scholarly journals A p-adic point counting algorithm for elliptic curves on legendre form

2005 ◽  
Vol 11 (1) ◽  
pp. 71-88 ◽  
Author(s):  
Marc Skov Madsen
Keyword(s):  
Elliptic Curves ◽  
Point Counting ◽  
Legendre Form ◽  
Counting Algorithm

scholarly journals On the distribution of Atkin and Elkies primes for reductions of elliptic curves on average

2015 ◽  
Vol 18 (1) ◽  
pp. 308-322 ◽  
Author(s):  
Igor E. Shparlinski ◽  
Andrew V. Sutherland
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Riemann Hypothesis ◽  
Complex Multiplication ◽  
Log P ◽  
Point Counting ◽  
Generalized Riemann Hypothesis ◽  
Expected Time ◽  
Almost All ◽  
Counting Algorithm

For an elliptic curve$E/\mathbb{Q}$without complex multiplication we study the distribution of Atkin and Elkies primes$\ell$, on average, over all good reductions of$E$modulo primes$p$. We show that, under the generalized Riemann hypothesis, for almost all primes$p$there are enough small Elkies primes$\ell$to ensure that the Schoof–Elkies–Atkin point-counting algorithm runs in$(\log p)^{4+o(1)}$expected time.

scholarly journals Computing cardinalities of -curve reductions over finite fields

2016 ◽  
Vol 19 (A) ◽  
pp. 115-129
Author(s):  
François Morain ◽  
Charlotte Scribot ◽  
Benjamin Smith
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Finite Fields ◽  
Lower Degree ◽  
Point Counting ◽  
Low Degree ◽  
Asymptotic Complexity ◽  
Specialized Point ◽  
Counting Algorithm

We present a specialized point-counting algorithm for a class of elliptic curves over $\mathbb{F}_{p^{2}}$ that includes reductions of quadratic $\mathbb{Q}$-curves modulo inert primes and, more generally, any elliptic curve over $\mathbb{F}_{p^{2}}$ with a low-degree isogeny to its Galois conjugate curve. These curves have interesting cryptographic applications. Our algorithm is a variant of the Schoof–Elkies–Atkin (SEA) algorithm, but with a new, lower-degree endomorphism in place of Frobenius. While it has the same asymptotic asymptotic complexity as SEA, our algorithm is much faster in practice.

scholarly journals Semistable models of elliptic curves over residue characteristic 2

2020 ◽  
pp. 1-9
Author(s):  
Jeffrey Yelton
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Mixed Characteristic ◽  
Semistable Model ◽  
Characteristic 2 ◽  
Residue Characteristic ◽  
Legendre Form

Abstract Given an elliptic curve E in Legendre form $y^2 = x(x - 1)(x - \lambda )$ over the fraction field of a Henselian ring R of mixed characteristic $(0, 2)$ , we present an algorithm for determining a semistable model of E over R that depends only on the valuation of $\lambda $ . We provide several examples along with an easy corollary concerning $2$ -torsion.

scholarly journals An Efficient Approach to Point-Counting on Elliptic Curves from a Prominent Family over the Prime Field Fp

Mathematics ◽  
2021 ◽  
Vol 9 (12) ◽  
pp. 1431
Author(s):  
Yuri Borissov ◽  
Miroslav Markov
Keyword(s):  
Elliptic Curves ◽  
Prime Number ◽  
Explicit Formula ◽  
Point Counting ◽  
Prime Field ◽  
Algorithmic Solution ◽  
The Family ◽  
Order Of Magnitude ◽  
Modulo P ◽  
Quantities Of Interest

Here, we elaborate an approach for determining the number of points on elliptic curves from the family Ep={Ea:y2=x3+a(modp),a≠0}, where p is a prime number >3. The essence of this approach consists in combining the well-known Hasse bound with an explicit formula for the quantities of interest-reduced modulo p. It allows to advance an efficient technique to compute the six cardinalities associated with the family Ep, for p≡1(mod3), whose complexity is O˜(log2p), thus improving the best-known algorithmic solution with almost an order of magnitude.

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