Related papers: Predicting the elliptic curve congruential generat…

Given a prime power q, for every pair of positive integers m and n with m dividing the GCD of n and q-1, we construct a modular curve over F_q that parametrizes elliptic curves over F_q along with F_q-defined points P and Q of order m and…

Let l be a prime number and let E and E' be l-isogenous elliptic curves defined over Q. In this paper we determine the proportion of primes p for which E(F_p) is isomorphic to E'(F_p). Our techniques are based on those developed in…

For the integer $ D=pq$ of the product of two distinct odd primes, we construct an elliptic curve $E_{2rD}:y^2=x^3-2rDx$ over $\mathbb Q$, where $r$ is a parameter dependent on the classes of $p$ and $q$ modulo 8, and show, under the parity…

This paper studies the task of two-sources randomness extractors for elliptic curves defined over finite fields $K$, where $K$ can be a prime or a binary field. In fact, we introduce new constructions of functions over elliptic curves which…

An elliptic curve E defined over \Q is an algebraic variety which forms a finitely generated abelian group, and the structure theorem then implies that E = \Z^r + \Z_{tors} for some r \geq 0; this value r is called the rank of E. It is a…

For a given elliptic curve $\mathbf{E}$ over a finite field of odd characteristic and a rational function $f$ on $\mathbf{E}$ we first study the linear complexity profiles of the sequences $f(nG)$, $n=1,2,\dots$ which complements earlier…

Let $E$ be an elliptic curve defined over $\mathbb Q$ and $\widetilde{E}_p$ denote the reduction of $E$ modulo a prime $p$ of good reduction for $E$. The divisibility of $|\widetilde{E}_{p}(\mathbb{F}_p)|$ by an integer $m\ge 2$ for a set…

By the Mordell-Weil theorem the group of Q(z)-rational points of an elliptic curve is finitely generated. It is not known whether the rank of this group can get arbitrary large as the curve varies. Mestre and Nagao have constructed examples…

For each $t\in\mathbb{Q}\setminus\{-1,0,1\}$, define an elliptic curve over $\mathbb{Q}$ by \begin{align*} E_t:y^2=x(x+1)(x+t^2). \end{align*} Using a formula for the root number $W(E_t)$ as a function of $t$ and assuming some standard…

In this paper, we will present how to find keys elliptic curve cryptosystems (ECC) with simple tools of Delphi 7 console application, using the software problem solving of the scalar point multiplication in the field GF(p), where p is an…

We develop a graph-theoretic algorithm to compute the $\varphi$-Selmer group of the elliptic curve $E_b: y^2 = x^3 + bx$ over $\mathbb{Q}(i)$, where $b \in \mathbb{Z}[i]$ and $\varphi$ is a degree 2 isogeny of $E_b$. We associate to $E_b$ a…

We express all the newforms of weight $2$ and levels $30$, $33$, $35$, $38$, $40$, $42$, $44$, $45$ as linear combinations of eta quotients and Eisenstein series, and list their corresponding strong Weil curves. Let $p$ denote a prime and…

Let F be the cubic field of discriminant -23 and O its ring of integers. Let Gamma be the arithmetic group GL_2 (O), and for any ideal n subset O let Gamma_0 (n) be the congruence subgroup of level n. In a previous paper, two of us (PG and…

Elliptic curves with a known number of points over a given prime field with n elements are often needed for use in cryptography. In the context of primality proving, Atkin and Morain suggested the use of the theory of complex multiplication…

A complex elliptic curve $E$ can be defined as the quotient of the analytic space $\mathbb{C}^*$ by a discrete action of the cyclic group $q^{\mathbb{Z}}$ for $\vert q\vert \neq 1$. We study the boundary case when $\vert q\vert =1$, which…

In this paper we consider elliptic divisibility sequences generated by a point on an elliptic curve over $\mathbb{Q}$ with $j$-invariant $1728$ given by an integral short Weierstrass equation. For several different such elliptic…

Loops and cycles play an important role in computing endomorphism rings of supersingular elliptic curves and related cryptosystems. For a supersingular elliptic curve $E$ defined over $\mathbb{F}_{p^2}$, if an imaginary quadratic order $O$…

Isogenies, the mappings of elliptic curves, have become a useful tool in cryptology. These mathematical objects have been proposed for use in computing pairings, constructing hash functions and random number generators, and analyzing the…

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…

Let $K$ be a field of characteristic different from $2$ and let $E$ be an elliptic curve over $K$, defined either by an equation of the form $y^{2} = f(x)$ with degree $3$ or as the Jacobian of a curve defined by an equation of the form…