Related papers: Endomorphism rings of supersingular elliptic curve…

Let $c<3p/16$ be a prime or $c=1$. Let $E$ be a $\mathbb{Z}[\sqrt{-cp}]$-oriented supersingular elliptic curve defined over $\mathbb{F}_{p^2}$. There exists a $c$-isogeny from $E$ to $E^p$ with kernel $G \subset E[c]$. Given an Eichler…

Computing endomorphism rings of supersingular elliptic curves is an important problem in computational number theory, and it is also closely connected to the security of some of the recently proposed isogeny-based cryptosystems. In this…

Let O be a maximal order in the quaternion algebra B_p over Q ramified at p and infinity. The paper is about the computational problem: Construct a supersingular elliptic curve E over F_p such that End(E) = O. We present an algorithm that…

We give an algorithm for computing an inseparable endomorphism of a supersingular elliptic curve $E$ defined over $\mathbb F_{p^2}$, which, conditional on GRH, runs in expected $O(p^{1/2}(\log p)^2(\log\log p)^3)$ bit operations and…

Let $p$ be a prime, $E$ be a supersingular elliptic curve defined over $\bar{\mathbb{F}}_p$, and $\mathscr{O}$ be its (geometric) endomorphism ring. Earlier results of Chevyrev-Galbraith and Goren-Love have shown that the successive minima…

An elliptic curve $E$ defined over a $p$-adic field $K$ with a $p$-isogeny $\phi:E\rightarrow E^\prime$ comes equipped with an invariant $\alpha_{\phi/K}$ that measures the valuation of the leading term of the formal group homomorphism…

It is well known that there is a one-to-one correspondence between supersingular $j$-invariants up to the action of $\text{Gal}(\mathbb{F}_{p^2}/\mathbb{F}_p)$ and type classes of maximal orders in $B_{p,\infty}$ by Deuring's theorem.…

We present a deterministic and explicit algorithm to compute the endomorphism rings of supersingular elliptic curves. As an example we compute the endomorphism rings of all supersingular elliptic curves defined over characteristic…

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 $O$…

For a prime $p{\,>\,}3$ and a supersingular elliptic curve $E$ defined over $\mathbb{F}_{p^2}$ with ${j(E)\notin\{0,1728\}}$, consider an endomorphism $\alpha$ of $E$ represented as a composition of $L$ isogenies of degree at most $d$. We…

Let $A=\mathbb{F}_q[T]$ be the polynomial ring over $\mathbb{F}_q$, and $F$ be the field of fractions of $A$. Let $\phi$ be a Drinfeld $A$-module of rank $r\geq 2$ over $F$. For all but finitely many primes $\mathfrak{p}\lhd A$, one can…

We describe the neighborhood of the vertex $[E_0]$ (resp. $[E_{1728}]$) in the $\ell$-isogeny graph $\mathcal{G}_\ell(\mathbb{F}_{p^2}, -2p)$ of supersingular elliptic curves over the finite field $\mathbb{F}_{p^2}$ when $p>3\ell^2$ (resp.…

We present two algorithms to compute the endomorphism ring of an ordinary elliptic curve E defined over a finite field F_q. Under suitable heuristic assumptions, both have subexponential complexity. We bound the complexity of the first…

Let $E$ be an elliptic curve over $\Q$. It is well known that the ring of endomorphisms of $E_p$, the reduction of $E$ modulo a prime $p$ of ordinary reduction, is an order of the quadratic imaginary field $Q(\pi_p)$ generated by the…

Let $p>3$ be a prime and $E$ be a supersingular elliptic curve defined over $\mathbb{F}_{p^2}$. Let $c$ be a prime with $c < 3p/16$ and $G$ be a subgroup of $E[c]$ of order $c$. The pair $(E,G)$ is called a supersingular elliptic curve with…

Let p>3 be a prime and let E, E' be supersingular elliptic curves over F_p. We want to construct an isogeny phi: E --> E'. The currently fastest algorithm for finding isogenies between supersingular elliptic curves solves this problem by…

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…

Let $E/\mathbb{Q}$ be a non-CM elliptic curve. Assuming GRH, we prove that, for a set of primes $p$ of density $1$, the absolute discriminant of the $\mathbb{F}_p$-endomorphism ring of the reduction of $E$ modulo $p$ is close to maximal.

In this paper, we investigate extreme values of $\omega(E(\mathbb{F}_p))$, where $E/\mathbb{Q}$ is an elliptic curve with complex multiplication and $\omega$ is the number-of-distinct-prime-divisors function. For fixed $\gamma > 1$, we…

Let E and E' be two elliptic curves over a number field. We prove that the reductions of E and E' at a finite place p are geometrically isogenous for infinitely many p, and draw consequences for the existence of supersingular primes. This…