address…
We present a probabilistic Las Vegas algorithm for computing the local zeta function of a genus-$g$ hyperelliptic curve defined over $\mathbb F_q$ with explicit real multiplication (RM) by an order $\Z[\eta]$ in a degree-$g$ totally real…
In this paper we develop techniques that eliminate the need of the Generalized Riemann Hypothesis (GRH) from various (almost all) known results about deterministic polynomial factoring over finite fields. Our main result shows that given a…
This paper presents explicit constructions of bases for Riemann-Roch spaces associated with arbitrary divisors on elliptic curves. In the context of algebraic geometry codes, the knowledge of an explicit basis for arbitrary divisors is…
This article gives an elementary computational proof of the group law for Edwards elliptic curves following Bernstein, Lange, et al., Edwards, and Friedl. The associative law is expressed as a polynomial identity over the integers that is…
We consider character sums determined by isogenies of elliptic curves over finite fields. We prove a congruence condition for character sums attached to arbitrary cyclic isogenies, and produce explicit formulas for isogenies of small…