On Elkies subgroups of l-torsion points in elliptic curves defined over a finite field [PDF]
arXiv, 2008 As a subproduct of the Schoof-Elkies-Atkin algorithm to count points on elliptic curves defined over finite fields of characteristic p, there exists an algorithm that computes, for l an Elkies prime, l-torsion points in an extension of degree l-1 at cost O(l max(l, \log q)^2) bit operations in the favorable case where l < p/2. We combine in this work
arxiv
Counting Points on Genus 2 Curves with Real Multiplication [PDF]
arXiv, 2011 We present an accelerated Schoof-type point-counting algorithm for curves of genus 2 equipped with an efficiently computable real multiplication endomorphism. Our new algorithm reduces the complexity of genus 2 point counting over a finite field (\F_{q}) of large characteristic from (\widetilde{O}(\log^8 q)) to (\widetilde{O}(\log^5 q)).
arxiv
Improved Complexity Bounds for Counting Points on Hyperelliptic Curves [PDF]
arXiv, 2017 We present a probabilistic Las Vegas algorithm for computing the local zeta function of a hyperelliptic curve of genus $g$ defined over $\mathbb{F}_q$. It is based on the approaches by Schoof and Pila combined with a modeling of the $\ell$-torsion by structured polynomial systems.
arxiv
On the Representation of Primes by Binary Quadratic Forms, and Elliptic Curves [PDF]
arXiv, 2016 It is shown that, under some mild technical conditions, representations of prime numbers by binary quadratic forms can be computed in polynomial complexity by exploiting Schoof's algorithm, which counts the number of $\mathbb F_q$-points of an elliptic curve over a finite field $\mathbb F_q$.
arxiv
Computing the cardinality of CM elliptic curves using torsion points [PDF]
arXiv, 2002 Let E be an elliptic curve having complex multiplication by a given quadratic order of an imaginary quadratic field K. The field of definition of E is the ring class field Omega of the order. If the prime p splits completely in Omega, then we can reduce E modulo one the factors of p and get a curve Ep defined over GF(p).
arxiv
AFDP: An Automated Function Description Prediction Approach to Improve Accuracy of Protein Function Predictions [PDF]
arXiv, 2019 With the rapid growth in high-throughput biological sequencing technologies and subsequently the amount of produced omics data, it is essential to develop automated methods to annotate the functionality of unknown genes and proteins. There are developed tools such as AHRD applying known proteins characterization to annotate unknown ones.
arxiv
Counting points on hyperelliptic curves with explicit real multiplication in arbitrary genus [PDF]
arXiv, 2018 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 number field.
arxiv
Cycles in the supersingular $\ell$-isogeny graph and corresponding endomorphisms [PDF]
arXiv, 2018 We study the problem of generating the endomorphism ring of a supersingular elliptic curve by two cycles in $\ell$-isogeny graphs. We prove a necessary and sufficient condition for the two endomorphisms corresponding to two cycles to be linearly independent, expanding on the work in Kohel's thesis.
arxiv
Modular forms invariant under non-split Cartan subgroups [PDF]
arXiv, 2018 In this paper we describe a method for computing a basis for the space of weight $2$ cusp forms invariant under a non-split Cartan subgroup of prime level $p$. As an application we compute, for certain small values of $p$, explicit equations over $\bf Q$ for the canonical embeddings of the associated modular curves.
arxiv