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Pseudoprime Reductions of Elliptic Curves

  Published:2011-06-27
 Printed: Feb 2012
  • C. David,
    Department of Mathematics and Statistics, Concordia University, Montréal, QC, H3G 1M8
  • J. Wu,
    Institut Elie Cartan Nancy, CNRS, Université Henri Poincaré (Nancy 1), INRIA, 54506 Vandœuvre-lès-Nancy, France
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Abstract

Let $E$ be an elliptic curve over $\mathbb Q$ without complex multiplication, and for each prime $p$ of good reduction, let $n_E(p) = | E(\mathbb F_p) |$. For any integer $b$, we consider elliptic pseudoprimes to the base $b$. More precisely, let $Q_{E,b}(x)$ be the number of primes $p \leq x$ such that $b^{n_E(p)} \equiv b\,({\rm mod}\,n_E(p))$, and let $\pi_{E, b}^{\operatorname{pseu}}(x)$ be the number of compositive $n_E(p)$ such that $b^{n_E(p)} \equiv b\,({\rm mod}\,n_E(p))$ (also called elliptic curve pseudoprimes). Motivated by cryptography applications, we address the problem of finding upper bounds for $Q_{E,b}(x)$ and $\pi_{E, b}^{\operatorname{pseu}}(x)$, generalising some of the literature for the classical pseudoprimes to this new setting.
Keywords: Rosser-Iwaniec sieve, group order of elliptic curves over finite fields, pseudoprimes Rosser-Iwaniec sieve, group order of elliptic curves over finite fields, pseudoprimes
MSC Classifications: 11N36, 14H52 show english descriptions Applications of sieve methods
Elliptic curves [See also 11G05, 11G07, 14Kxx]
11N36 - Applications of sieve methods
14H52 - Elliptic curves [See also 11G05, 11G07, 14Kxx]
 

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